proposition@alttheorem lemma@alttheorem corollary@alttheorem
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# Lipschitz continuity and Bochner-Eells-Sampson inequality for harmonic maps
from $\operatorname{RCD}(K,N)$ spaces to ${\sf CAT}(0)$ spaces

Andrea Mondino  and  Daniele Semola

###### Abstract.

We establish Lipschitz regularity of harmonic maps from
$\operatorname{RCD}(K,N)$ metric measure spaces with lower Ricci curvature
bounds and dimension upper bounds in synthetic sense with values into ${\sf
CAT}(0)$ metric spaces with non-positive sectional curvature. Under the same
assumptions, we obtain a Bochner-Eells-Sampson inequality with a Hessian type-
term. This gives a fairly complete generalization of the classical theory for
smooth source and target spaces to their natural synthetic counterparts and an
affirmative answer to a question raised several times in the recent
literature.
The proofs build on a new interpretation of the interplay between Optimal
Transport and the Heat Flow on the source space and on an original
perturbation argument in the spirit of the viscosity theory of PDEs.

Andrea Mondino: Mathematical Institute, University of Oxford, Radcliffe
Observatory, Andrew Wiles Building, Woodstock Rd, Oxford OX2 6GG, UK,
<EMAIL_ADDRESS>
Daniele Semola: Mathematical Institute, University of Oxford, Radcliffe
Observatory, Andrew Wiles Building, Woodstock Rd, Oxford OX2 6GG, UK,
<EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Geometric Analysis tools on $\operatorname{RCD}(K,N)$ spaces
    2. 2.2 Energy and non-linear harmonic maps
  3. 3 Auxiliary results
  4. 4 Approximate maximum principles and perturbation arguments
  5. 5 The key propagation theorem
  6. 6 Lipschitz continuity
  7. 7 Bochner inequality with Hessian-type term

##  1\. Introduction

In this paper we establish local Lipschitz continuity of harmonic maps from
non-smooth metric measure spaces with synthetic Ricci curvature lower bounds
in the sense of the $\operatorname{RCD}$ theory with values into ${\sf
CAT}(0)$ metric spaces with non-positive sectional curvature. This answers to
a question raised several times in the recent literature, see [43, 29, 45, 46,
50]. Building on the top of this regularity result, we prove a Bochner-Eells-
Sampson type inequality with Hessian term, which is expected to be fundamental
for future applications.

A smooth map $u:M^{n}\to N^{k}$ between Riemannian manifolds is called
harmonic when the _tension field_

$\Delta u:=\mathrm{tr}\nabla\left(\mathop{}\\!\mathrm{d}u\right)$

vanishes identically, as a section of the pull-back bundle $u^{*}TN$. There
are several examples of harmonic maps: harmonic functions when the target
space is $\mathbb{R}$, geodesics when the source space is $\mathbb{R}$,
isometries, conformal maps, holomorphic maps between Kähler manifolds and
inclusions of volume minimizing submanifolds. Their role is ubiquitous in
Geometric Analysis.
The basic question becomes then the existence of harmonic maps, under suitable
assumptions. The problem was approached from a parabolic perspective by Eells-
Sampson [30] and subsequently by Hamilton [51], based on the long time
behaviour of the non-linear heat equation

$\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}u(x,t)=\Delta
u(x,t)\,.$

Later on, the variational perspective was put forward by Hildebrandt-Kaul-
Widman [52, 53] (see also the subsequent work of Schoen [87]), building on top
of the interpretation of harmonic maps as critical points of the energy
functional

$E(u):=\int_{M}\left\lvert\mathop{}\\!\mathrm{d}u\right\rvert^{2}\mathop{}\\!\mathrm{d}\mathrm{vol}_{M}\,.$

Very much intertwined with the question of existence, there is the issue of
regularity.
If we denote by $\mathrm{Ric}_{M}$ and $\mathrm{R}^{N}$ the Ricci curvature
tensor of $M$ and the Riemann curvature tensor of $N$, respectively, and by
$\\{e_{\alpha}\\}_{1\leq\alpha\leq n}$ an orthonormal base of $TM$, the
Bochner-Eells-Sampson formula for harmonic maps

$\Delta\frac{1}{2}\left\lvert\mathop{}\\!\mathrm{d}u\right\rvert^{2}=\left\lvert\nabla\mathop{}\\!\mathrm{d}u\right\rvert^{2}+\mathrm{Ric}_{M}(\nabla
u,\nabla
u)-\sum_{\alpha,\beta=1,\dots,n}\langle\mathrm{R}^{N}\left(u_{*}e_{\alpha},u_{*}e_{\beta}\right)u_{*}e_{\alpha},u_{*}e_{\beta}\rangle$
(1.1)

hints towards a prominent role of lower bounds on the Ricci curvature of $M$
and upper bounds on the sectional curvature of $N$ in developing a regularity
theory. Indeed, if we assume that $\mathrm{Ric}_{M}\geq K$ and
$\mathrm{R}^{N}\leq 0$, then from (1.1) we obtain that

$\Delta\frac{1}{2}\left\lvert\mathop{}\\!\mathrm{d}u\right\rvert^{2}\geq
K\left\lvert\mathop{}\\!\mathrm{d}u\right\rvert^{2}\,.$ (1.2)

A priori, local $L^{\infty}$-estimates for
$\left\lvert\mathop{}\\!\mathrm{d}u\right\rvert$ can be derived from (1.2) via
the classical De Giorgi-Moser iteration. Smoothness then follows from elliptic
regularity, see [30, 88]. We refer also to the more recent work of Sturm [91]
for a different, probabilistic interpretation of the curvature conditions on
source and target spaces in the theory of harmonic maps, deeply related to the
developments of the present note.

In the last thirty years, starting from the work of Gromov-Schoen [49], there
has been growing interest in developing a theory of harmonic maps between
spaces more general than Riemannian manifolds and possibly non-smooth. This
has required a completely new set of ideas and techniques, as neither
isometric embeddings into Euclidean spaces, nor local charts are in general
available.
The analysis in [49] was dedicated to maps from smooth source spaces with
values into locally finite Riemannian simplicial complexes, with striking
applications in Geometric Group Theory. Korevaar-Schoen [67, 68], and
independently Jost [57, 58], later developed a general theory of Sobolev and
harmonic maps with values into metric spaces with non-positive curvature in
the sense of Alexandrov. From the variational perspective, the curvature
assumption on the target guarantees convexity of the energy functional.
In [67, 68] source spaces are smooth manifolds and the authors obtain local
Lipschitz continuity of harmonic maps. In [57, 58] source spaces are locally
compact metric spaces with a Dirichlet form and the author obtains local
Hölder continuity, assuming a uniform scale invariant Poincaré inequality.

For the sake of the applications, Lipschitz continuity and suitable versions
of the Bochner-Eells-Sampson inequality (1.1) are two cornerstones. An example
from [69] shows that, in general, doubling and Poincaré assumptions on the
source space do not guarantee Lipschitz regularity of harmonic functions, even
in the scalar-valued case. Meanwhile, the developments of the theory of
Alexandrov spaces motivated the conjecture that harmonic maps from metric
spaces with sectional curvature bounded from below with values into metric
spaces with non-positive curvature should be locally Lipschitz. The conjecture
was formulated by Lin [74] and, in a more open form, by Jost [61], and it has
been recently settled by Zhang-Zhu [99].
We mention [32, 48, 72, 35, 24, 26, 27] for previous, related developments of
the theory of harmonic maps between metric spaces, without pretending of being
complete in this list. Also, more recently, there has been progress in the
existence and regularity of harmonic maps into ${\sf CAT}(1)$ target spaces by
Breiner-Fraser-Huang-Mese-Sargent-Zhang [15, 16]

Nowadays, there is a well-established theory of metric measure spaces with
lower bounds on the Ricci curvature in the synthetic sense, the so-called
$\operatorname{RCD}(K,N)$ metric measure spaces $(X,\mathsf{d},\mathfrak{m})$.
Here $K\in\mathbb{R}$ plays the role of a synthetic lower bound on the Ricci
curvature and $1\leq N<\infty$ plays the role of a synthetic upper bound on
the dimension, in the sense of the Lott-Sturm-Villani “Curvature-Dimension”
condition [92, 93, 75]. The “Riemannian” assumption, formulated in terms of
linearity of the heat flow, is added to force Hilbertian behaviour in the much
broader class of Finsler geometries, that the Curvature-Dimension condition
does not rule out a priori.
We address the reader to subsection 2.1 (see also the survey [1] and
references therein) for the relevant background on the theory of
$\operatorname{RCD}(K,N)$ metric measure spaces. Here we just remark that the
$\operatorname{RCD}$ theory is fully consistent with the theory of smooth
(weighted) Riemannian manifolds with (weighted) Ricci curvature bounded from
below and with the theory of Alexandrov spaces with sectional curvature
bounded from below. Furthermore, the $\operatorname{RCD}(K,N)$ condition is
stable with respect to the measured Gromov-Hausdorff topology, under cone and
spherical suspension constructions, and under quotients by actions of groups
of measure preserving isometries.

The role of the lower Ricci curvature bound on the source space for the
regularity of harmonic maps in the classical theory, together with the
remarkable developments of Geometric Analysis on $\operatorname{RCD}(K,N)$
spaces in recent years, give strong motivations for a theory of harmonic maps
from $\operatorname{RCD}(K,N)$ spaces with values into ${\sf CAT}(0)$ metric
spaces with non-positive curvature. A theory of Sobolev maps in this framework
has been developed by Gigli-Tyulenev in [46], where the existence and
uniqueness of solutions of the Dirichlet problem have been achieved too. Local
Hölder regularity of harmonic maps has been recently obtained by Guo in [50],
along the lines of the previous [59, 74].
The question of local Lipschitz regularity of harmonic maps from
$\operatorname{RCD}(K,N)$ spaces to ${\sf CAT}(0)$ spaces has been raised
several times in the recent literature, starting from [43] and later in [29,
45, 46, 50].

The first main result of this paper is a positive answer to this question in
full generality, which can be considered also as a complete answer to the
question raised in [61]. Indeed, we fully generalize the Lipschitz regularity
result for smooth manifolds by Eells-Sampson [30] to the natural synthetic
framework.
We address the reader to subsection 2.2 for the introduction of the relevant
background and terminology about harmonic maps in this setting. Below, with
the notation $\mathrm{osc}\,u$ we indicate the oscillation of the harmonic map
$u$, which is locally bounded thanks to [50].

######  Theorem 1.1 (cf. with Theorem 6.1).

Let $(X,\mathsf{d},\mathfrak{m})$ be an $\operatorname{RCD}(K,N)$ metric
measure space for some $K\in\mathbb{R}$ and $1\leq N<\infty$. Let
$(Y,\mathsf{d}_{Y})$ be a ${\sf CAT}(0)$ space and let $\Omega\subset X$ be an
open domain. Assume that $u:\Omega\to Y$ is a harmonic map. Then for any
$0<R\leq 1$ there exists a constant $C=C(K,N,R)>0$ such that if
$B_{2R}(q)\Subset\Omega$ for some point $q\in X$, then for any $x,y\in
B_{R/16}(q)$ it holds

$\mathsf{d}_{Y}(u(x),u(y))\leq
C(K,N,R)\left(\left(\fint_{B_{R}(q)}\left\lvert\mathop{}\\!\mathrm{d}u(z)\right\rvert^{2}\mathop{}\\!\mathrm{d}\mathfrak{m}(z)\right)^{\frac{1}{2}}+\mathrm{osc}_{\overline{B}_{R}(q)}u\right)\mathsf{d}(x,y)\,.$
(1.3)

We postpone to the second part of the introduction a description of the
strategy of the proof of Theorem 1.1. As we shall see, the key step will be
establishing a (very) weak form of the Bochner-Eells-Sampson inequality (1.2).
Towards this goal, we will need to introduce several original ideas with
respect to the previous literature.

As of today, there seems to be no notion of Hessian available for harmonic
maps from $\operatorname{RCD}(K,N)$ metric measure spaces with values into
${\sf CAT}(0)$ metric spaces. This would be required to give meaning to a
version of (1.1) in this setting, where only the curvature terms are
disregarded. Nevertheless, we are able to prove a Bochner-Eells-Sampson
inequality for harmonic maps where a Hessian-type term

$\left\lvert\nabla\left\lvert\mathop{}\\!\mathrm{d}u\right\rvert\right\rvert\leq\left\lvert\nabla\mathop{}\\!\mathrm{d}u\right\rvert$

appears. Below we shall denote by $\operatorname{lip}u$ the pointwise
Lipschitz constant of a harmonic map $u:\Omega\to Y$, defined by

$\operatorname{lip}u(x):=\limsup_{y\to
x}\frac{\mathsf{d}_{Y}(u(x),u(y))}{\mathsf{d}(x,y)}\,.$

We remark that, by Theorem 1.1, the pointwise Lipschitz constant of a harmonic
map is locally bounded.

######  Theorem 1.2 (cf. with Theorem 7.1).

Let $(X,\mathsf{d},\mathfrak{m})$ be an $\operatorname{RCD}(K,N)$ metric
measure space for some $K\in\mathbb{R}$, $1\leq N<\infty$, and let
$(Y,\mathsf{d}_{Y})$ be a ${\sf CAT}(0)$ space. Let $\Omega\subset X$ be an
open domain and let $u:\Omega\to\mathbb{R}$ be a harmonic map. Then
$\operatorname{lip}u\in W^{1,2}_{{\rm loc}}(\Omega)\cap L^{\infty}_{{\rm
loc}}(\Omega)$ and

$\Delta\frac{\left\lvert\operatorname{lip}u\right\rvert^{2}}{2}\geq\left\lvert\nabla\operatorname{lip}u\right\rvert^{2}+K\left\lvert\operatorname{lip}u\right\rvert^{2}\,,\quad\text{on
$\Omega$}\,,$ (1.4)

in the sense of distributions.

We refer to the very recent work [100] by Zhang-Zhong-Zhu for an analogous
statement under the assumption that the source space is a smooth
$N$-dimensional Riemannian manifold with Ricci curvature bounded from below by
$K$ and to [33, 34] for previous instances of Bochner-Eells-Sampson formulas
(without Hessian-type terms) for maps with values into ${\sf CAT}(k)$ spaces
and with source smooth Riemannian manifolds and polyhedra, respectively.

The validity of a Bochner inequality for scalar-valued maps defined on a non-
smooth $\operatorname{RCD}$ space (even without Hessian term) is a very deep
result: it was proved for $\operatorname{RCD}(K,\infty)$ spaces by Ambrosio-
Gigli-Savaré [5] (see also [6] by the same authors for the reverse
implication). The dimensional improvement for $\operatorname{RCD}^{*}(K,N)$
spaces was established independently by Erbar-Kuwada-Sturm [31] and by
Ambrosio-Savaré and the first author [10] (together with the reverse
implication). The fact that the scalar Bochner inequality (without Hessian)
“self-improves” to estimate the norm of the Hessian was noticed in the setting
of $\Gamma$-calculus by Bakry [14] and subsequently proved in the non-smooth
setting of $\operatorname{RCD}$ spaces by Savaré [86] and Gigli [37].

To the best of our knowledge, Theorem 1.2 is the first instance of a Bochner-
Eells-Sampson inequality with Hessian-type term for harmonic maps when both
the source and the target spaces are non-smooth.
We remark that the appearance of the Hessian-type term in (1.4) is expected to
have fundamental importance in the future developments of the theory, see for
instance the discussion in the introduction of [100].

### Strategy of the proof

There are two fundamental difficulties in the proof of the local Lipschitz
continuity for harmonic maps in this setting. The first one is the need to
deduce the information coming from the combination of the Bochner-Eells-
Sampson formula (1.1) with the curvature constraints on the source space from
“lower order considerations”, independent of any regularity. This is a
fundamental issue in Geometric Analysis of non-smooth spaces, which requires a
new argument in the present situation and it is well illustrated already at
the level of scalar-valued harmonic functions. The second key point is the
need to turn the combination of harmonicity, which is understood here in
variational terms, with the curvature constraints of the target into a
differential inequality. This difficulty is tied with the non-linearity of the
variational problem and it requires an original idea.

We illustrate the first idea in the case of linear harmonic functions
$u:\Omega\to\mathbb{R}$, where $\Omega\subset X$ is an open domain and
$(X,\mathsf{d},\mathfrak{m})$ is an $\operatorname{RCD}(0,N)$ metric measure
space, for the sake of simplicity. The case of general lower Ricci curvature
bounds $K\in\mathbb{R}$ would introduce additional error terms without
affecting the general strategy.
Notice that local Lipschitz estimates in this case follow from Harnack’s
inequality as soon as one is able to prove that

$\Delta\left\lvert\nabla u\right\rvert^{2}\geq 0\,,$ (1.5)

in the sense of distributions. For smooth Riemannian manifolds, and under the
assumption that $u$ is smooth, the estimate (1.5) follows from Bochner’s
inequality. For $\operatorname{RCD}(0,N)$ spaces we refer to [56, 98] for a
distributional approach, building on the top of weak versions of Bochner’s
inequality (see also [6, 31, 10]).

Here we present a different strategy, more suitable to be generalized to ${\sf
CAT}(0)$-valued harmonic maps. This approach was developed in the case of
source spaces with sectional curvature bounded from below in the Alexandrov
sense by Petrunin and Zhang-Zhu in [81, 97]. We remark that the strategy has
strong analogies with the so-called two points maximum principle, see for
instance [66, 70] and [12] for a recent survey.

We assume that $u$ is continuous (a property that is usually proved via De
Giorgi-Moser’s iteration and Harnack’s inequality) and we wish to bootstrap
the continuity to local Lipschitz continuity. We introduce the evolution via
the Hopf-Lax semigroup (up to a sign)

$\mathcal{Q}^{t}u(x):=\sup_{y\in\Omega}\left\\{u(y)-\frac{\mathsf{d}^{2}(x,y)}{2t}\right\\}\,.$

Then we claim that $\Delta\mathcal{Q}^{t}u\geq 0$, locally and for $t>0$
sufficiently small. On a smooth Riemannian manifold, neglecting the regularity
issues, this inequality can be proved with a computation using the second
variation of the arc length and Jacobi fields, see for instance the survey by
Andrews [12] for similar arguments. On Alexandrov spaces with lower sectional
curvature bounds, the statement is proved by Zhang-Zhu [97] (following an
argument proposed by Petrunin in the unpublished [81]) relying on several
perturbation arguments and on Petrunin’s second variation formula for the arc-
length [82]. None of these approaches is available in the framework of
$\operatorname{RCD}(K,N)$ metric measure spaces. Roughly speaking, the main
reason is that they rely on estimates for second-order variations “in single
directions”, from which Laplacian estimates are deduced, as a second step, by
average.

In turn, on $\operatorname{RCD}$ spaces there is a completely alternative
strategy, where the need to estimate second-order variations in single
directions is completely overcome. The sub-harmonicity
$\Delta\mathcal{Q}^{t}u\geq 0$ follows from the interplay between Optimal
Transport and the Heat Flow in this setting [94, 71, 5], by further developing
an argument found by the authors in [80]. Denoting by $P_{s}$ the Heat Flow

$\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}s}P_{s}u=\Delta
P_{s}u\,,$

the so-called Kuwada duality [71] guarantees that

$P_{s}\mathcal{Q}^{t}u(x)\geq P_{s}u(y)-\frac{\mathsf{d}^{2}(x,y)}{2t}\,,$

for any $x,y\in X$ and for any $s>0$. Therefore, formally

$\liminf_{s\to
0}\frac{P_{s}\mathcal{Q}^{t}u(x)-\mathcal{Q}^{t}u(x)}{s}\geq\liminf_{s\to
0}\frac{P_{s}u(x_{t})-u(x_{t})}{s}=\Delta u(x_{t})=0\,,$ (1.6)

where $x_{t}\in\Omega$ is any point such that

$\mathcal{Q}^{t}u(x)=u(x_{t})-\frac{\mathsf{d}^{2}(x,x_{t})}{2t}\,.$

This (formally) shows that $\Delta\mathcal{Q}^{t}u\geq 0$. As $u$ is harmonic,
also

$\Delta\left(\frac{\mathcal{Q}^{t}u-u}{t}\right)\geq 0\,.$ (1.7)

If we recall that the Hopf-Lax semigroup solves the Hamilton-Jacobi equation

$\frac{\mathop{}\\!\mathrm{d}\mathcal{Q}^{t}u}{\mathop{}\\!\mathrm{d}t}-\frac{1}{2}\left\lvert\nabla\mathcal{Q}^{t}u\right\rvert^{2}=0\,,$

see for instance [4], then we easily infer that a local gradient estimate for
$u$ follows, again formally, from a uniform (as $t\downarrow 0$) local
$L^{\infty}$-estimate for

$\frac{\mathcal{Q}^{t}u-u}{t}\,.$ (1.8)

Indeed, taking the limit as $t\downarrow 0$, it holds
$(\mathcal{Q}^{t}u-u)/t\to\frac{1}{2}\left\lvert\nabla u\right\rvert^{2}$. In
order to get the sought local, uniform estimate for (1.8), it is now
sufficient to use Harnack’s inequality for sub-harmonic functions, thanks to
(1.7).

It is possible to interpret the strategy outlined above in terms of the two-
variables functions $F_{t}:\Omega\times\Omega\to\mathbb{R}$,

$F_{t}(x,y):=u(y)-\frac{\mathsf{d}^{2}(x,y)}{2t}\,.$ (1.9)

Second variation arguments exploit the assumption that $u$ is harmonic,
therefore the first term above solves the Laplace equation with respect to the
$y$ variable, and the lower Ricci curvature bound of the ambient space,
encoded into the behaviour of the distance squared on the product $X\times X$.

In order to deal with harmonic maps with values into ${\sf CAT}(0)$ spaces
$(Y,\mathsf{d}_{Y})$, the non-linearity of the target is a fundamental issue,
as there is no clear counterpart of (1.9).
A key idea borrowed from [99] (see also [12, Section 5]) is to consider the
function of two variables

$G_{t}(x,y):=\mathsf{d}_{Y}(u(x),u(y))-\frac{\mathsf{d}^{2}(x,y)}{2t}\,.$
(1.10)

The strategy is to exploit the curvature constraints of the source and the
target spaces and the harmonicity of the map $u$ in terms of the behaviour of
the function $G_{t}$ on the product $X\times X$.

The lower Ricci curvature bound enters into play as in the scalar-valued case,
through the Wasserstein contractivity of the Heat Flow and it controls the
second term at the right-hand side in (1.10). Notice that, again, this is a
fundamentally different approach with respect to the previous [30, 67, 99] and
with respect to the strategy illustrated in [12]. It shares some similarities
with [91] where, however, the perspective on harmonic maps was parabolic
rather than variational.

The ${\sf CAT}(0)$ condition on the target is combined with the assumption
that $u$ is harmonic to control the first term at the right-hand side in
(1.10). In particular, the combination of these two assumptions, neglecting
the regularity issues, leads to the inequality

$\Delta\left(\mathsf{d}_{Y}^{2}(u(\cdot),u(x_{0}))-\mathsf{d}^{2}_{Y}(u(\cdot),P)\right)(x_{0})\leq
0\,,$ (1.11)

for any $x_{0}\in\Omega$ and for any $P\in Y$. Moreover, the non-positive
curvature assumption allows a decoupling of the two variables in (1.10), via a
quadrilateral comparison finding its roots in [85] (see section 3 for the
precise statement), and it leads from (1.11) to the differential inequality

$\Delta_{x}f(x_{0})+\Delta_{y}g(y_{0})\geq 0\,,$ (1.12)

for suitably constructed auxiliary functions $f:X\to\mathbb{R}$ and
$g:X\to\mathbb{R}$ such that

$f(x)+g(y)\leq\mathsf{d}_{Y}(u(x),u(y))\,,\quad\text{for any $x,y\in X$}\,$

and

$f(x_{0})+g(y_{0})=\mathsf{d}_{Y}(u(x_{0}),u(y_{0}))\,,$

see (5.10) and the subsequent discussion for the details of the construction.
The combination of (1.12), together with the Wasserstein contractivity of the
Heat Flow to control the second term in (1.10), proves the sub-harmonicity of
the function

$\mathcal{G}^{t}(x):=\sup_{y\in\Omega}\left\\{\mathsf{d}_{Y}(u(x),u(y))-\frac{\mathsf{d}^{2}(x,y)}{2t}\right\\}\,.$

This will be rigorously proved in section 5. The local Lipschitz regularity of
$u$ will be established in section 6, via a variant of the aforementioned
argument presented for scalar valued harmonic maps, where the sub-harmonicity
of the function $\mathcal{G}^{t}$ plays the role of (1.7).

A major difficulty that we encounter in making rigorous the strategy above is
that we are able to verify (1.11) (and therefore also (1.12)) only away from a
set of negligible measure in the source domain, see section 3. This is in
perfect analogy with similar situations in the classical viscosity theory of
Partial Differential Equations, see for instance [19], and with the case of
Alexandrov spaces [81, 99]. We overcome this issue with an original
perturbation argument of independent interest, in the spirit of Jensen’s
approximate maximum principle [55] for semi-concave functions.

In the Euclidean theory, perturbation arguments usually rely on the affine
structure. In [81, 99] the authors employ a combination of two perturbation
arguments: the first one is required to move the minimum of a given function
near to a regular point in the sense of the Alexandrov theory and it is
achieved with a small additive perturbation using Perelman’s concave
functions. The second one uses the existence of concave bi-Lipschitz
coordinates as a replacement for the Euclidean affine functions. In the
present situation, these techniques seem out of reach. Indeed, the existence
of concave auxiliary functions heavily relies on the synthetic lower bound on
the sectional curvature and the existence of bi-Lipschitz coordinates goes
much beyond the present regularity theory of spaces with lower Ricci curvature
bounds, even in the “non-collapsed” case.
Perturbations will be constructed using distance functions, by further
developing an idea introduced by Cabré [18], with a different aim, in the
setting of smooth Riemannian manifolds with lower sectional curvature bounds
(see also the subsequent [63, 95]). The proof of the key estimate will require
several new ingredients with respect to the case of Riemannian manifolds and
will rely, again, on a new interpretation of the interplay between Optimal
Transport and Heat Flow through Kuwada’s duality, see section 4. For more
details, we refer the reader to the first few pages of section 4, where we
present the general perturbation strategy, the difficulties of the present
setting, and the new ideas developed in the paper.

Once local Lipschitz continuity has been established, the Bochner-Eells-
Sampson inequality with Hessian type term (1.4) will be proved via bootstrap.
The main idea, borrowed from the recent [100], is to run the same arguments
above changing the squared distance $\mathsf{d}^{2}(x,y)$ with any power
$\mathsf{d}^{p}(x,y)$ for $1<p<\infty$ and then to let $p\to\infty$. While
[100] considers smooth Riemannian manifolds, following the strategy of [99],
in the present context the analysis is possible thanks to the Wasserstein
contractivity of the Heat Flow in any Wasserstein space with distance $W_{p}$
for $1\leq p\leq\infty$, see [86]. Combined with a version of Kirchheim’s
metric differentiability theorem [65], recently established in [46], this will
lead to the sought Bochner-Eells-Sampson inequality in section 7.

Two months after posting the preprint of this work on arXiv, the preprint of
[38] appeared on the same archive, containing partially overlapping results.

Acknowledgements. The authors are supported by the European Research Council
(ERC), under the European Union Horizon 2020 research and innovation
programme, via the ERC Starting Grant “CURVATURE”, grant agreement No. 802689.
They are grateful to the reviewer for the careful reading and useful comments
on a preliminary version of the paper.

##  2\. Preliminaries

This preliminary section is meant to introduce some basic material and to set
the terminology that we shall adopt in the paper. In subsection 2.1 we collect
some mostly well known preliminaries about Geometric Analysis on
$\operatorname{RCD}(K,N)$ metric measure spaces. In subsection 2.2 we
introduce the relevant background and terminology about Sobolev and harmonic
maps from $\operatorname{RCD}(K,N)$ spaces into ${\sf CAT}(0)$ spaces.

###  2.1. Geometric Analysis tools on $\operatorname{RCD}(K,N)$ spaces

Throughout the paper, $(X,\mathsf{d},\mathfrak{m})$ will be a metric measure
space, i.e. $(X,\mathsf{d})$ is a complete and separable metric space endowed
with a non-negative Borel measure which is finite on bounded sets.

Given $f:X\to\mathbb{R}$, we denote with $\operatorname{lip}f$ the slope of
$f$ defined as

$\operatorname{lip}f(x_{0}):=\limsup_{x\to
x_{0}}\frac{|f(x)-f(x_{0})|}{\mathsf{d}(x,x_{0})}\;\text{ if $x_{0}$ is not
isolated}\,,\quad\operatorname{lip}f(x_{0})=0\;\text{ otherwise}\,.$

We denote by $C(X)$ the space of continuous functions and with
$\operatorname{Lip}(X)$ (resp. $\operatorname{Lip_{b}}(X),$
$\operatorname{Lip_{\rm bs}}(X)$) the space of Lipschitz functions on
$(X,\mathsf{d})$ (resp. bounded Lipschitz functions, and Lipschitz functions
with bounded support). Analogous notations will be used for the spaces of
continuous and Lipschitz functions on an open domain $\Omega\subset X$.
We will indicate by $B_{r}(x)$ the open ball $B_{r}(x):=\\{y\in
X\,:\,\mathsf{d}(x,y)<r\\}$ for $r>0$ and $x\in X$ and by
$\overline{B_{r}(x)}:=\\{y\in X\,:\,\mathsf{d}(x,y)\leq r\\}$ the closed ball,
for $x\in X$ and $r>0$.

The Cheeger energy (introduced in [23] and further studied in [4]) is defined
as the $L^{2}$-lower semicontinuous envelope of the functional
$f\mapsto\frac{1}{2}\int_{X}(\operatorname{lip}f)^{2}\,\mathop{}\\!\mathrm{d}\mathfrak{m}$,
i.e.:

${\sf
Ch}(f):=\inf\left\\{\liminf_{n\to\infty}\frac{1}{2}\int_{X}(\operatorname{lip}f_{n})^{2}\,\mathop{}\\!\mathrm{d}\mathfrak{m}\,:\,f_{n}\in\operatorname{Lip}(X),\;f_{n}\to
f\text{ in }L^{2}(X,\mathfrak{m})\right\\}\,.$

If ${\sf Ch}(f)<\infty$ it was proved in [23, 4] that the set

$G(f):=\left\\{g\in
L^{2}(X,\mathfrak{m})\,:\,\exists\,f_{n}\in\operatorname{Lip}(X),\,f_{n}\to
f,\,\operatorname{lip}f_{n}\rightharpoonup h\geq g\text{ in
}L^{2}(X,\mathfrak{m})\right\\}$

is closed and convex, therefore it admits a unique element of minimal norm
called minimal weak upper gradient and denoted by $|\nabla f|$. The Cheeger
energy can be then represented by integration as

${\sf Ch}(f):=\frac{1}{2}\int_{X}|\nabla
f|^{2}\mathop{}\\!\mathrm{d}\mathfrak{m}\,.$

It is not difficult to see that ${\sf Ch}$ is a $2$-homogeneous, lower semi-
continuous, convex functional on $L^{2}(X,\mathfrak{m})$, whose proper domain
${\rm Dom}({\sf Ch}):=\\{f\in L^{2}(X,\mathfrak{m})\,:\,{\sf Ch}(f)<\infty\\}$
is a dense linear subspace of $L^{2}(X,\mathfrak{m})$. It then admits an
$L^{2}$-gradient flow which is a continuous semigroup of contractions
$(P_{t})_{t\geq 0}$ in $L^{2}(X,\mathfrak{m})$, whose continuous trajectories
$t\mapsto P_{t}f$, for $f\in L^{2}(X,\mathfrak{m})$, are locally Lipschitz
curves from $(0,\infty)$ with values into $L^{2}(X,\mathfrak{m})$.

Throughout the paper, we will assume that ${\sf Ch}:{\rm Dom}({\sf
Ch})\to\mathbb{R}$ satisfies the parallelogram identity (i.e. it is a
quadratic form) or, equivalently, that $P_{t}:L^{2}(X,\mathfrak{m})\to
L^{2}(X,\mathfrak{m})$ is a linear operator for every $t\geq 0$. This
condition is known in the literature as infinitesimal Hilbertianity, after [5,
36]. If $(X,\mathsf{d},\mathfrak{m})$ is infinitesimally Hilbertian, then
${\rm Dom}({\sf Ch})$ endowed with the norm
$\|f\|_{H^{1,2}}^{2}:=\|f\|_{L^{2}}+2{\sf Ch}(f)$ is a Hilbert space (in
general it is only a Banach space) that will be denoted by
$W^{1,2}(X,\mathsf{d},\mathfrak{m})$.

The main subject of our investigation will be the so-called
$\operatorname{RCD}(K,N)$ metric measure spaces $(X,\mathsf{d},\mathfrak{m})$,
i.e. infinitesimally Hilbertian metric measure spaces with Ricci curvature
bounded from below and dimension bounded from above, in synthetic sense.

The Riemannian Curvature Dimension condition $\operatorname{RCD}(K,\infty)$
was introduced in [5] (see also [36, 3]) coupling the Curvature Dimension
condition $\operatorname{CD}(K,\infty)$, previously proposed in [92, 93] and
independently in [75], with the infinitesimally Hilbertian assumption,
corresponding to the Sobolev space $W^{1,2}$ being Hilbert.
The finite dimensional counterparts led to the notions of
$\operatorname{RCD}(K,N)$ and $\operatorname{RCD}^{*}(K,N)$ spaces,
corresponding to $\operatorname{CD}(K,N)$ (resp. $\operatorname{CD}^{*}(K,N)$,
see [13]) satisfying the infinitesimally Hilbertian assumption. The class
$\operatorname{RCD}(K,N)$ was proposed in [36]. The (a priori more general)
$\operatorname{RCD}^{*}(K,N)$ condition was thoroughly analyzed in [31] and
(subsequently and independently) in [10] (see also [20] for the equivalence
between $\operatorname{RCD}^{*}$ and $\operatorname{RCD}$ in the case of
finite reference measure, and [73] for the extension to the case of
$\sigma$-finite measures).

We avoid giving a detailed introduction to this notion, addressing the reader
to the survey [1] and references therein for the relevant background. Below we
recall some of the main properties that will be relevant for our purposes.

Unless otherwise stated, from now on we assume that
$(X,\mathsf{d},\mathfrak{m})$ is an $\operatorname{RCD}(K,N)$ metric measure
space for some $K\in\mathbb{R}$ and $1\leq N<\infty$.

We begin by recalling the notion of Laplacian.

######  Definition .

The Laplacian $\Delta:D(\Delta)\to L^{2}(X,\mathfrak{m})$ is a densely defined
linear operator whose domain consists of all functions $f\in
W^{1,2}(X,\mathsf{d},\mathfrak{m})$ satisfying

$\int hg\mathop{}\\!\mathrm{d}\mathfrak{m}=-\int\nabla h\cdot\nabla
f\mathop{}\\!\mathrm{d}\mathfrak{m}\quad\text{for any $h\in
W^{1,2}(X,\mathsf{d},\mathfrak{m})$}$

for some $g\in L^{2}(X,\mathfrak{m})$. The unique $g$ with this property is
denoted by $\Delta f$.

As a consequence of the infinitesimal hilbertianity, it is easily checked that
$\Delta$ is an (unbounded) linear operator. More generally, we say that $f\in
W^{1,2}_{{\rm loc}}(X,\mathsf{d},\mathfrak{m})$ is in the domain of the
measure valued Laplacian, and we write $f\in D(\bm{\Delta})$, if there exists
a Radon measure $\mu$ on $X$ such that, for every
$\psi\in\operatorname{Lip}_{c}(X)$, it holds

$\int\psi\mathop{}\\!\mathrm{d}\mu=-\int\nabla
f\cdot\nabla\psi\mathop{}\\!\mathrm{d}\mathfrak{m}\,.$

In this case we write $\bm{\Delta}f:=\mu$. If moreover
$\bm{\Delta}f\ll\mathfrak{m}$ with $L^{2}_{{\rm loc}}$ density we denote by
$\Delta f$ the unique function in $L^{2}_{{\rm loc}}(X,\mathfrak{m})$ such
that $\bm{\Delta}f=\Delta f\,\mathfrak{m}$ and we write $f\in D_{{\rm
loc}}(\Delta)$. When there is no risk of confusion, we will adopt the simpler
notation $\Delta$ even for the measure valued Laplacian.

Notice that the definition makes sense even under the assumption that $f\in
W^{1,p}_{{\rm loc}}(X,\mathsf{d},\mathfrak{m})$ for some $1\leq p<\infty$, and
we will rely on this observation later.

We shall also consider the Laplacian on open sets, imposing Dirichlet boundary
conditions. Let us first introduce the local Sobolev space with Dirichlet
boundary conditions.

######  Definition .

Let $(X,\mathsf{d},\mathfrak{m})$ be an $\operatorname{RCD}(K,N)$ metric
measure space and let $\Omega\subset X$ be an open and bounded domain. Then we
let $W^{1,2}_{0}(\Omega)$ be the $W^{1,2}(X,\mathsf{d},\mathfrak{m})$ closure
of $\operatorname{Lip}_{c}(\Omega,\mathsf{d})$.

We also introduce the local Sobolev space (i.e. without imposing Dirichlet
boundary conditions).

######  Definition .

Let $(X,\mathsf{d},\mathfrak{m})$ be an $\operatorname{RCD}(K,N)$ metric
measure space and let $\Omega\subset X$ be an open and bounded domain. We say
that a function $f\in L^{2}(\Omega,\mathfrak{m})$ belongs to the local Sobolev
space $W^{1,2}(\Omega,\mathsf{d},\mathfrak{m})$ if

  * (i)

$f\varphi\in W^{1,2}(X,\mathsf{d},\mathfrak{m})$ for any
$\varphi\in\operatorname{Lip}_{c}(\Omega,\mathsf{d})$;

  * (ii)

$\left\lvert\nabla f\right\rvert\in L^{2}(X,\mathfrak{m})$.

Above, we intend that $f\varphi$ is set to be $0$ outside of $\Omega$. Notice
that $\left\lvert\nabla f\right\rvert$ is well defined on any
$\Omega^{\prime}\subset\Omega$ (and hence on $\Omega$) as $\left\lvert\nabla
f\right\rvert:=\left\lvert\nabla(f\varphi)\right\rvert$ for some
$\varphi\in\operatorname{Lip}_{c}(\Omega)$ such that $\varphi\equiv 1$ on
$\Omega^{\prime}$. The definition does not depend on the cut-off function by
locality.

######  Definition .

Let $f\in W^{1,2}(\Omega)$. We say that $f\in D(\Delta,\Omega)$ if there
exists a function $h\in L^{2}(\Omega,\mathfrak{m})$ such that

$\int_{\Omega}gh\mathop{}\\!\mathrm{d}\mathfrak{m}=-\int_{\Omega}\nabla
g\cdot\nabla f\mathop{}\\!\mathrm{d}\mathfrak{m}\,,\quad\text{for any $g\in
W^{1,2}_{0}(\Omega,\mathsf{d},\mathfrak{m})$}\,.$

The Heat Flow $P_{t}$, previously defined as the
$L^{2}(X,\mathfrak{m})$-gradient flow of ${\sf Ch}$, can be equivalently
characterised by the following property: for any $u\in L^{2}(X,\mathfrak{m})$,
the curve $t\mapsto P_{t}u\in L^{2}(X,\mathfrak{m})$ is locally absolutely
continuous in $(0,+\infty)$ and satisfies

$\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}P_{t}u=\Delta
P_{t}u\quad\text{for $\mathscr{L}^{1}$-a.e. $t\in(0,\infty)$}\,.$

Under our assumptions the Heat Flow provides a linear, continuous and self-
adjoint contraction semigroup in $L^{2}(X,\mathfrak{m})$. Moreover $P_{t}$
extends to a linear, continuous and mass preserving operator, still denoted by
$P_{t}$, in all the $L^{p}$ spaces for $1\leq p<+\infty$.

It has been proved in [5, 3] that, on $\operatorname{RCD}(K,\infty)$ metric
measure spaces, the dual heat semigroup
$\bar{P}_{t}:\mathcal{P}_{2}(X)\to\mathcal{P}_{2}(X)$ of $P_{t}$, defined by

$\int_{X}f\mathop{}\\!\mathrm{d}\bar{P}_{t}\mu:=\int_{X}P_{t}f\mathop{}\\!\mathrm{d}\mu\qquad\quad\forall\mu\in\mathcal{P}_{2}(X),\quad\forall
f\in\operatorname{Lip_{b}}(X)\,,$

is $K$-contractive (w.r.t. the $W_{2}$-distance) and, for $t>0$, maps
probability measures into probability measures absolutely continuous w.r.t.
$\mathfrak{m}$. Then, for any $t>0$, we can introduce the so called heat
kernel $p_{t}:X\times X\to[0,+\infty)$ by

$p_{t}(x,\cdot)\mathfrak{m}:=\bar{P}_{t}\delta_{x}\,.$

As there is no risk of confusion, we will mostly adopt the notation $P_{t}$
also for the dual Heat Flow defined on probability measures with finite second
order moment. We shall denote by $P_{t}\delta_{x}$ the heat kernel (measure)
centred at $x\in X$ at time $t>0$.

A key property of the heat kernel follows, namely the so-called stochastic
completeness: for any $x\in X$ and for any $t>0$ it holds

$\int_{X}p_{t}(x,y)\mathop{}\\!\mathrm{d}\mathfrak{m}(y)=1\,.$ (2.1)

Let us recall a classical regularity result for solutions of the Poisson
equation, see [56].

######  Proposition .

Let $(X,\mathsf{d},\mathfrak{m})$ be an $\operatorname{RCD}(K,N)$ metric
measure space. Let $\Omega\subset X$ be an open and bounded domain. If $g\in
L^{\infty}(\Omega)$ and $f\in W^{1,2}(\Omega)$ verifies $\Delta f=g$ on
$\Omega$, then $f$ is locally Lipschitz.

Following [36] we introduce the notion of Laplacian bound in the sense of
distributions.

######  Definition .

Let $(X,\mathsf{d},\mathfrak{m})$ be an $\operatorname{RCD}(K,N)$ metric
measure space and let $\Omega\subset X$ be an open domain. Let $f\in
W^{1,2}(\Omega)$ and $\eta\in L^{\infty}(\Omega)$. Then we say that $\Delta
f\leq\eta$ in the sense of distributions if the following holds. For any non-
negative function $\varphi\in\operatorname{Lip}_{c}(\Omega)$,

$-\int_{\Omega}\nabla
f\cdot\nabla\varphi\mathop{}\\!\mathrm{d}\mathfrak{m}\leq\int_{\Omega}\varphi\eta\mathop{}\\!\mathrm{d}\mathfrak{m}\,.$

The following can be obtained with minor modifications from [64]. We refer to
[99, Corollary 3.5] for the case of Alexandrov spaces, the proof works
verbatim in the present setting.

######  Proposition .

Let $(X,\mathsf{d},\mathfrak{m})$ be an $\operatorname{RCD}(K,N)$ metric
measure space. Let $\Omega\subset X$ be an open domain, $g\in
L^{\infty}(\Omega)$ and $f\in W^{1,2}_{{\rm loc}}(\Omega)\cap C(\Omega)$. Then
the following are equivalent:

  * (i)

$\Delta f\leq g$ in the sense of distributions;

  * (ii)

for any open domain $\Omega^{\prime}\Subset\Omega$, if $v\in
W^{1,2}(\Omega^{\prime})$ solves $\Delta v=g$ on $\Omega^{\prime}$ and $f-v\in
W^{1,2}_{0}(\Omega)$, then $v\leq f$ in $\Omega^{\prime}$.

The following is a slight extension of [80, Proposition 3.27]. We consider
only constant upper bounds for the Laplacian, but we remove the Lipschitz
continuity assumption.

######  Proposition .

Let $(X,\mathsf{d},\mathfrak{m})$ be an $\operatorname{RCD}(K,N)$ metric
measure space for some $K\in\mathbb{R}$ and $1\leq N<\infty$. Let
$\Omega\subset X$ be an open domain and let $f:\Omega\to\mathbb{R}$ be
continuous and bounded. If $f\in W^{1,2}(\Omega)$ admits measure valued
Laplacian on $\Omega$, with

$\Delta f\leq C\,,\quad\text{on $\Omega$}$ (2.2)

in the sense of distributions for some constant $C\in\mathbb{R}$, then the
following holds. For any domain $\Omega^{\prime}\Subset\Omega$ and for any
function $g:X\to\mathbb{R}$ with polynomial growth and such that $f\equiv g$
on $\Omega^{\prime}$, it holds

$\limsup_{t\to 0}\frac{P_{t}g(x)-g(x)}{t}\leq C\,,\quad\text{for any
$x\in\Omega^{\prime}$}\,.$ (2.3)

###### Proof.

Thanks to [80, Lemma 2.53] the value of the $\limsup$ in (2.3) is independent
of the chosen extension $g$ of $f$. In particular, thanks to [79, 9], we can
choose a regular cut-off function $\varphi:X\to[0,1]$ with compact support
inside $\Omega$ and such that $\varphi\equiv 1$ on $\Omega^{\prime}$,
Lipschitz and with bounded Laplacian. Then we consider $g:=f\varphi$, where it
is understood that $g\equiv 0$ outside of $\Omega$.

By the standard Leibniz rule for the measure-valued Laplacian, $g$ admits
measure-valued Laplacian on $X$. Moreover,

$\Delta g=f\Delta\varphi+2\nabla f\cdot\nabla\varphi+\varphi\Delta f\,.$

Hence, for any $x\in X$ and for any $t\geq 0$,

$P_{t}g(x)-g(x)\leq\int_{0}^{t}P_{s}\left(f\Delta\varphi+2\nabla
f\cdot\nabla\varphi+\varphi\Delta f\right)(x)\mathop{}\\!\mathrm{d}s\,.$ (2.4)

The inequality above can be checked by using the continuity of $g$ and
integrating against Test cut-off functions. More precisely, for any
sufficiently regular and compactly supported function $\psi:X\to[0,\infty)$ it
holds

$\displaystyle\int_{X}(P_{t}g-g)\psi\mathop{}\\!\mathrm{d}\mathfrak{m}=$
$\displaystyle\int_{X}g(P_{t}\psi-\psi)\mathop{}\\!\mathrm{d}\mathfrak{m}$
(2.5) $\displaystyle=$ $\displaystyle\int_{X}\int_{0}^{t}g\Delta
P_{s}\psi\mathop{}\\!\mathrm{d}s\mathop{}\\!\mathrm{d}\mathfrak{m}$ (2.6)
$\displaystyle\leq$
$\displaystyle\int_{X}\left(\int_{0}^{t}P_{s}\left(f\Delta\varphi+2\nabla
f\cdot\nabla\varphi+\varphi\Delta
f\right)\right)\psi\mathop{}\\!\mathrm{d}\mathfrak{m}\,,$ (2.7)

which is enough to get (2.4) by approximation.

Applying [80, Lemma 2.53] to the first two terms above, we easily obtain that

$\limsup_{t\to 0}\frac{P_{t}g(x)-g(x)}{t}\leq\limsup_{t\to
0}\frac{1}{t}\int_{0}^{t}P_{s}\left(\varphi\Delta
f\right)(x)\mathop{}\\!\mathrm{d}s\leq C\,,$

where we employed (2.2) and the comparison principle for the Heat Flow to
obtain the last inequality. ∎

We recall that $\operatorname{RCD}(K,N)$ metric measure spaces
$(X,\mathsf{d},\mathfrak{m})$ are _strongly rectifiable spaces_ , according to
[46, Definition 2.18] (see also the previous [42]). This condition means that
for any $\varepsilon>0$ there exists a countable covering of $X$ with Borel
sets $U_{i}^{\varepsilon}$, away from a set of $\mathfrak{m}$-measure $0$,
such that all these sets are $(1+\varepsilon)$-biLipschitz to Borel subsets of
$\mathbb{R}^{n}$ with charts $\varphi_{i}^{\varepsilon}$, for a given
$n\in\mathbb{N}$ independent of $i$, and it holds

$c_{i}\mathscr{L}^{n}|_{\varphi_{i}^{\varepsilon}(U_{i}^{\varepsilon})}\leq\left(\varphi_{i}^{\varepsilon}\right)_{\sharp}\left(\mathfrak{m}|_{U_{i}^{\varepsilon}}\right)\leq(c_{i}+\varepsilon)\mathscr{L}^{n}|_{\varphi_{i}^{\varepsilon}(U_{i}^{\varepsilon})}\,,$

for some constant $c_{i}>0$, where we denoted by
$\left(\varphi_{i}^{\varepsilon}\right)_{\sharp}$ the pushforward operator for
measures through $\varphi_{i}^{\varepsilon}$.
This statement follows from the combination of [79], [62, 28, 42] and [17].

We refer again to [46, Definition 2.18] for the notion of _aligned_ family of
atlas $\mathcal{A}^{\varepsilon_{n}}$, for a given sequence
$\varepsilon_{n}\downarrow 0$, of a strongly rectifiable space that will be
relevant for the subsequent developments of the note.

The unique natural number $1\leq n\leq N$ such that the above hold will be
denoted by _essential dimension_ of the $\operatorname{RCD}(K,N)$ metric
measure space $(X,\mathsf{d},\mathfrak{m})$.

###  2.2. Energy and non-linear harmonic maps

The subject of this paper are harmonic maps from open subsets of
$\operatorname{RCD}(K,N)$ metric measure spaces to ${\sf CAT}(0)$ spaces. Let
us recall the relevant background and terminology. The main reference for this
presentation is [46]. We refer to [67] for the original notions and results in
the case when the source space is a smooth Riemannian manifold.

######  Definition .

Let $(X,\mathsf{d},\mathfrak{m})$ be a metric measure space,
$Y_{\bar{y}}=(Y,\mathsf{d}_{Y},\bar{y})$ a pointed complete space,
$\Omega\subset X$ open and $u\in L^{2}(\Omega,Y_{\bar{y}})$.
For every $r>0$ we define $\mathrm{ks}_{2,r}[u,\Omega]:\Omega\to[0,\infty]$ as

${\sf
ks}_{2,r}[u,\Omega](x):=\left\\{\begin{array}[]{ll}\displaystyle{\Big{|}\fint_{B_{r}(x)}\frac{\mathsf{d}^{2}_{Y}(u(x),u(y))}{r^{2}}\,\mathop{}\\!\mathrm{d}\mathfrak{m}(y)\Big{|}^{1/2}}&\qquad\text{if
}B_{r}(x)\subset\Omega,\\\ 0&\qquad\text{otherwise}\end{array}\right.$

and say that $u\in{\sf KS}^{1,2}(\Omega,Y_{\bar{y}})$ provided

$\mathrm{E}_{2}^{\Omega}(u):=\sup\limsup_{r\downarrow
0}\int_{\Omega}\phi\,{\sf
ks}^{2}_{2,r}[u,\Omega]\,\mathop{}\\!\mathrm{d}\mathfrak{m}<\infty,$ (2.8)

where the $\sup$ is taken among all $\phi:X\to[0,1]$ continuous and such that
$\operatorname{supp}(\phi)$ is compact and contained in $\Omega$.

We recall that the notion of metric differentiability, originally formulated
in [65] for maps $u:\mathbb{R}^{n}\to Y$, where $(Y,\mathsf{d}_{y})$ is a
metric space, has been extended to the case when the source space is a
strongly rectifiable metric measure space in [46, Definition 3.3], with the
introduction of the notion of approximately metrically differentiable map.
Moreover, in [46] it is proved that maps $u\in{\sf
KS}^{1,2}(\Omega,Y_{\bar{y}})$ are $\mathfrak{m}$-a.e. approximately
metrically differentiable. We shall denote the (approximate) metric
differential of $u$ at a point $x\in X$ as $\mathrm{md}_{x}u$. The metric
differential is a semi-norm on $\mathbb{R}^{n}$ whenever
$(X,\mathsf{d},\mathfrak{m})$ is an $\operatorname{RCD}(K,N)$ metric measure
space with essential dimension $1\leq n\leq N$. Below we report the relevant
definition of metric differentiability in the present setting and address the
reader to [46, Section 3] for the more general notion of approximate metric
differentiability.

######  Definition .

Let $(X,\mathsf{d},\mathfrak{m})$ be an $\operatorname{RCD}(K,N)$ metric
measure space with essential dimension $1\leq n\leq N$ and let
$(Y,\mathsf{d}_{Y})$ be a complete metric space. Let $\Omega\subset X$ be an
open domain and let $u:\Omega\to Y$ be a Lipschitz map. Given a point
$x\in\Omega$, we say that $u$ is metrically differentiable at $x$ if the
following hold:

  * (i)

$x$ is an $n$-regular point of $(X,\mathsf{d},\mathfrak{m})$;

  * (ii)

the functions $g_{i}:X\to[0,\infty)$ defined by
$g_{i}(y):=\mathsf{d}_{Y}(u(y),u(x))/r_{i}$ considered along the pmGH
converging sequence
$X_{i}:=(X,r_{i}^{-1}\mathsf{d},\left(\mathfrak{m}(B_{r_{i}}(x))\right)^{-1}\mathfrak{m},x)$
for $r_{i}\downarrow 0$ converge locally uniformly to a semi-norm
$\mathrm{md}_{x}u:\mathbb{R}^{n}\to[0,\infty)$, which is independent of the
chosen sequence, up to composition with Euclidean isometries.

######  Definition .

Given a seminorm $\left\lVert\cdot\right\rVert$ on $\mathbb{R}^{n}$, we define
its $2$-size as

$S_{2}^{2}(\left\lVert\cdot\right\rVert):=\fint_{B_{1}(0^{n})}\left\lVert
v\right\rVert^{2}\mathop{}\\!\mathrm{d}\mathscr{L}^{n}(v)\,.$

In the next statement, we recall the existence of the energy density and its
representation in terms of the $2$-size of the metric differential (proved in
[46, Theorem 3.13]), and the identification result between the metric
differential and differentials $\mathop{}\\!\mathrm{d}u$ of metric valued
Sobolev maps in the sense of [43] (see also [46, Definition 4.5]) obtained in
[46, Theorem 4.12]. Such identification plays an important role in some of the
subsequent developments of the theory, see for instance Theorem 2.2 below.

######  Theorem 2.1.

Let $(X,\mathsf{d},\mathfrak{m})$ be locally uniformly doubling, supporting a
Poincaré inequality and strongly rectifiable, $\Omega\subset X$ open and
$Y_{\bar{y}}=(Y,\mathsf{d}_{Y},{\bar{y}})$ a pointed and complete space.

Then the following hold:

  * (i)

${\sf KS}^{1,2}(\Omega,Y_{\bar{y}})=W^{1,2}(\Omega,Y_{\bar{y}})$ as sets.

  * (ii)

For any $u\in{\sf KS}^{1,2}(\Omega,Y_{\bar{y}})$, there is a function
$\mathrm{e}_{2}[u]\in L^{2}(X)$, called _$2$ -energy density_ of $u$, such
that

${\sf ks}_{2,r}[u,\Omega]\quad\to\quad\mathrm{e}_{2}[u]\qquad\text{
$\mathfrak{m}$-a.e.\ and in $L^{2}_{{\rm loc}}(\Omega)$ as $r\downarrow
0$}\,.$

  * (iii)

Any $u\in{\sf KS}^{1,2}(\Omega,Y_{\bar{y}})$ is approximately metrically
differentiable $\mathfrak{m}$-a.e. in $\Omega$ (here we extend $u$ on the
whole $X$ declaring it to be constant outside $\Omega$ to apply the definition
of approximate metric differentiability) and it holds

$\mathrm{e}_{2}[u](x)=S_{2}(\mathrm{md}_{x}(u))=S_{2}(\mathop{}\\!\mathrm{d}u)(x)\qquad\mathfrak{m}\text{-a.e.}\
x\in\Omega.$ (2.9)

  * (iv)

The functional
$\mathrm{E}^{\Omega}_{2}:L^{2}(\Omega,Y_{\bar{y}})\to[0,+\infty]$ defined by
(2.8) is lower semicontinuous and can be written as

$\mathrm{E}_{2}^{\Omega}(u):=\left\\{\begin{array}[]{ll}\displaystyle{\int_{\Omega}\mathrm{e}_{2}^{2}[u]\,\mathop{}\\!\mathrm{d}\mathfrak{m}},&\qquad\text{
if }u\in{\sf KS}^{1,2}(\Omega,Y_{\bar{y}}),\\\ +\infty,&\qquad\text{
otherwise}.\end{array}\right.$

We recall that a metric space $(Y,\mathsf{d}_{Y})$ satisfies the ${\sf
CAT}(0)$ condition if for any points $x_{0},x_{1}\in Y$ and for any minimizing
geodesic $\gamma:[0,1]\to Y$ connecting them the parallelogram inequality

$\mathsf{d}_{Y}^{2}(\gamma_{t},y)\leq(1-t)\mathsf{d}_{Y}^{2}(x_{0},y)+t\mathsf{d}_{Y}^{2}(x_{1},y)-t(1-t)\mathsf{d}^{2}_{Y}(x_{0},x_{1})$
(2.10)

holds for any $t\in[0,1]$ and for any $y\in Y$.

An outcome of the _universal infinitesimal Hilbertianity_ of ${\sf CAT}(0)$
spaces, previously proved in [29], is the following representation of the
energy density as Hilbert-Schmidt norm of the differential, obtained in [46,
Proposition 6.7].

######  Proposition  (Energy density as Hilbert-Schmidt norm).

Let $(X,\mathsf{d},\mathfrak{m})$ be a strongly rectifiable space of dimension
$n\in\mathbb{N}$ with uniformly locally doubling measure and supporting a
Poincaré inequality (in particular these assumptions hold if it is a
$\operatorname{RCD}(K,N)$ space for some $K\in\mathbb{R}$ and
$N\in[1,\infty)$) and $\Omega\subset X$ an open set. Let
$Y_{\bar{y}}=(Y,\mathsf{d}_{Y},{\bar{y}})$ be a pointed ${\sf CAT}(0)$ space
and $u\in{\sf KS}^{1,2}(\Omega,Y_{\bar{y}})$.
Then the energy density $\mathrm{e}_{2}[u]$ admits the representation formula

$\mathrm{e}_{2}[u]=(n+2)^{-\frac{1}{2}}|\mathop{}\\!\mathrm{d}u|_{\sf
HS}\quad\mathfrak{m}\text{-a.e. },$

where $n$ is the dimension of $X$.

It is possible to consider Sobolev spaces with prescribed boundary values also
in the metric-valued case. This is key in order to establish existence of
harmonic maps, defined as minimizers of the energy functional with prescribed
boundary conditions.

######  Definition .

Let $(X,\mathsf{d},\mathfrak{m})$ be a metric measure space,
$Y_{\bar{y}}=(Y,\mathsf{d}_{Y},{\bar{y}})$ a pointed complete metric space,
$\Omega\subset X$ open and $\bar{u}\in L^{2}(\Omega,Y_{\bar{y}})$. Then the
space ${\sf KS}^{1,2}_{\bar{u}}(\Omega,Y_{\bar{y}})\subset{\sf
KS}^{1,2}(\Omega,Y_{\bar{y}})$ is defined as

${\sf KS}^{1,2}_{\bar{u}}(\Omega,Y_{\bar{y}}):=\left\\{u\in{\sf
KS}^{1,2}(\Omega,Y_{\bar{y}})\,:\,\mathsf{d}_{Y}(\bar{u},u)\in
W^{1,2}_{0}(\Omega)\right\\}\,.$

Moreover, the energy functional
$\mathrm{E}_{2,\bar{u}}^{\Omega}:L^{2}(\Omega,Y)\to[0,\infty]$ is defined as

$\mathrm{E}_{2,\bar{u}}^{\Omega}=\left\\{\begin{array}[]{ll}\int_{\Omega}\mathrm{e}_{2}^{2}[u]\,\mathop{}\\!\mathrm{d}\mathfrak{m}&\qquad\text{if
$u\in{\sf KS}^{1,2}_{\bar{u}}(\Omega,Y_{\bar{y}})$},\\\
+\infty&\qquad\text{otherwise .}\end{array}\right.$

We recall from [46] that if $(X,\mathsf{d},\mathfrak{m})$ is an
$\operatorname{RCD}(K,N)$ metric measure space,
$Y_{\bar{y}}=(Y,\mathsf{d}_{Y},{\bar{y}})$ is a pointed ${\sf CAT}(0)$ space,
$\Omega\subset X$ is an open domain such that
$\mathfrak{m}(X\setminus\Omega)>0$ and $\bar{u}\in{\sf
KS}^{1,2}(\Omega,Y_{\bar{y}})$, then the energy functional
$\mathrm{E}_{2,\bar{u}}^{\Omega}$ is convex and lower semicontinuous from
$L^{2}(\Omega,Y)$ to $[0,\infty]$ and it admits a unique minimizer, see [46,
Theorem 6.4]. The statement generalizes the previous [67, Theorem 2.2],
dealing with smooth source spaces. We shall call any such minimizer, for a
given boundary datum, a harmonic map.

When both $(X,\mathsf{d})$ and $(Y,\mathsf{d}_{Y})$ are isometric to smooth
Riemannian manifolds, $u:X\to Y$ is harmonic and $f:Y\to\mathbb{R}$ is a
$\lambda$-convex function, then the chain rule easily yields that

$\Delta(f\circ
u)\geq\lambda\left\lvert\mathop{}\\!\mathrm{d}u\right\rvert_{\sf HS}^{2}\,,$

see for instance [60, Chapter 8].
This is generalized to the present setting in [41, Theorem 4.1]. See also [76]
for the case of maps with Euclidean source and ${\sf CAT}(0)$ target.

######  Theorem 2.2.

Let $(X,\mathsf{d},\mathfrak{m})$ be an $\operatorname{RCD}(K,N)$ metric
measure space with essential dimension $1\leq n\leq N$. Let
$(Y,\mathsf{d}_{Y})$ be a ${\sf CAT}(0)$ space and $\Omega\subset X$ be open
and bounded. Let $u\in{\sf KS}^{1,2}(\Omega,Y)$ be harmonic and let
$f:Y\to\mathbb{R}$ be a Lipschitz and $\lambda$-convex function, for some
$\lambda\in\mathbb{R}$. Then $f\circ u\in D(\bm{\Delta},\Omega)$ and
$\bm{\Delta}(f\circ u)$ is a signed Radon measure such that

$\bm{\Delta}(f\circ
u)\geq\lambda\left\lvert\mathop{}\\!\mathrm{d}u\right\rvert^{2}_{\sf
HS}\mathfrak{m}\,.$ (2.11)

With respect to [41, Theorem 4.1], there is a significant modification at the
right-hand side of (2.11), namely in [41, Equation (4.17)] the bound

$\bm{\Delta}(f\circ
u)\geq\frac{\lambda}{n+2}\left\lvert\mathop{}\\!\mathrm{d}u\right\rvert^{2}_{\sf
HS}\mathfrak{m}\,$

is obtained under the same assumptions. We report below the detailed
computation showing that the stronger bound (2.11) actually holds and fixing a
typo in [41].

Borrowing the notation from [41], we start from the bound

$\left\lvert\mathop{}\\!\mathrm{d}u_{t}\right\rvert_{\sf HS}^{2}\leq
e^{-2\lambda tg}\left(\left\lvert\mathop{}\\!\mathrm{d}u\right\rvert_{\sf
HS}^{2}-2t\langle\mathop{}\\!\mathrm{d}g,\mathop{}\\!\mathrm{d}f\circ
g\rangle+Ct^{2}\right)\,,\quad\text{$\mathfrak{m}$-a.e. in $\Omega$}\,.$

Above, a nonnegative function $g\in\operatorname{Lip}_{\rm{bs}}(X)$ has been
fixed and $u_{t}(x)\in Y$ denotes the gradient flow of $f$ starting from
$u(x)$ at time $tg(x)$, which is well defined for $\mathfrak{m}$-a.e. $x$.

Integrating over $\Omega$, subtracting and taking the limsup as $t\downarrow
0$, we get

$\limsup_{t\to
0}\frac{\mathrm{E}_{2}^{\Omega}(u_{t})-\mathrm{E}_{2}^{\Omega}(u)}{t}\leq\frac{1}{n+2}\int_{\Omega}\left(\lambda
g\left\lvert\mathop{}\\!\mathrm{d}u\right\rvert_{\sf
HS}^{2}+\langle\mathop{}\\!\mathrm{d}g,\mathop{}\\!\mathrm{d}f\circ
u\rangle\right)\mathop{}\\!\mathrm{d}\mathfrak{m}\,,$

where we notice that, with respect to [41, Equation (4.12)], the denominator
$(n+2)$ is _in front of all the terms in the integral_. Then, since $u$ is
harmonic and $u_{t}$ is a competitor for the variational problem in the
definition of harmonic maps,

$\mathrm{E}_{2}^{\Omega}(u_{t})-\mathrm{E}_{2}^{\Omega}(u)\geq
0\,\quad\text{for any $t\geq 0$}\,,$

therefore

$\frac{1}{n+2}\int_{\Omega}\left(\lambda
g\left\lvert\mathop{}\\!\mathrm{d}u\right\rvert_{\sf
HS}^{2}+\langle\mathop{}\\!\mathrm{d}g,\mathop{}\\!\mathrm{d}f\circ
u\rangle\right)\mathop{}\\!\mathrm{d}\mathfrak{m}\geq 0\,,\quad\text{for any
$g\in\operatorname{Lip_{\rm bs}}(X)$, $g\geq 0$}\,.$

Hence

$\Delta f\circ u\geq\lambda\left\lvert\mathop{}\\!\mathrm{d}u\right\rvert_{\sf
HS}^{2}\,,$

in the sense of distributions on $\Omega$.

######  Remark .

Let $(Y,\mathsf{d}_{Y})$ be a ${\sf CAT}(0)$ space. Then, for any point $P\in
Y$, the function $f_{P}(\cdot):=\mathsf{d}_{Y}^{2}(P,\cdot)$ is $2$-convex. It
follows from Theorem 2.2 that

$\bm{\Delta}(f_{P}\circ u)\geq 2(n+2)\mathrm{e}_{2}^{2}[u]\,\mathfrak{m}\,.$

Let us recall that non-linear harmonic maps from domains inside
$\operatorname{RCD}(K,N)$ metric measure spaces to ${\sf CAT}(0)$ spaces are
locally Hölder continuous in the interior. This has been proved in full
generality in [50, Corollary 1.7] extending previous results from [59, 74].

######  Theorem 2.3.

Let $(X,\mathsf{d},\mathfrak{m})$ be an $\operatorname{RCD}(K,N)$ metric
measure space. Let $(Y,\mathsf{d}_{Y})$ be a ${\sf CAT}(0)$ space and
$\Omega\subset X$ be open and bounded. Let $u\in{\sf KS}^{1,2}(\Omega,Y)$ be
harmonic. Then $u$ is locally Hölder continuous on $\Omega$.

We recall the notion of pointwise Lipschitz constant, for a continuous
function $u:\Omega\to Y$ defined as

$\operatorname{lip}u(x):=\limsup_{y\to
x}\frac{\mathsf{d}_{Y}(u(x),u(y))}{\mathsf{d}(x,y)}=\limsup_{r\to 0}\sup_{y\in
B_{r}(x)}\frac{\mathsf{d}_{Y}(u(x),u(y))}{r}\,.$

Notice that since $u$ is continuous, the pointwise Lipschitz constant
coincides with the approximate local Lipschitz constant, that, in turn, can be
identified with the norm of the metric differential, see [46].

######  Proposition .

Let $(X,\mathsf{d},\mathfrak{m})$ be an $\operatorname{RCD}(K,N)$ metric
measure space and let $(Y,\mathsf{d}_{Y})$ be a ${\sf CAT}(0)$ space. Let
$\Omega\subset X$ be an open and bounded domain and let $u\in{\sf
KS}^{1,2}(\Omega,Y)$ be a harmonic map. There exists a constant $c=c(n)$,
where $n$ is the essential dimension of $(X,\mathsf{d},\mathfrak{m})$ such
that

$\operatorname{lip}u(x)=\left\lVert\mathrm{md}_{x}u\right\rVert\leq
c(n)\left\lvert\mathop{}\\!\mathrm{d}u\right\rvert(x)\,,\quad\text{for
$\mathfrak{m}$-a.e. $x\in\Omega$}\,.$

###### Proof.

The statement has been proved in [46]. Notice that the assumption that $u$ is
harmonic here plays only a role in the identification between approximate
local Lipschitz constant and pointwise Lipschitz constant. Otherwise the
statement holds for any Sobolev function. ∎

##  3\. Auxiliary results

The aim of this section is to collect some technical results that will be
useful for the proof of the main theorems. Often, the statements are the
natural counterpart of analogous results proved in [99]. The main difference
is the use of the Heat Flow as a substitute for averages on balls. This choice
makes the connection with Laplacian estimates more transparent and it allows
to cover the _weighted_ case, corresponding to $\operatorname{RCD}(K,N)$
spaces with essential dimension $1\leq n<N$, which is not considered in [99].

Let us recall a special case of the quadrilateral comparison from [67,
Corollary 2.1.3] (see also the previous [85]).

######  Lemma .

Let $(Y,\mathsf{d}_{Y})$ be a ${\sf CAT}(0)$ space. Let $\\{P,Q,R,S\\}$ be an
ordered set of points in $Y$ and let us denote by $Q_{m}$ the midpoint of
$QR$. Then

$\displaystyle\left(\mathsf{d}_{Y}(P,S)-\mathsf{d}_{Y}(Q,R)\right)\mathsf{d}_{Y}(Q,R)\geq$
$\displaystyle\left(\mathsf{d}_{Y}^{2}(P,Q_{m})-\mathsf{d}_{Y}^{2}(P,Q)-\mathsf{d}_{Y}^{2}(Q_{m},Q)\right)$
$\displaystyle+\left(\mathsf{d}_{Y}^{2}(S,Q_{m})-\mathsf{d}_{Y}^{2}(S,R)-\mathsf{d}_{Y}^{2}(Q_{m},R)\right)\,.$

The following statement corresponds to [99, Proposition 5.4] originally proved
for an Alexandrov space with curvature bounded below as source. We report here
the strategy of the proof and indicate the minor changes that are needed in
the present more general setting. We consider an $\operatorname{RCD}(K,N)$
metric measure space $(X,\mathsf{d},\mathfrak{m})$ and endow $X\times X$ with
the canonical product structure.

######  Proposition .

Let $(X,\mathsf{d},\mathfrak{m})$ be an $\operatorname{RCD}(K,N)$ metric
measure space and let $(Y,\mathsf{d}_{Y})$ be a ${\sf CAT}(0)$ space. Let
$\Omega\subset X$ be an open and bounded domain and let $u\in{\sf
KS}^{1,2}(\Omega,Y)$ be a harmonic map. Then, denoting

$f(x,y)=\mathsf{d}_{Y}(u(x),u(y))\,,$

it holds that $f\in W^{1,2}(\Omega\times\Omega)$ and

$\bm{\Delta}_{X\times X}f\geq 0\,.$

###### Proof.

The proof is divided into three steps. We first prove that, for any $p\in Y$,
the function $\Omega\ni x\mapsto\mathsf{d}(p,u(x))$ is sub-harmonic. Then we
check that $f\in W^{1,2}(\Omega\times\Omega)$. Eventually, we combine the two
statements to infer that $f$ is sub-harmonic.

Step 1. The fact that, for any point $p\in Y$, the function $\Omega\ni
x\mapsto\mathsf{d}_{Y}(u(x),p)$ is sub-harmonic follows from the convexity of
the function $\mathsf{d}_{Y}(\cdot,p)$ on $Y$, ensured by the ${\sf CAT}(0)$
condition and Theorem 2.2.

Step 2. In order to verify that $f\in W^{1,2}(\Omega\times\Omega)$ we just
need to observe that, when $x\in\Omega$ is fixed, $f_{x}(y):=f(x,y)$ is in
$W^{1,2}(\Omega)$ and

$\int_{\Omega}\left\lvert\nabla
f_{x}(y)\right\rvert^{2}\mathop{}\\!\mathrm{d}\mathfrak{m}(y)\leq\int_{\Omega}\left\lvert\mathop{}\\!\mathrm{d}u\right\rvert^{2}\mathop{}\\!\mathrm{d}\mathfrak{m}\,,$

since $q\mapsto\mathsf{d}_{Y}(q,p)$ is a $1$-Lipschitz function. Analogously,
for any $y\in\Omega$ fixed, the function $f^{y}(x):=f(x,y)$ is in
$W^{1,2}(\Omega)$ and it holds

$\int_{\Omega}\left\lvert\nabla
f^{y}(x)\right\rvert^{2}\mathop{}\\!\mathrm{d}\mathfrak{m}\leq\int_{\Omega}\left\lvert\mathop{}\\!\mathrm{d}u\right\rvert^{2}\mathop{}\\!\mathrm{d}\mathfrak{m}\,.$

The conclusion that $f\in W^{1,2}(\Omega\times\Omega)$ follows from the
tensorization of Cheeger energies and Sobolev spaces, see for instance [6].

Step 3. In order to verify that $f$ is sub-harmonic on $\Omega\times\Omega$ we
rely again on the tensorization of the Cheeger energies. We consider any non-
negative Lipschitz function $\varphi$ with compact support in
$\Omega\times\Omega$. Then

$\nabla_{X\times X}\varphi\cdot\nabla_{X\times
X}f(x,y)=\nabla_{x}\varphi\cdot\nabla_{x}f(x,y)\,+\,\nabla_{y}\varphi\cdot\nabla_{y}f(x,y)\,,\quad\text{$\mathfrak{m}\otimes\mathfrak{m}$-a.e.
on $\Omega\times\Omega$}\,.$

Then we compute

$\displaystyle\int_{\Omega\times\Omega}\nabla_{X\times
X}\varphi\cdot\nabla_{X\times
X}f(x,y)\mathop{}\\!\mathrm{d}\mathfrak{m}(x)\mathop{}\\!\mathrm{d}\mathfrak{m}(y)=$
$\displaystyle\int_{\Omega}\int_{\Omega}\left(\nabla_{x}\varphi\cdot\nabla_{x}f(x,y)\right)\mathop{}\\!\mathrm{d}\mathfrak{m}(x)\mathop{}\\!\mathrm{d}\mathfrak{m}(y)$
$\displaystyle+\int_{\Omega}\int_{\Omega}\left(\nabla_{y}\varphi\cdot\nabla_{y}f(x,y)\right)\mathop{}\\!\mathrm{d}\mathfrak{m}(x)\mathop{}\\!\mathrm{d}\mathfrak{m}(y)$
$\displaystyle=$
$\displaystyle-\int_{\Omega}\int_{\Omega}\varphi(x,y)\mathop{}\\!\mathrm{d}\Delta_{x}f(x,y)$
$\displaystyle-\int_{\Omega}\int_{\Omega}\varphi(x,y)\mathop{}\\!\mathrm{d}\Delta_{y}f(x,y)$
$\displaystyle\leq$ $\displaystyle\,0\,,$

where the last inequality follows from Step 1. The sub-harmonicity of $f$ on
the product follows.

∎

Below we will apply the (global) Heat Flow to functions that are only locally
defined on some open domain $\Omega\subset X$, where
$(X,\mathsf{d},\mathfrak{m})$ is an $\operatorname{RCD}(K,N)$ metric measure
space. It is understood that a point $x\in\Omega$ is fixed and we consider any
global extension with polynomial growth of $f|_{U_{x}}$, where $U_{x}$ is a
neighbourhood of $x$. All the statements are not affected by the specific
choice of the extension, thanks to [80, Lemma 2.53, Lemma 2.54]

We will need some asymptotic mean value inequalities, playing the counterpart
of [99, Proposition 3.2, Corollary 4.7, Corollary 5.6, Lemma 6.4] in the
present setting. Here a key difference between the strategy in [99] is that we
will consider the short time asymptotic of the Heat Flow, rather than the
asymptotic of averages on balls for small radii. We make a brief digression in
order to motivate this choice.

On a smooth $n$-dimensional Riemannian manifold $(M,g)$, for any smooth
function $f:M\to\mathbb{R}$ it holds

$\Delta f(x)=\frac{1}{2(n+2)}\left(\lim_{r\to
0}\frac{\fint_{B_{r}(x)}f\mathop{}\\!\mathrm{d}\mathrm{vol}-f(x)}{r^{2}}\right)\,,\quad\text{for
any $x\in M$}\,,$

where $\Delta$ denotes the Laplace-Beltrami operator.
The connection between asymptotics of averages on balls and the Laplacian is
more delicate on $\operatorname{RCD}(K,N)$ metric measure spaces. This is well
illustrated already at the level of smooth, weighted Riemannian manifolds.
Indeed, if $(M,g,e^{-\varphi}\mathrm{vol})$ is a smooth weighted Riemannian
manifold and, as above, $f:M\to\mathbb{R}$ is a smooth function, then,
denoting by $\Delta_{\varphi}$ the weighted Laplacian associated to the metric
measure structure $(M,\mathsf{d},e^{-\varphi}\mathrm{vol})$ and by $\Delta$
the Laplace-Beltrami operator on $(M,g)$, it holds

$\Delta_{\varphi}f(x)=\Delta f(x)-\nabla\varphi(x)\cdot\nabla f(x)\,,$

while

$\lim_{r\to
0}\frac{\fint_{B_{r}(x)}f\mathop{}\\!\mathrm{d}(e^{-\varphi}\mathrm{vol})-f(x)}{r^{2}}=\frac{1}{n+2}\left[\frac{1}{2}\Delta
f(x)-\nabla f(x)\cdot\nabla\varphi(x)\right]\,.$

We notice that an extra factor depending on the gradient of the function at
$x$ appears in the weighted case. Nevertheless, the identification between
asymptotic of averages on balls and Laplacian, up to dimensional constants,
holds at critical points. In particular it holds at local minima.

######  Proposition .

Let $(X,\mathsf{d},\mathfrak{m})$ be an $\operatorname{RCD}(K,N)$ metric
measure space for some $K\in\mathbb{R}$ and $1\leq N<\infty$ with essential
dimension $1\leq n\leq N$. Let $(Y,\mathsf{d}_{Y})$ be a ${\sf CAT}(0)$ space
and let $\Omega\subset X$ be an open and bounded domain. Let $u\in{\sf
KS}^{1,2}(\Omega,Y)$ be harmonic. Then, for $\mathfrak{m}$-a.e. $x\in\Omega$,

$P_{t}\mathsf{d}_{Y}^{2}(u(\cdot),u(x))(x)=2(n+2)\mathrm{e}^{2}_{2}[u](x)t+o(t)\,,\quad\text{as
$t\to 0$}\,.$ (3.1)

###### Proof.

We rely on the theory developed in [46] and on the convergence and stability
theory from [7]. The strategy is to employ a blow-up argument. The rescalings
of the map $u$ (more specifically, of the function
$\mathsf{d}_{Y}(u(\cdot),u(x))$) are controlled thanks to [46], while the
short time behaviour of the heat kernel is controlled through a classical
blow-up argument near regular points.

Step 1. We consider $n$-regular points $x\in X$. In particular,
$\mathfrak{m}$-a.e. point $x\in X$ is $n$-regular and it holds that

$X_{i}:=\left(X,r_{i}^{-1}\mathsf{d},\frac{1}{\mathfrak{m}(B_{r_{i}}(x))}\mathfrak{m},x\right)\to\left(\mathbb{R}^{n},\mathsf{d}_{\mathrm{eucl}},c_{n}\mathscr{L}^{n},0^{n}\right)\,,$
(3.2)

in the pointed measured Gromov-Hausdorff sense for any sequence $(r_{i})_{i}$
such that $r_{i}\to 0$.

One of the outcomes of [46] is the fact that, given $u\in{\sf
KS}^{1,2}(\Omega,Y)$ as in the statement, for $\mathfrak{m}$-a.e. $x\in\Omega$
there exists a semi-norm $\mathrm{md}_{x}u:\mathbb{R}^{n}\to[0,\infty)$ such
that the sequence of functions $f_{i}:=\mathsf{d}_{Y}(u(\cdot),u(x))/r_{i}$,
considered along the sequence $X_{i}$ defined in (3.2), converges strongly in
$L^{2}_{{\rm loc}}$ to $\mathrm{md}_{x}u:\mathbb{R}^{n}\to[0,\infty)$. This
statement can be verified combining [46, Proposition 3.6] (see also [46,
Definition 3.3] for the definition of approximate metric differentiability)
with the continuity of $u$ from Theorem 2.3 to turn approximate limits into
full limits.
This argument extends [65], dealing with the case of Euclidean source space,
and [23], dealing with the case of scalar valued functions (see also [8] for a
proof tailored to the $\operatorname{RCD}$ setting and the recent [54]).

Step 2. In this second step we compute

$\lim_{t\to
0}\frac{1}{t}P_{t}\left(\mathsf{d}_{Y}^{2}(u(x),u(\cdot))\right)(x)$

in terms of $\mathrm{md}_{x}u$ and relate it with

$\displaystyle\lim_{r\to
0}\fint_{B_{r}(x)}\frac{\mathsf{d}_{Y}^{2}(u(y),u(x))}{r^{2}}\mathop{}\\!\mathrm{d}\mathfrak{m}(y)=$
$\displaystyle\,\mathrm{e}^{2}_{2}[u](x)=S^{2}_{2}(\mathrm{md}_{x}(u))$ (3.3)
$\displaystyle=$
$\displaystyle\fint_{B_{1}(0^{n})}\left\lvert\mathrm{md}_{x}u(v)\right\rvert^{2}\mathop{}\\!\mathrm{d}\mathscr{L}^{n}(v)\,.$
(3.4)

Let us notice that, denoting by $P_{t}^{\mathbb{R}^{n}}$ the standard Heat
Flow on $\mathbb{R}^{n}$,

$P_{1}^{\mathbb{R}^{n}}\left\lvert\mathrm{md}_{x}u(\cdot)\right\rvert^{2}(0^{n})=\frac{1}{(4\pi)^{\frac{n}{2}}}\int_{\mathbb{R}^{n}}e^{-\frac{\left\lvert
v\right\rvert^{2}}{4}}\left\lvert\mathrm{md}_{x}u(v)\right\rvert^{2}\mathop{}\\!\mathrm{d}\mathscr{L}^{n}(v)\,.$
(3.5)

Moreover, $\mathrm{md}_{x}u$ is a seminorm, hence

$\fint_{B_{r}(0^{n})}\left\lvert\mathrm{md}_{x}u(v)\right\rvert^{2}\mathop{}\\!\mathrm{d}\mathscr{L}^{n}(v)=r^{2}\fint_{B_{1}(0^{n})}\left\lvert\mathrm{md}_{x}u(v)\right\rvert^{2}\mathop{}\\!\mathrm{d}\mathscr{L}^{n}(v)\,,\quad\text{for
any $r>0$}\,.$ (3.6)

The combination of (3.5) with (3.6) gives that

$P_{1}^{\mathbb{R}^{n}}\left\lvert\mathrm{md}_{x}u(\cdot)\right\rvert^{2}(0^{n})=2(n+2)\fint_{B_{1}(0^{n})}\left\lvert\mathrm{md}_{x}u(v)\right\rvert^{2}\mathop{}\\!\mathrm{d}\mathscr{L}^{n}(v)\,.$
(3.7)

Notice that, for any sequence $(r_{i})_{i\in\mathbb{N}}$ such that
$r_{i}\downarrow 0$, as $f_{i}$ converge locally in $L^{2}$ to
$\mathrm{md}_{x}u(\cdot)$, $f_{i}^{2}$ converge locally in $L^{1}$ to
$\left\lvert\mathrm{md}_{x}u(\cdot)\right\rvert^{2}$ along the sequence
$X_{i}$. Therefore, since

$\frac{1}{t}P_{t}\mathsf{d}_{Y}^{2}(u(\cdot),u(x))(x)=P_{1}^{X_{i}}f_{i}^{2}(x)\,$

for $t=r_{i}^{2}$ by the scaling properties of the Heat Flow and of the heat
kernel, a stability argument using [2, Lemma 4.11] (see also [7, 8]) proves
that

$\lim_{t\to
0}\frac{1}{t}P_{t}\mathsf{d}_{Y}^{2}(u(\cdot),u(x))(x)=P_{1}^{\mathbb{R}^{n}}\left\lvert\mathrm{md}_{x}u(\cdot)\right\rvert^{2}(0^{n})\,.$
(3.8)

The combination of (3.8) with (3.7) and (3.3) proves (3.1). ∎

######  Remark .

As a consistency check, we notice that, thanks to subsection 2.2,

$\Delta\mathsf{d}_{Y}^{2}(u(\cdot),u(x))\geq
2(n+2)\mathrm{e}^{2}_{2}[u]\mathfrak{m}\,,$

in the sense of distributions on $\Omega$. Hence, by a slight variant of
subsection 2.1, we can verify that

$\displaystyle\liminf_{t\to
0}\frac{1}{t}P_{t}\mathsf{d}_{Y}^{2}(u(\cdot),u(x))(x)\geq$
$\displaystyle\liminf_{t\to
0}\frac{1}{t}\int_{0}^{t}\left(P_{s}\Delta\mathsf{d}_{Y}^{2}(u(\cdot),u(x))\right)(x)\mathop{}\\!\mathrm{d}s$
$\displaystyle\geq$ $\displaystyle\liminf_{t\to
0}\frac{2(n+2)}{t}\int_{0}^{t}P_{s}\left(\mathrm{e}^{2}_{2}[u]\right)(x)\mathop{}\\!\mathrm{d}s\,.$

If $x$ is a Lebesgue point for $\mathrm{e}^{2}_{2}[u]$, which is true for
$\mathfrak{m}$-a.e. $x\in\Omega$, then by [80, Lemma 2.54]

$P_{s}\left(\mathrm{e}^{2}_{2}[u]\right)(x)\to\mathrm{e}^{2}_{2}[u](x)\,,\quad\text{as
$s\to 0$}\,.$

Hence

$\int_{0}^{t}P_{s}\left(\mathrm{e}^{2}_{2}[u]\right)(x)\mathop{}\\!\mathrm{d}s=t\mathrm{e}^{2}_{2}[u](x)+o(t)\,,\quad\text{as
$t\to 0$}\,.$

Therefore

$P_{t}\mathsf{d}_{Y}^{2}(u(\cdot),u(x))(x)\geq
2(n+2)t\mathrm{e}^{2}_{2}[u](x)+o(t)\,,\quad\text{as $t\to 0$}\,,$

which yields one of the inequalities in (3.1).

The following proposition generalises [99, Corollary 5.6] to an
$\operatorname{RCD}$ source space.

######  Proposition .

Let $(X,\mathsf{d},\mathfrak{m})$ be an $\operatorname{RCD}(K,N)$ metric
measure space with essential dimension $1\leq n\leq N$. Let
$(Y,\mathsf{d}_{Y})$ be a ${\sf CAT}(0)$ space and let $\Omega\subset X$ be an
open and bounded domain. Let $u\in{\sf KS}^{1,2}(\Omega,Y)$ be harmonic.
Then for $\mathfrak{m}$-a.e. $x_{0}\in\Omega$ it holds

$-P_{t}\mathsf{d}_{Y}^{2}(u(\cdot),P)(x_{0})+\mathsf{d}_{Y}^{2}(u(x_{0}),P)\leq-2(n+2)\mathrm{e}^{2}_{2}[u](x_{0})t+o(t)\,,\quad\text{as
$t\to 0$}\,,$ (3.9)

for any $P\in Y$.

###### Proof.

We consider the function

$x\mapsto-\mathsf{d}_{Y}^{2}(u(\cdot),P)(x_{0})+\mathsf{d}_{Y}^{2}(u(x_{0}),P)$
(3.10)

and notice that it vanishes at $x_{0}$, by its very definition.

We claim that the statement holds for any point $x_{0}$ such that subsection
2.2 holds, section 3 holds and $x_{0}$ is a Lebesgue point for the energy
density $\mathrm{e}^{2}_{2}[u]$.
Indeed, since the function $-\mathsf{d}_{Y}(\cdot,P)^{2}$ appearing in (3.10)
is $(-2)$–concave by the ${\sf CAT}(0)$ condition, by Theorem 2.2 we have

$\Delta\left(-\mathsf{d}_{Y}^{2}(u(\cdot),P)(x_{0})+\mathsf{d}_{Y}^{2}(u(x_{0}),P)\right)\leq-2(n+2)\mathrm{e}^{2}_{2}[u]\,\mathfrak{m}\,.$

Hence, applying the Heat Flow and taking into account the considerations
above, arguing as in the proof of subsection 2.1 we get

$-P_{t}\mathsf{d}_{Y}^{2}(u(\cdot),P)(x_{0})+\mathsf{d}_{Y}^{2}(u(x_{0}),P)\leq-2(n+2)tP_{t}\left(\mathrm{e}^{2}_{2}[u]\right)(x_{0})+o(t)\,,\quad\text{as
$t\to 0$}\,.$

If $x_{0}$ is a Lebesgue point for $\mathrm{e}^{2}_{2}[u]$, then by [80, Lemma
2.54]

$-2(n+2)tP_{t}\left(\mathrm{e}^{2}_{2}[u]\right)(x_{0})+o(t)=-2(n+2)t(\mathrm{e}^{2}_{2}[u](x_{0})+o(1))+o(t)\,,\quad\quad\text{as
$t\to 0$}\,.$

The claimed (3.9) follows.

∎

The following asymptotic inequality generalises [99, Lemma 6.4] to the present
setting.

######  Proposition .

Let $(X,\mathsf{d},\mathfrak{m})$ be an $\operatorname{RCD}(K,N)$ metric
measure space. Let $(Y,\mathsf{d}_{Y})$ be a ${\sf CAT}(0)$ space and let
$\Omega\subset X$ be an open and bounded domain. Let $u\in{\sf
KS}^{1,2}(\Omega,Y)$ be harmonic.
For any $z\in\Omega$ and $P\in Y$, let us set

$w_{z,P}(\cdot):=\mathsf{d}_{Y}^{2}(u(\cdot),u(z))-\mathsf{d}_{Y}^{2}(u(\cdot),P)+\mathsf{d}_{Y}^{2}(P,u(z))\,.$

Then for $\mathfrak{m}$-a.e. $x_{0}\in\Omega$ it holds that

$\limsup_{t\to 0}\frac{1}{t}P_{t}\left(w_{x_{0},P}(\cdot)\right)(x_{0})\leq
0\,,$ (3.11)

for every $P\in Y$.

###### Proof.

The statement follows from the combination of section 3 with section 3. ∎

Let us recall the Laplacian comparison for $\operatorname{RCD}(K,N)$ spaces
from [36].

######  Theorem 3.1.

Let $(X,\mathsf{d},\mathfrak{m})$ be an $\operatorname{RCD}(K,N)$ metric
measure space for some $K\in\mathbb{R}$ and $1\leq N<\infty$. Fix $p\in X$.
Then the function $\mathsf{d}^{2}(p,\cdot)$ admits locally measure valued
Laplacian bounded from above

$\Delta\mathsf{d}^{2}(p,\cdot)\leq
f_{K,N}(\mathsf{d}(\cdot,p))\mathfrak{m}\,,$ (3.12)

where $f_{K,N}:[0,\infty)$ is a continuous function. When $K=0$, we can take
$f_{0,N}:=2N$ and, more in general $f_{K,N}(0)=2N$ for any $K\in\mathbb{R}$.

The following observations will be useful for our future purposes.

######  Lemma .

The following elementary identity holds

$\mathsf{d}_{X}^{2}(x,y)=2\mathsf{d}^{2}_{X\times
X}\left((x,y),D\right)\,,\quad\text{for any $x,y\in X$}\,,$

where

$D:=\\{(z,z)\,:\,z\in X\\}$

is the diagonal in the product space $X\times X$ and

$\mathsf{d}^{2}_{X\times X}\left((x,y),D\right):=\inf_{w\in
X}\left\\{\mathsf{d}^{2}\left((x,y),(w,w)\right)\right\\}\,.$

In particular if $(X,\mathsf{d},\mathfrak{m})$ is an $\operatorname{RCD}(K,N)$
metric measure space, then

$(x,y)\mapsto\mathsf{d}_{X}^{2}(x,y)\,,$

as a function on $X\times X$, has locally measure valued Laplacian locally
bounded from above by a continuous function.

###### Proof.

The first part of the statement is completely elementary.

In order to prove the second part of the statement, we observe that $X\times
X$ is an $\operatorname{RCD}(K,2N)$ metric measure space. Then we recall that
the Laplacian comparison for distance functions (squared) from points extends
naturally to a Laplacian comparison for distance functions (squared) from
closed sets in this setting, see for instance [22]. Hence

$\Delta_{X\times X}\mathsf{d}^{2}(\cdot,\cdot)\leq
f_{K,2N}\left(\mathsf{d}(\cdot,\cdot)/2\right)\,$

in the sense of distributions on $X\times X$, where $f_{K,2N}$ is the function
appearing in the classical Laplacian comparison for distance functions, see
Theorem 3.1. ∎

We recall that, given probability measures $\mu$ and $\nu$ on $X$, an
admissible transport plan between $\mu$ and $\nu$ is a probability measure
$\Pi$ on $X\times X$ whose push-forwards via the projections on the first and
second marginal are $\mu$ and $\nu$, respectively. More precisely, if
$\pi_{1}:X\times X\to X$ is defined by $\pi_{1}(x,y)=x$ and $\pi_{2}:X\times
X\to X$ is defined by $\pi_{2}(x,y)=y$, then

$\Pi\left(\pi_{1}^{-1}(A)\right)=\mu(A)\,,\quad\Pi\left(\pi_{2}^{-1}(B)\right)=\nu(B)\,,$
(3.13)

for any Borel sets $A,B\subset X$.

######  Lemma .

Let $(X,\mathsf{d},\mathfrak{m})$ be an $\operatorname{RCD}(K,N)$ metric
measure space for some $K\in\mathbb{R}$ and $1\leq N<\infty$. Let $(p,q)\in
X\times X$ and $(x_{0},y_{0})\in X\times X$. For any $r>0$, let $\Pi_{r}$ be
an admissible transport plan between the heat kernels $P_{r}\delta_{x_{0}}$
and $P_{r}\delta_{y_{0}}$. Then

$\displaystyle\limsup_{r\to 0}$
$\displaystyle\frac{\int\mathsf{d}^{2}_{X\times
X}\left((x,y),(p,q)\right)\mathop{}\\!\mathrm{d}\Pi_{r}(x,y)-\mathsf{d}^{2}_{X\times
X}\left((x_{0},y_{0}),(p,q)\right)}{r}$ $\displaystyle\leq
2f_{(K,N)}\left(\max\\{\mathsf{d}(x_{0},p),\mathsf{d}(y_{0},q)\\}\right)\,,$

where $f_{K,N}:[0,\infty)\to(0,\infty)$ is defined in (3.12).

###### Proof.

By the very definition of the product metric measure structure on $X\times X$
it holds

$\mathsf{d}^{2}_{X\times
X}\left((x,y),(p,q)\right)=\mathsf{d}^{2}_{X}(x,p)+\mathsf{d}^{2}_{X}(y,q)\,.$

Therefore, denoting by $\Pi$ an admissible plan between probabilities
$\mu,\nu\in\mathcal{P}(X)$, it holds

$\displaystyle\int_{X\times X}\mathsf{d}^{2}_{X\times
X}\left((x,y),(p,q)\right)\mathop{}\\!\mathrm{d}\Pi(x,y)$
$\displaystyle=\int_{X\times
X}\left(\mathsf{d}^{2}_{X}(x,p)+\mathsf{d}^{2}_{X}(y,q)\right)\mathop{}\\!\mathrm{d}\Pi(x,y)$
$\displaystyle=\int_{X}\mathsf{d}^{2}_{X}(x,p)\mathop{}\\!\mathrm{d}\mu(x)+\int_{X}\mathsf{d}^{2}(y,q)\mathop{}\\!\mathrm{d}\nu(y)\,.$

In particular, with the notation of the statement,

$\displaystyle\int_{X\times X}\mathsf{d}^{2}_{X\times
X}\left((x,y),(p,q)\right)\mathop{}\\!\mathrm{d}\Pi_{r}(x,y)$
$\displaystyle-\mathsf{d}^{2}_{X\times X}\left((x_{0},y_{0}),(p,q)\right)$
$\displaystyle=P_{r}\left(\mathsf{d}^{2}(\cdot,p)\right)(x_{0})-$
$\displaystyle\mathsf{d}^{2}(x_{0},p)+P_{r}\left(\mathsf{d}^{2}(\cdot,q)\right)(y_{0})-\mathsf{d}^{2}(y_{0},q)\,.$

Hence

$\displaystyle\limsup_{r\to 0}$ $\displaystyle\frac{1}{r}\left[\int_{X\times
X}\mathsf{d}^{2}_{X\times
X}\left((x,y),(p,q)\right)\mathop{}\\!\mathrm{d}\Pi_{r}(x,y)-\mathsf{d}^{2}_{X\times
X}\left((x_{0},y_{0}),(p,q)\right)\right]$ $\displaystyle\leq\,$
$\displaystyle\limsup_{t\to
0}\frac{1}{r}\left[P_{r}\left(\mathsf{d}^{2}(\cdot,p)\right)(x_{0})-\mathsf{d}^{2}(x_{0},p)\right]+\limsup_{r\to
0}\frac{1}{r}\left[P_{r}\left(\mathsf{d}^{2}(\cdot,q)\right)(y_{0})-\mathsf{d}^{2}(y_{0},q)\right]$
$\displaystyle\leq\,$ $\displaystyle
2f_{(K,N)}\left(\max\\{\mathsf{d}(x_{0},p),\mathsf{d}(y_{0},q)\\}\right)\,,$

where the last inequality follows from (3.12). ∎

######  Lemma .

Let $(X,\mathsf{d},\mathfrak{m})$ be an $\operatorname{RCD}(K,N)$ metric
measure space. Let $p,q\in X$ and let $\Pi_{r}$ be an admissible transport
plan between the heat kernels at time $r>0$ from $p$ and $q$,
$P_{r}\delta_{p}$ and $P_{r}\delta_{q}$, respectively. Then it holds

$\displaystyle 0\leq$ $\displaystyle\liminf_{r\to 0}\frac{\int_{X\times
X}\left(\mathsf{d}^{2}(x,p)+\mathsf{d}^{2}(y,q)\right)\mathop{}\\!\mathrm{d}\Pi_{r}(x,y)}{r}$
(3.14) $\displaystyle\leq$ $\displaystyle\limsup_{r\to 0}\frac{\int_{X\times
X}\left(\mathsf{d}^{2}(x,p)+\mathsf{d}^{2}(y,q)\right)\mathop{}\\!\mathrm{d}\Pi_{r}(x,y)}{r}\leq
2N\,.$ (3.15)

###### Proof.

The inequality (3.14) is trivial, since at the right hand side there is the
integral of a non-negative function.

In order to prove (3.15) we notice that, since $\Pi_{r}$ is an admissible
transport plan between $P_{r}\delta_{p}$ and $P_{r}\delta_{q}$, it holds

$\int_{X\times
X}\left(\mathsf{d}^{2}(x,p)+\mathsf{d}^{2}(y,q)\right)\mathop{}\\!\mathrm{d}\Pi_{r}(x,y)=P_{r}\left(\mathsf{d}^{2}(\cdot,p)\right)(p)+P_{r}\left(\mathsf{d}^{2}(\cdot,q)\right)(q)\,.$
(3.16)

The conclusion follows from the Laplacian comparison Theorem 3.1. ∎

##  4\. Approximate maximum principles and perturbation arguments

In this section we construct perturbations of functions with measure-valued
Laplacian uniformly bounded from above, with the aim of slightly moving around
their minimum points. This ability will be fundamental for the subsequent
developments of the note. The idea finds its roots in Jensen’s approximate
maximum principle for semiconcave functions [55]. In the Euclidean space,
perturbations are constructed thanks to affine functions, that perturb the
function at the first order without affecting it at the second order, as they
have vanishing Hessian.
The principle was later extended to smooth manifolds with sectional curvature
bounded from below in [18], where the fundamental novelty is that
perturbations are constructed through distance functions (squared), as affine
functions do not have a natural counterpart. The idea was further developed in
[63] and [95], where manifolds with non-negative Ricci curvature and general
lower Ricci curvature bounds, respectively, were considered.
A different strategy to construct well-behaved perturbations on Alexandrov
spaces with lower sectional curvature bounds was proposed independently in the
unpublished manuscript [81] and later studied in [97]. The idea is to combine
two perturbation arguments. The first one is used to move the minimum at a
regular point in the sense of Alexandrov geometry. Then the concave,
biLipschitz coordinate functions are employed as a replacement of affine
functions in the proof of the Euclidean approximate maximum principle. A key
observation is that the concave coordinate functions might affect the
behaviour of the original function at the second order (while affine functions
do not) but the error has the right sign.

Here we partially generalize the strategy put forward in [18, 63, 95] to the
setting of $\operatorname{RCD}(K,N)$ spaces metric measure spaces.
Before doing so, we recall the smooth Riemannian Alexandroff-Bakelman-Pucci
(ABP) estimate and briefly outline its proof from [95].

Let us introduce some notation: given a sufficiently regular function
$u:\Omega\to\mathbb{R}$, where $\Omega$ is an open domain inside a smooth
Riemannian manifold and $E$ is a compact set, for any $a>0$ we let

$A_{a}(E,\Omega,u):=\left\\{x\in\overline{\Omega}\,:\,\exists\,y\in
E\,:\,\inf_{\overline{\Omega}}\left(u+\frac{a}{2}\mathsf{d}^{2}_{y}\right)=u(x)+\frac{a}{2}\mathsf{d}^{2}_{y}(x)\right\\}\,.$
(4.1)

######  Theorem 4.1.

Let $(X,\mathsf{d},\mathfrak{m})$ be a smooth metric measure space with
weighted Ricci curvature bounded from below by $-K\leq 0$. Let $\Omega\subset
X$ be an open domain and let $u:\Omega\to\mathbb{R}$ be a $C^{2}$ function.
Then, for any compact set $E\subset X$ and for any $a>0$ such that

$A_{a}(E,\Omega,u)\subset\Omega\,,$ (4.2)

it holds:

$\mathfrak{m}(E)\leq\int_{A_{a}}\exp\left(\frac{1}{2}K\left(\frac{\left\lvert\nabla
u\right\rvert^{2}}{a}\right)+\frac{\Delta
u}{a}\right)\mathop{}\\!\mathrm{d}\mathfrak{m}\,.$ (4.3)

We outline the strategy, borrowed from [95] (see also the previous [63, 18,
78, 25]). First, we claim that the map $T:X\to X$ defined by
$T(x):=\exp_{x}(a^{-1}\nabla u(x))$ is surjective from $A_{a}(E,\Omega,u)$ to
$E$. This can be easily verified since for any $y\in E$ there exists $x\in
A_{a}(E,\Omega,u)$ such that

$\inf_{\overline{\Omega}}\left(u+\frac{a}{2}\mathsf{d}^{2}_{y}\right)=u(x)+\frac{a}{2}\mathsf{d}^{2}_{y}(x)\,.$

The first variation implies that

$y=\exp_{x}(a^{-1}\nabla u(x))\,.$

Then, we interpolate between the identity map and the map $T$ via the maps
$T^{t}$ defined by

$T^{t}(x):=\exp(ta^{-1}\nabla u(x))\,,\quad\text{for all }t\in[0,1]\,.$

Since $T=T^{1}$ is surjective from $A_{a}(E,\Omega,u)$ onto $E$, in order to
get (4.3) it is sufficient to estimate its Jacobian determinant. We set

$J(t,x):=\lim_{r\to
0}\frac{\mathfrak{m}(T^{t}(B_{r}(x)))}{\mathfrak{m}(B_{r}(x))}\,.$

Since the weighted Ricci curvature of $(X,\mathsf{d},\mathfrak{m})$ is bounded
from below by $-K$, setting $l(t,x):=\log J(t,x)$, $l$ is well defined for any
$t\in[0,1)$ and it satisfies the second order differential inequality

$l^{\prime\prime}(t,x)\leq\frac{K}{2}\left(\frac{\left\lvert\nabla
u(x)\right\rvert}{a}\right)^{2}\,.$

Moreover, the initial conditions

$l(0,x)=0\,,\quad l^{\prime}(0,x)=a^{-1}\Delta u(x)$ (4.4)

are met. By ODE comparison, we get a bound for $l$ at all later times and
therefore we bound the Jacobian $J(t,x)$ for all later times $t\in[0,1]$. Then
(4.3) follows by integration of the Jacobian bound thanks to the area formula.

######  Remark .

For suitably chosen $a$ and $y$, the function

$u_{a,y}:=u+\frac{a}{2}\mathsf{d}^{2}_{y}\,,$

is a small perturbation of $u$. In particular, for $a$ very small the minima
should converge to the minima of the function $u$. Moreover, if $x$ is a
minimum point of $u_{y,a}$, then, neglecting the regularity issues,

$0\leq\Delta u_{y,a}(x)=\Delta
u(x)+\frac{a}{2}\Delta\mathsf{d}^{2}_{y}(x)\leq\Delta
u(x)+\frac{a}{2}C_{K,N}\,.$

Hence

$\Delta u(x)\geq-\frac{a}{2}C_{K,N}\,.$

This formal computation suggests that Theorem 4.1 could be useful to move
around minimum points of $u$ via the perturbation $u_{a,y}$, still controlling
the Laplacian from below. Notice that, at a qualitative level, (4.3) shows
that if $E$ has positive measure, the set of touching points
$A_{a}(E,\Omega,u)$ has positive measure too.

A couple of deep difficulties to implement such a strategy in the non-smooth
setting of $\operatorname{RCD}(K,N)$ spaces are that:

  * •

In a first step, one proves estimates that are _pointwise along the geodesics_
(using Jacobi fields computations on each single geodesic) and then, in a
second step, such estimates are integrated to get the desired integral bounds.
In the non-smooth setting, Jacobi fields computations are not available and
typically one works with Wasserstein geodesics (which correspond to “sets of
positive measure” in the space of geodesics) in order to take advantage of
optimal transport tools.

  * •

The initial conditions (4.4) (which play a key role in the argument) are met
since the differential at the origin of the exponential map in a smooth
Riemannian manifold is the identity map of the tangent space. This observation
has no clear counterpart in the non-smooth setting.

To overcome these difficulties, we introduce several new ingredients. A
fundamental role is played by the Hopf-Lax semigroup and by the fact that it
preserves Laplacian upper bounds (compare with the recent [80] by the
authors). A key observation is that the Hopf-Lax semigroup plays a similar
role of the exponential map, with the two-fold advantage of not needing
smoothness of the ambient space and of communicating well with the synthetic
lower Ricci bounds (thanks to a deep duality discovered by Kuwada [71], see
also [6] for the extension to the $\operatorname{RCD}$ setting). The theory of
Regular Lagrangian Flows [11] (see also [47] for some useful localised
versions) is then a key tool in order to develop an Eulerian approach based on
the continuity equation (well suited for the non-smooth $\operatorname{RCD}$
setting) of the smooth Lagrangian perspective given by classical Jacobi fields
computation along geodesics.

We will not look for the sharpest possible estimate, regarding the dependence
on the various parameters, but rather for a quantitative one sufficient for
the subsequent purposes of the present note.

######  Theorem 4.2.

Let $(X,\mathsf{d},\mathfrak{m})$ be an $\operatorname{RCD}(K,N)$ metric
measure space for some $K\in\mathbb{R}$ and $1\leq N<\infty$. Let
$\Omega\subset X$ be an open domain and let $u:\Omega\to\mathbb{R}$ belong to
$W^{1,2}_{{\rm loc}}(\Omega)\cap C(\Omega)$ with a strict minimum point on
$\Omega$. Let us assume that $u$ admits measure valued Laplacian on $\Omega$
with

$\Delta u\leq L\mathfrak{m}\,\quad\text{on $\Omega$}\,,$ (4.5)

for some constant $L>0$. Then, for any compact set $E\subset X$ and for any
$a>0$ sufficiently small so that

$A_{a}(E,\Omega,u)\subset\Omega\,,$ (4.6)

the following estimate holds:

$\mathfrak{m}(E)\leq C(K,\Omega,a,L)\,\mathfrak{m}(A_{a}(E,\Omega,u))\,,$
(4.7)

for some explicit constant $C(K,\Omega,a,L)>0$.
In particular, if $\mathfrak{m}(E)>0$, then
$\mathfrak{m}(A_{a}(E,\Omega,u))>0$.

###### Proof.

The proof is divided into four steps. We are going to consider a
$W_{2}$-geodesic formally induced by the map $T^{t}(x):=\exp_{x}(ta^{-1}\nabla
u(x))$ between a suitable probability measure concentrated on
$A_{a}(E,\Omega,u)\subset\Omega$ and the normalized restriction of
$\mathfrak{m}$ to $E$ and then reversing time. We view this Wasserstein
geodesic as a solution of the continuity equation, where the vector field is
explicitely determined through the Hopf-Lax semigroup from $u$, see [39]. The
propagation of Laplacian bounds via the Hopf-Lax semigroup (see the previous
[80]) provides uniform one-sided estimates for the divergence of the vector
field along the solution of the continuity equation. To conclude, we will
combine these one sided estimates with a regularization procedure from [40]
and with [47, Proposition 5.3], which is a local version of the estimates
obtained in [11], to get uniform bounds for the density $\rho_{t}$ of the
interpolant $\mu_{t}$, that will ultimately show (4.7).

Step 1. Let us consider a ball $B_{R}(q)\Subset\Omega$, where $q\in\Omega$ is
the strict minimum point of $u$. Up to adding a constant, which does not
affect the statement, we assume that $u(q)<0$.

Then, we consider a continuous extension with compact support of $u$ from
$B_{R}(q)$ to $X$ such that $\inf_{X}u=u(q)$. Since there is no risk of
confusion, we shall keep the notation $u$ also for the global extension of the
original function $u:\Omega\to\mathbb{R}$. Then we let

$u^{c}_{a}(y):=\inf_{x\in
X}\left\\{u(x)+\frac{a}{2}\mathsf{d}^{2}(x,y)\right\\}\,.$ (4.8)

For $a>0$ sufficiently small and for $y\in B_{R}(q)$, it is easy to verify
that the infimum can be restricted to the original set of definition, i.e.

$u^{c}_{a}(y):=\inf_{x\in\overline{\Omega}}\left\\{u(x)+\frac{a}{2}\mathsf{d}^{2}(x,y)\right\\}\,.$
(4.9)

Hence

$\frac{u^{c}_{a}(y)}{a}=\inf_{x\in\overline{\Omega}}\left\\{\frac{1}{2}\mathsf{d}^{2}(x,y)-\left(\frac{-u(x)}{a}\right)\right\\}=\inf_{x\in
X}\left\\{\frac{1}{2}\mathsf{d}^{2}(x,y)-\left(\frac{-u(x)}{a}\right)\right\\}\,.$
(4.10)

In particular, the couple $(\varphi,\psi):=(-a^{-1}u,a^{-1}u^{c}_{a})$
verifies

$\varphi(x)+\psi(y)\leq\frac{1}{2}\mathsf{d}^{2}(x,y)\,,\quad\text{for any
$x,y\in X$}\,,$ (4.11)

the function $\psi$ is $c$-concave and for any $y\in E$ there exists $x\in
A_{a}(E,\Omega,u)$ such that

$\varphi(x)+\psi(y)=\frac{1}{2}\mathsf{d}^{2}(x,y)\,.$ (4.12)

Let $\mu_{0}:=\frac{1}{\mathfrak{m}(E)}\mathfrak{m}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits E$ be the probability measure
with constant density w.r.t. $\mathfrak{m}$ concentrated on $E$.
Since $\psi$ is a $c$-concave function and $\mu_{0}$ is absolutely continuous
with respect to $\mathfrak{m}$ with bounded density and bounded support, we
can consider the Wasserstein geodesic $(\mu_{s})_{s\in[0,1]}$ induced by
exponentiation from $\mu_{0}$ by $\psi$, borrowing the terminology from [44].
We shall denote by $\psi^{c}$ the function obtained from $\psi$ by
$c$-duality, i.e.

$\psi^{c}(x):=\inf_{y\in
X}\left\\{\frac{\mathsf{d}^{2}(x,y)}{2}-\psi(y)\right\\}\,,\quad\text{for any
$x\in X$}\,.$ (4.13)

It is easy to verify that $\psi$ and $\psi^{c}$ are Lipschitz functions with
compact support. Moreover,

$\psi^{c}(x)+\psi(y)\leq\frac{1}{2}\mathsf{d}^{2}(x,y)\,,\quad\text{for any
$x,y\in X$}$ (4.14)

and for any $y\in E$ there exists $x\in A_{a}(E,\Omega,u)$ such that

$\psi^{c}(x)+\psi(y)=\frac{1}{2}\mathsf{d}^{2}(x,y)\,.$ (4.15)

The above imply that $-a^{-1}u\leq\psi^{c}$ everywhere and $-a^{-1}u=\psi^{c}$
on $A_{a}(E,\Omega,u)$.
Furthermore, $\mu_{1}$ is concentrated on $A_{a}(E,\Omega,u)$ and, by [44] and
(4.15), we get that $(\psi,\psi^{c})$ is an optimal couple of Kantorovich
potentials for the geodesic $(\mu_{t})_{t\in[0,1]}$. Again by [44],
$\mu_{t}\ll\mathfrak{m}$ for every $t\in[0,1)$.

By the general theory of Wasserstein geodesics on geodesic metric spaces, it
holds

$\displaystyle\mathcal{Q}_{t}(-\psi)+\mathcal{Q}_{1-t}(-\psi^{c})\geq
0,\quad\text{everywhere}$
$\displaystyle\mathcal{Q}_{t}(-\psi)+\mathcal{Q}_{1-t}(-\psi^{c})=0\,,\quad\text{on
$\mathrm{supp}\,\mu_{t}$, for any $t\in[0,1]$.}$

Above we employ the standard notation for the semigroup $\mathcal{Q}_{t}$
which is defined by

$\mathcal{Q}_{t}f(x):=\inf_{y\in
X}\left\\{\frac{\mathsf{d}^{2}(x,y)}{2t}+f(y)\right\\}\,.$ (4.16)

Moreover, it is easy to verify that

$\displaystyle\mathcal{Q}_{1-t}(a^{-1}u)\geq\mathcal{Q}_{1-t}(-\psi^{c})\,,\quad\text{everywhere}$
(4.17)
$\displaystyle\mathcal{Q}_{1-t}(a^{-1}u)=\mathcal{Q}_{1-t}(-\psi^{c})\,,\quad\text{on
$\mathrm{supp}\,\mu_{t}$ for every $t\in[0,1]$.}$ (4.18)

Step 2. In this step we introduce regularized potentials that induce the
Wasserstein geodesic $(\mu_{t})_{t\in[0,1]}$, whose existence is guaranteed by
[40]. Indeed, for any $t\in(0,1)$ by [40, Theorem 3.13] there exists a
Lipschitz function with compact support $\eta_{t}:X\to\mathbb{R}$ such that

$-\mathcal{Q}_{t}(-\psi)\leq\eta_{t}\leq\mathcal{Q}_{1-t}(-\psi^{c})$ (4.19)

and $\eta_{t}\in D(\Delta)$ with

$\left\lVert\Delta\eta_{t}\right\rVert_{L^{\infty}}\leq C(t)<\infty\,,$ (4.20)

where $C:(0,1)\to(0,\infty)$ is a continuous function depending on
$\left\lVert\varphi\right\rVert_{\infty}$, $K$ and $N$, blowing up near to the
boundary points.
The combination of (4.19) with (4.17) and (4.18) proves that

$\eta_{t}\leq\mathcal{Q}_{1-t}(a^{-1}u)\,,$ (4.21)

everywhere and

$\eta_{t}=\mathcal{Q}_{1-t}(a^{-1}u)\,,\quad\text{on
$\mathrm{supp}\,\mu_{t}\,,$ for every $t\in[0,1]$.}$ (4.22)

Arguing as in the proof of [17, Proposition 3.7] it is possible to verify that
the couple $(\mu_{t},-\nabla\eta_{t})$ is a solution of the continuity
equation

$\frac{\partial\mu_{t}}{\partial
t}+\operatorname{div}(-\nabla\eta_{t}\mu_{t})=0\,,\quad\text{for every
$t\in(0,1)$}\,.$ (4.23)

Thanks to (4.20), the theory of Regular Lagrangian Flows on
$\operatorname{RCD}$ spaces from [11] can be applied between any intermediate
times $0<s<r<1$ along the solution of the continuity equation
$(\mu_{t},-\nabla\eta_{t})$.

Step 3. The goal of this step is to uniformly control the positive part of the
Laplacian of $\eta_{t}$ for any $t\in[0,1]$.
This follows indeed from the assumption that $u$ has measure valued Laplacian
with uniformly bounded positive part combined with (4.21), (4.22), and with a
variant of the argument introduced in [80, Section 4], in turn building on top
of [71, 6].

We will first obtain a uniform bound for the positive part of the Laplacian of
$\mathcal{Q}_{s}(a^{-1}u)$, for any $s\in[0,1]$.
Let us consider $s\in[0,1]$ and for any $x\in B_{R}(q)$ we let $x_{s}\in
B_{R}(q)$ be a point such that

$\mathcal{Q}_{s}(a^{-1}u)(x)=\inf_{y\in
X}\left\\{\frac{\mathsf{d}^{2}(x,y)}{2s}+a^{-1}u(y)\right\\}=\frac{\mathsf{d}^{2}(x,x_{s})}{2s}+a^{-1}u(x_{s})\,$

and $x_{s}$ minimizes the distance from $x$ among all points with the property
above. Then

$\mathcal{Q}_{s}(a^{-1}u)(z)\leq\frac{\mathsf{d}^{2}(z,y)}{2s}+a^{-1}u(y)\,,\quad\text{for
any $z,y\in X$}$ (4.24)

and

$\mathcal{Q}_{s}(a^{-1}u)(x)=\frac{\mathsf{d}^{2}(x,x_{s})}{2s}+a^{-1}u(x_{s})\,.$
(4.25)

In particular, we can easily deduce the classical estimate for the Hopf-Lax
semigroup

$\frac{\mathsf{d}^{2}(x,x_{s})}{s}\leq a^{-1}\left\lvert
u(x)-u(x_{s})\right\rvert\leq a^{-1}\mathrm{osc}_{\Omega}u\,,\quad\text{for
any $x\in B_{R}(q)$}\,.$ (4.26)

For any $r\geq 0$, we let $\Pi_{r}$ be the optimal transport plan for
quadratic cost between probability measures $P_{r}\delta_{x}$ and
$P_{r}\delta_{x_{s}}$, where we denoted by $P_{r}\delta_{p}$ the heat kernel
measure with centre $p\in X$ at time $r$. Then we integrate both sides of
(4.24) with respect to $\Pi_{r}$ on $X\times X$ and get

$\displaystyle P_{r}\mathcal{Q}_{s}\left(a^{-1}u\right)(x)\leq$
$\displaystyle\,\frac{W_{2}^{2}(P_{r}\delta_{x},P_{r}\delta_{x_{s}})}{2s}+a^{-1}P_{r}u(x_{s})$
(4.27) $\displaystyle\leq$
$\displaystyle\,\frac{e^{-2Kr}\mathsf{d}^{2}(x,x_{s})}{2s}+a^{-1}P_{r}u(x_{s})\,.$
(4.28)

Subtracting (4.25) from both sides, dividing by $r$ and taking the $\limsup$
as $r\downarrow 0$, taking into account the Wasserstein contractivity of the
Heat Flow in the Wasserstein space under the $\operatorname{RCD}(K,\infty)$
condition [5, 3], we formally get

$\Delta\mathcal{Q}_{s}\left(a^{-1}u\right)(x)\leq-\frac{Kr\mathsf{d}^{2}(x,x_{s})}{s}+a^{-1}\Delta
u(x_{s})\,,$

which gives a uniform upper bound for the positive part of the Laplacian of
$\mathcal{Q}_{s}\left(a^{-1}u\right)$ as soon as we can uniformly bound
$\mathsf{d}^{2}(x,x_{s})/s$ with respect to $x$ and $s$ and uniformly bound
the positive part of $\Delta u$. Next, we make the above argument rigorous.

Recall that by construction of the regularized Kantorovich potentials,
$\eta_{t}$ belongs to the domain of the Laplacian. In particular, see [80,
Lemma 2.56] for instance, for $\mathfrak{m}$-a.e. $x\in X$ it holds

$\Delta\eta_{t}(x)=\lim_{r\to 0}\frac{P_{r}\eta_{t}(x)-\eta_{t}(x)}{r}\,.$

Since $\eta_{t}(x)=\mathcal{Q}_{1-t}(a^{-1}u)(x)$ for every
$x\in\operatorname{supp}(\mu_{t})$, we can use (4.27) and (4.25) to get

$\frac{P_{r}\eta_{t}(x)-\eta_{t}(x)}{r}\leq\left(\frac{e^{-2Kr}-1}{2r}\right)\frac{\mathsf{d}^{2}(x,x_{1-t})}{1-t}+a^{-1}\frac{P_{r}u(x_{1-t})-u(x_{1-t})}{r}\,,\quad\text{for
any $r>0$}\,.$

Hence, taking the $\limsup$ as $r\to 0$ we infer that

$\Delta\eta_{t}(x)\leq-K\frac{\mathsf{d}^{2}(x,x_{1-t})}{1-t}+a^{-1}\limsup_{r\to
0}\frac{P_{r}u(x_{s})-u(x_{s})}{r}\,,\quad\text{for $\mathfrak{m}$-a.e.
$x\in\operatorname{supp}(\mu_{t})$}\,.$

Since by assumption $\Delta u\leq L$ on $\Omega$, employing subsection 2.1 and
then (4.26), we obtain that

$\Delta\eta_{t}(x)\leq-K\frac{\mathsf{d}^{2}(x,x_{t})}{1-t}+a^{-1}L\leq-
Ka^{-1}\mathrm{osc}_{\Omega}u+a^{-1}L\,,$ (4.29)

for $\mathfrak{m}$-a.e. $x\in\operatorname{supp}(\mu_{t})$ and for every
$t\in(0,1]$.

Step 4. Let us complete the proof of (4.7) combining the previous ingredients
with a limiting argument and [47, Proposition 5.3].

We assume $K\leq 0$ and set

$C:=-Ka^{-1}\mathrm{osc}_{\Omega}u+a^{-1}L\,>\,0\,.$ (4.30)

Notice that, by the very definition of $\mu_{0}:=\rho_{0}\mathfrak{m}$ it
holds

$\rho_{0}\equiv\frac{1}{\mathfrak{m}(E)}\,,\quad\text{ $\mathfrak{m}$-a.e. on
$E$}\,,\quad\rho_{0}\equiv 0\,,\quad\text{$\mathfrak{m}$-a.e. outside of
$E$}\,,$ (4.31)

hence $\left\lVert\rho_{0}\right\rVert_{\infty}=1/\mathfrak{m}(E)$.
By the $\operatorname{RCD}(K,N)$ condition (actually essentially non-branching
plus $\operatorname{MCP}(K,N)$ would suffice, see [21, Theorem 1.1], after
[84]),

$\left\lVert\rho_{t}\right\rVert_{L^{\infty}}\leq
1/\mathfrak{m}(E)+o(1)\,,\quad\text{as $t\to 0$.}$ (4.32)

On the other hand, we can apply [47, Proposition 5.3] (see in particular
equation (5.8) therein) in combination with the upper bound for the positive
part of the Laplacian of $\eta_{t}$ in (4.29) to obtain that

$\sup_{t\in[s,r]}\left\lVert\rho_{t}\right\rVert_{L^{\infty}}\leq\left\lVert\rho_{s}\right\rVert_{L^{\infty}}e^{(r-s)C}\,,\quad\text{for
any $0<s<r<1$}\,.$ (4.33)

Taking the limit as $s\to 0$ and taking into account (4.32), we get

$\sup_{t\in[0,r]}\left\lVert\rho_{t}\right\rVert_{L^{\infty}}\leq\left\lVert\rho_{0}\right\rVert_{L^{\infty}}e^{rC}\leq\left\lVert\rho_{0}\right\rVert_{L^{\infty}}e^{C}\,,\quad\text{for
any $0\leq r<1$}\,.$ (4.34)

To conclude, we observe that the probability measures $\mu_{t}$ weakly
converge to $\mu_{1}$ as $t\to 1$, as they converge in Wasserstein distance.
Then (4.34) implies that $\mu_{1}\ll\mathfrak{m}$ and, setting
$\mu_{1}=\rho_{1}\mathfrak{m}$, it holds

$\left\lVert\rho_{1}\right\rVert_{L^{\infty}}\leq\frac{e^{C}}{\mathfrak{m}(E)}\,.$
(4.35)

As $\mu_{1}$ is concentrated on $A_{a}(E,\Omega,u)$, we conclude that

$\mathfrak{m}(E)\leq\mathfrak{m}(A)\,e^{C}\,.$ (4.36)

∎

######  Remark .

It seems likely that a refinement of the proof of Theorem 4.2 could lead to a
sharper estimate more in the spirit of (4.3), when we additionally assume that
$u$ has Laplacian locally in $L^{2}$. However this is not needed for the main
results of the present note and thus, for the sake or brevity, we leave it for
future investigation.

######  Corollary .

Let $(X,\mathsf{d},\mathfrak{m})$ be an $\operatorname{RCD}(K,N)$ metric
measure space. Let $\Omega\subset X$ be an open domain and let $u\in
W^{1,2}_{{\rm loc}}(\Omega)\cap C(\Omega)$ be such that:

  * (i)

$u$ admits locally measure valued Laplacian on $\Omega$ with $\Delta u\leq
L\mathfrak{m}$ in the sense of distributions on $\Omega$ for some constant
$L\geq 0$;

  * (ii)

$u$ has a strict local minimum at some $x\in\Omega$.

Then for any set of full $\mathfrak{m}$-measure $B\subset\Omega$ and for any
natural number $n>0$ there exist $0<a_{n}<1/n$ and $y_{n}\in\Omega$ with
$\mathsf{d}(x,y_{n})<1/n$ such that the function

$\Omega\ni z\mapsto u(z)+a_{n}\mathsf{d}^{2}(z,y_{n})$ (4.37)

admits a minimum at a point $\bar{x}_{n}\in B$ with
$\mathsf{d}(\bar{x}_{n},x)<1/n$.

###### Proof.

It is sufficient to apply Theorem 4.2 with $E=B_{1/n}(x)$ and $a$ sufficiently
small so that, for any $y\in B_{1/n}(x)$ it holds $\mathsf{d}(x,\bar{x})<1/n$
for any minimum point of the function

$\Omega\ni z\mapsto u(z)+a\,\mathsf{d}^{2}(z,y)\,.$ (4.38)

As Theorem 4.2 shows that the set of all possible minimum points when $y$
varies in $B_{1/n}(x)$ has positive measure, clearly it intersects any set of
full measure. ∎

##  5\. The key propagation theorem

In this section we consider an $\operatorname{RCD}(K,N)$ metric measure space
$(X,\mathsf{d},\mathfrak{m})$, for some $K\in\mathbb{R}$ and $1\leq N<\infty$,
a ${\sf CAT}(0)$ space $(Y,\mathsf{d}_{Y})$, an open domain $\Omega\subset X$
and a harmonic map $u:\Omega\to Y$, as in the statement of Theorem 1.1. The
goal is to prove a propagation estimate analogous to [99, Lemma 6.7], see
section 5. This is a key technical step for the proof of the Lipschitz
continuity of harmonic maps from $\operatorname{RCD}(K,N)$ spaces to ${\sf
CAT}(0)$ spaces.

Let us stress some fundamental differences between our proof and the one in
[99]. The proof in [99] uses the second variation formula and parallel
transport for Alexandrov spaces [82]. It builds on a strategy introduced in
[81] (see also [97]). Moreover, it relies on a delicate perturbation argument
finding its roots again in [81] in the Alexandrov case and similar to the one
used in the Euclidean viscosity theory of PDEs to prove the approximate
maximum principle for semiconvex functions, see [55, 19].
In order to prove the estimate on non smooth $\operatorname{RCD}$ spaces, we
give a completely new argument. The substitute for the second variation
formula is the analysis of the interplay between optimal transport and the
Heat Flow under lower Ricci curvature bounds finding its roots in [94, 71, 6]
and further explored by the authors in [80]. The substitute of the
perturbation argument in the Alexandrov theory has been obtained in section 4.

We first introduce some terminology.
For any domain $\Omega^{\prime}$ compactly contained in $\Omega\subset X$, for
any $t>0$ and for any $0\leq\lambda\leq 1$ we set

$f_{t}(x,\lambda):=\inf_{y\in\Omega^{\prime}}\left\\{\frac{e^{-2K\lambda}\mathsf{d}^{2}(x,y)}{2t}-\mathsf{d}_{Y}(u(x),u(y))\right\\}\,.$
(5.1)

We denote by $S_{t}(x,\lambda)$ the set of those points where the infimum is
achieved, i.e.

$S_{t}(x,\lambda):=\left\\{y\in\Omega^{\prime}\,:\,f_{t}(x,\lambda)=\frac{e^{-2K\lambda}\mathsf{d}^{2}(x,y)}{2t}-\mathsf{d}_{Y}(u(x),u(y))\right\\}\,.$

Notice that

$0\geq
f_{t}(x,\lambda)\geq-\mathrm{osc}_{\overline{\Omega^{\prime}}}u=-\max_{x,y\in\overline{\Omega^{\prime}}}\mathsf{d}_{Y}(u(x),u(y))\,,$
(5.2)

where we recall that $u$ is continuous, see Theorem 2.3, and thus its
oscillation is finite on any bounded set.

The following is obtained in [99, Lemma 6.1] and the proof works without any
modification in the present setting, so we omit it. It is a variant of the
classical mild regularity properties for the evolution via the Hopf-Lax
semigroup (see for instance [4]) in this non-linear setting.

######  Lemma .

With the notation above, let us set for any open domain
$\Omega^{\prime\prime}\Subset\Omega^{\prime}$,

$C_{*}:=2\,\mathrm{osc}_{\overline{\Omega^{\prime}}}\,u+2\,,\quad
t_{0}:=\frac{\mathsf{d}^{2}(\Omega^{\prime\prime},\partial\Omega^{\prime})}{4C_{*}}\,.$
(5.3)

Then, for each $t\in(0,t_{0})$, the following hold:

  * (i)

For each $\lambda\in[0,1]$ and $x\in\Omega^{\prime\prime}$, it holds
$S_{t}(x,\lambda)\neq\emptyset$ and

$f_{t}(x,\lambda)=\min_{\overline{B_{\sqrt{C_{*}t}}(x)}}\left\\{\frac{e^{-2K\lambda}\mathsf{d}^{2}(x,y)}{2t}-\mathsf{d}_{Y}(u(x),u(y))\right\\}\,.$

  * (ii)

For each $\lambda\in[0,1]$, the function $x\mapsto f_{t}(x,\lambda)$ is in
$C(\Omega^{\prime\prime})\cap W^{1,2}(\Omega^{\prime\prime})$ and satisfies
the following energy estimate

$\int_{\Omega^{\prime\prime}}\left\lvert\nabla_{x}f_{t}(x,\lambda)\right\rvert^{2}\mathop{}\\!\mathrm{d}\mathfrak{m}\leq
C(K,N)\frac{{\rm
diam}^{2}(\Omega^{\prime})}{t^{2}}\mathfrak{m}(\Omega^{\prime\prime})+C(K,N)\int_{\Omega^{\prime\prime}}\left\lvert\mathop{}\\!\mathrm{d}u\right\rvert_{\mathrm{HS}}^{2}\mathop{}\\!\mathrm{d}\mathfrak{m}\,.$

  * (iii)

For any $x\in\Omega^{\prime\prime}$, the function $\lambda\mapsto
f_{t}(x,\lambda)$ is Lipschitz with

$\left\lvert f_{t}(\lambda,x)-f_{t}(\lambda^{\prime},x)\right\rvert\leq
C_{*}e^{-2K}\left\lvert\lambda-\lambda^{\prime}\right\rvert\,,\quad\text{for
any $\lambda,\lambda^{\prime}\in[0,1]$}\,.$

  * (iv)

The function $(x,\lambda)\mapsto f_{t}(x,\lambda)$ is in
$C(\Omega^{\prime\prime}\times[0,1])$ and
$W^{1,2}(\Omega^{\prime\prime}\times[0,1])$, where $X\times[0,1]$ is endowed
with the canonical product structure.

For $\Omega^{\prime\prime}\Subset\Omega^{\prime}$, let $t_{0}>0$ be given by
(5.3) and, for all $t\in(0,t_{0})$, consider the function
$L_{t,\lambda}:\Omega^{\prime\prime}\to[0,\infty)$ defined by

$L_{t,\lambda}(x):=\mathsf{d}(x,S_{t}(x,\lambda))=\min_{y\in
S_{t}(x,\lambda)}\mathsf{d}(x,y)\,.$

The following corresponds to [99, Lemma 6.2], whose proof works with no
modifications in the present setting, as it relies only on metric arguments,
therefore it is omitted.

######  Lemma .

Let $\Omega^{\prime\prime}\Subset\Omega^{\prime}$ and $t_{0}>0$ be given as
above. Then for any $t\in(0,t_{0})$ it holds that

  * (i)

The function $(x,\lambda)\mapsto L_{t,\lambda}(x)$ is lower semicontinuous on
$\Omega^{\prime\prime}\times[0,1]$.

  * (ii)

For each $\lambda\in[0,1]$,

$\left\lVert
L_{t,\lambda}\right\rVert_{L^{\infty}(\Omega^{\prime\prime})}\leq\sqrt{C_{*}t}\,.$

Below we report [99, Lemma 6.3], whose proof works again with no modifications
in the present setting, therefore it is omitted.

######  Lemma .

Let $\Omega^{\prime\prime}\Subset\Omega^{\prime}$ and $t_{0}>0$ be given as
above. Then for any $t\in(0,t_{0})$ it holds that

$\liminf_{\mu\to
0^{+}}\frac{f_{t}(x,\lambda+\mu)-f_{t}(x,\lambda)}{\mu}\geq-e^{-2K\lambda}\frac{K}{t}L^{2}_{t,\lambda}(x)\,,$
(5.4)

for any $\lambda\in[0,1)$ and any $x\in\Omega^{\prime\prime}$.

######  Remark .

Since $\lambda\mapsto f_{t}(x,\lambda)$ is a Lipschitz function for every
$x\in\Omega^{\prime\prime}$, (5.4) can be turned into an inequality between
derivatives valid $\mathscr{L}^{1}$-a.e. on $(0,1)$.

In the case when $(X,\mathsf{d},\mathfrak{m})$ is an $\operatorname{RCD}(0,N)$
metric measure space, we can remove the dependence on the additional parameter
$\lambda$ and set

$f(t,x):=\inf_{y\in\Omega^{\prime}}\left\\{\frac{\mathsf{d}^{2}(x,y)}{2t}-\mathsf{d}_{Y}(u(x),u(y))\right\\}\,.$

Our goal is to prove that for any $p\in\Omega^{\prime}$ there exists a
neighbourhood $U_{p}$ of $p$ and a positive time $t_{p}>0$ such that $f_{t}$
is superharmonic on $U_{p}$ for any $0<t<t_{p}$. See section 5 below for the
general statement when $(X,\mathsf{d},\mathfrak{m})$ is an
$\operatorname{RCD}(K,N)$ space for general $K\in\mathbb{R}$.

As mentioned in the discussion at the beginning of the section, the proof of
the propagation estimate section 5 needs several new ideas with respect to
[99].

We introduce the first new ingredient avoiding the technicalities for the sake
of this presentation. In particular, for simplicity of presentation, we do not
care about the maps being defined only on open domains and assume harmonicity
to hold globally.
Assume that

$f(t,x_{0})=\frac{\mathsf{d}^{2}(x_{0},y_{0})}{2t}-\mathsf{d}_{Y}(u(x_{0}),u(y_{0}))\,,$
(5.5)

and

$f(t,x)\leq\frac{\mathsf{d}^{2}(x,y)}{2t}-\mathsf{d}_{Y}(u(x),u(y))\,,\quad\text{for
any $x,y\in X$}\,.$ (5.6)

Let us consider then the evolutions of Dirac deltas through the Heat Flow at
points $x_{0}$ and $y_{0}$ and denote them by $P_{s}\delta_{x_{0}}$,
$P_{s}\delta_{y_{0}}$ respectively. Let us also denote by
$\Pi_{s,x_{0},y_{0}}$ an optimal transport plan for quadratic cost between
$P_{s}\delta_{x_{0}}$ and $P_{s}\delta_{y_{0}}$. Notice that
$\Pi_{s,x_{0},y_{0}}$ is a probability measure on $X\times X$ whose first and
second marginals are $P_{s}\delta_{x_{0}}$ and $P_{s}\delta_{y_{0}}$,
respectively.

Let us integrate both sides in (5.6) with respect to $\Pi_{s,x_{0},y_{0}}$.
Then we obtain

$\int_{X\times
X}f(t,x)\mathop{}\\!\mathrm{d}\Pi_{s,x_{0},y_{0}}(x,y)\leq\int_{X\times
X}\left(\frac{\mathsf{d}^{2}(x,y)}{2s}-\mathsf{d}_{Y}(u(x),u(y))\right)\mathop{}\\!\mathrm{d}\Pi_{s,x_{0},y_{0}}(x,y)\,.$
(5.7)

Notice now that the integrand at the left hand side is independent of $y$. The
first marginal of $\Pi_{s,x_{0},y_{0}}$ is $P_{s}\delta_{x_{0}}$, hence

$\int_{X\times
X}f(t,x)\mathop{}\\!\mathrm{d}\Pi_{s,x_{0},y_{0}}(x,y)=\int_{X}f(t,x)\mathop{}\\!\mathrm{d}P_{s}\delta_{x_{0}}=P_{s}(f(t,\cdot))(x_{0})\,.$

Since $\Pi_{s_{0},x_{0},y_{0}}$ is an optimal transport plan between
$P_{s}\delta_{x_{0}}$ and $P_{s}\delta_{y_{0}}$ for quadratic cost, the
$\operatorname{RCD}(0,\infty)$ condition implies that

$\int_{X\times
X}\frac{\mathsf{d}^{2}(x,y)}{2t}\mathop{}\\!\mathrm{d}\Pi_{s,x_{0},y_{0}}(x,y)=\frac{1}{2t}W_{2}^{2}(P_{s}\delta_{x_{0}},P_{s}\delta_{y_{0}})\leq\frac{1}{2t}\mathsf{d}^{2}(x_{0},y_{0})\,,$
(5.8)

for any $s>0$.

We are left to bound the term

$-\int_{X\times
X}\mathsf{d}_{Y}(u(x),u(y))\mathop{}\\!\mathrm{d}\Pi_{s,x_{0},y_{0}}(x,y)\,.$
(5.9)

There are two possibilities. Either $u(x_{0})=u(y_{0})$ and then (5.9) is
trivially bounded above by $0$, or $u(x_{0})\neq u(y_{0})$ in which case we
can argue as follows.

Let us apply section 3 with points $\\{u(x),u(x_{0}),u(y_{0}),u(y)\\}$. Then,
denoting by $z_{0}$ the midpoint of $u(x_{0})u(y_{0})$, we obtain

$\displaystyle\left(\mathsf{d}_{Y}(u(x),u(y))-\mathsf{d}_{Y}(u(x_{0}),u(y_{0}))\right)\mathsf{d}_{Y}(u(x_{0}),u(y_{0}))$
$\displaystyle\qquad\geq\left(\mathsf{d}_{Y}^{2}(u(x),z_{0})-\mathsf{d}_{Y}^{2}(u(x),u(x_{0}))-\mathsf{d}_{Y}^{2}(z_{0},u(x_{0}))\right)$
$\displaystyle\qquad\quad+\left(\mathsf{d}_{Y}^{2}(u(y),z_{0})-\mathsf{d}_{Y}^{2}(u(y),u(y_{0}))-\mathsf{d}_{Y}^{2}(z_{0},u(y_{0}))\right)\,,$

for any $x,y$.

With the notation introduced in section 3, this can be rewritten as

$\left(\mathsf{d}_{Y}(u(x),u(y))-\mathsf{d}_{Y}(u(x_{0}),u(y_{0}))\right)\geq-\frac{1}{\mathsf{d}_{Y}(u(x_{0}),u(y_{0}))}\left(w_{x_{0},z_{0}}(x)+w_{y_{0},z_{0}}(y)\right)\,,$

for any $x,y\in X$. Hence

$-\mathsf{d}_{Y}(u(x),u(y))\leq-\mathsf{d}_{Y}(u(x_{0}),u(y_{0}))+\frac{1}{\mathsf{d}_{Y}(u(x_{0}),u(y_{0}))}\left(w_{x_{0},z_{0}}(x)+w_{y_{0},z_{0}}(y)\right)\,.$
(5.10)

Integrating both sides of (5.10) w.r.t. $\Pi_{s,x_{0},y_{0}}$, we can estimate

$\displaystyle-\int_{X\times X}$
$\displaystyle\mathsf{d}_{Y}(u(x),u(y))\mathop{}\\!\mathrm{d}\Pi_{s,x_{0},y_{0}}(x,y)\leq-\mathsf{d}_{Y}(u(x_{0}),u(y_{0}))$
$\displaystyle+\frac{1}{\mathsf{d}_{Y}(u(x_{0}),u(y_{0}))}\int_{X\times
X}\left(w_{x_{0},z_{0}}(x)+w_{y_{0},z_{0}}(y)\right)\mathop{}\\!\mathrm{d}\Pi_{s,x_{0},y_{0}}$
$\displaystyle\leq-\mathsf{d}_{Y}$
$\displaystyle(u(x_{0}),u(y_{0}))+\frac{1}{\mathsf{d}_{Y}(u(x_{0}),u(y_{0}))}\left(P_{s}w_{x_{0},z_{0}}(\cdot)(x_{0})+P_{s}w_{y_{0},z_{0}}(\cdot)(y_{0})\right)\,,$
(5.11)

where the last inequality is due to the fact that the marginals of
$\Pi_{s,x_{0},y_{0}}$ are the heat kernel measures centred at $x_{0}$ and
$y_{0}$ at time $s$.

The combination of (5.7) with (5.8), (5.11) and section 3, assuming for the
sake of this presentation that $x_{0}$ and $y_{0}$ are such that the
asymptotic estimate (3.11) holds, proves that

$\limsup_{s\to 0}\frac{\left(P_{s}f(t,\cdot)(x_{0})-f(t,x_{0})\right)}{s}\leq
0\,,$ (5.12)

that yields, formally, super-harmonicity of $f(t,\cdot)$.

We will encounter a key additional difficulty to make rigorous the strategy
above, due to the fact that the asymptotic in section 3 is not available for
any point, but rather for $\mathfrak{m}$-a.e. point. In order to deal with
this issue, we will rely on Theorem 4.2 and section 4. Moreover, there will be
error terms to deal with in the case of an $\operatorname{RCD}(K,N)$ space,
with general $K\in\mathbb{R}$.

For the proof of the key propagation results we will rely on the following
technical statement.

######  Lemma .

Let $(X,\mathsf{d},\mathfrak{m})$ be an $\operatorname{RCD}(K,N)$ metric
measure space for some $K\in\mathbb{R}$ and $1\leq N<\infty$. Let $(x,y)\in
X\times X$ and $U\subset X\times X$ be an open neighbourhood of $(x,y)$. Let
$F:U\to\mathbb{R}$ be a continuous function with a local minimum at $(x,y)$.
For any $r>0$, let us denote by $\Pi_{r}$ an admissible transport plan between
$P_{r}\delta_{x}$ and $P_{r}\delta_{y}$ (which is a probability measure on
$X\times X$). Then, for any open set $V\Subset U$ and for any function
$G:X\times X\to\mathbb{R}$ continuous and bounded such that $F\equiv G$ on
$V$, it holds

$\liminf_{r\to 0}\frac{\int_{X\times
X}G(z,w)\mathop{}\\!\mathrm{d}\Pi_{r}(z,w)-G(x,y)}{r}\geq 0\,.$ (5.13)

###### Proof.

We divide the proof in two steps. First we verify that the value of the
$\liminf$ in (5.13) depends only on the behaviour of $G$ in a neighbourhood of
$(x,y)$. Then we complete the proof with a particular choice of the global
bounded and continuous extension of $F$.

Step 1. It is sufficient to prove the following claim. If
$H\in\operatorname{C_{b}}(X\times X)$ and $H\equiv 0$ in a neighbourhood of
$(x,y)$, then

$\lim_{r\to 0}\frac{\int_{X\times
X}H(z,w)\mathop{}\\!\mathrm{d}\Pi_{r}(z,w)}{r}=0\,.$ (5.14)

We can assume without loss of generality that $H\geq 0$ on $X\times X$, up to
substituting $H$ with $\left\lvert H\right\rvert$. If $H$ is non-negative,
continuous, bounded and it vanishes in a neighbourhood of $(x,y)$, then there
exist bounded and continuous, non-negative functions
$H_{1},H_{2}:X\to[0,\infty)$ identically vanishing in a neighbourhood of $x$
and $y$ respectively, such that

$0\leq H(z,w)\leq H_{1}(z)+H_{2}(w)\,,\quad\text{for any $z,w\in X$}\,.$

Since $\Pi_{r}$ has marginals $P_{r}\delta_{x}$ and $P_{r}\delta_{y}$ on the
first and second component respectively, it holds

$\displaystyle\int_{X\times X}H(z,w)\mathop{}\\!\mathrm{d}\Pi_{r}(z,w)\leq$
$\displaystyle\int_{X\times
X}\left(H_{1}(z)+H_{2}(w)\right)\mathop{}\\!\mathrm{d}\Pi_{r}(z,w)$ (5.15)
$\displaystyle=$ $\displaystyle\,P_{r}H_{1}(x)+P_{r}H_{2}(y)\,.$ (5.16)

Since $H_{1}$ vanishes identically in a neighbourhood of $x$ and $H_{2}$
vanishes identically in a neighbourhood of $y$, it follows from [80, Lemma
2.53] that

$\lim_{r\to 0}\frac{P_{r}H_{1}(x)}{r}=\lim_{r\to
0}\frac{P_{r}H_{2}(y)}{r}=0\,.$ (5.17)

Combining (5.15) with (5.17) we obtain (5.14).

Step 2. Given what we obtained in the previous step, it is sufficient to
choose any open domain $W\Subset U$ such that

$F(x,y)=\min_{(p,q)\in W}F(p,q)\,.$

Then, for any $n\in\mathbb{N}$ we set $F_{n}:W\to\mathbb{R}$ by

$F_{n}(p,q):=F(p,q)+\frac{1}{n}\left(\mathsf{d}^{2}(x,p)+\mathsf{d}^{2}(y,q)\right)\,.$

For any $n\in\mathbb{N}$, $F_{n}$ admits a strict minimum at $(x,y)$ on $W$.
Then we can extend $F_{n}$ to a global function $G_{n}:X\times X\to\mathbb{R}$
such that $G_{n}$ admits a minimum at $(x,y)$. In particular

$\liminf_{r\to 0}\frac{\int_{X\times
X}G_{n}(z,w)\mathop{}\\!\mathrm{d}\Pi_{r}(z,w)-G_{n}(x,y)}{r}\geq 0\,,$ (5.18)

since $G_{n}(z,w)\geq G_{n}(x,y)$ for any $z,w\in X$ and $\Pi_{r}$ is a
probability measure.
Taking into account section 3, (5.18) shows that

$\liminf_{r\to 0}\frac{\int_{X\times
X}G(z,w)\mathop{}\\!\mathrm{d}\Pi_{r}(z,w)-G(x,y)}{r}\geq-\frac{2N}{n}\,,\quad\text{for
any $n\in\mathbb{N}$, $n\geq 1$}\,,$

where $G$ denotes any bounded and continuous extension of $F$ to $X\times X$.
Taking the limit as $n\to\infty$ we obtain (5.13).

∎

The statement below is the counterpart of [99, Lemma 6.7] in the present
setting. It is the main technical tool for the proof of the Lipschitz
continuity of harmonic maps from $\operatorname{RCD}(K,N)$ spaces to ${\sf
CAT}(0)$ spaces.
As we anticipated, there are some fundamental differences between our proof
and the proof in [99]. The first one is that in [99] the authors build on the
parallel transport and second variation formula for Alexandrov spaces from
[82], in order to estimate second variations in each direction and then
average up the estimates. We will estimate averages w.r.t the heat kernel
directly (i.e. Laplacians) relying on the interplay between Heat Flow and
optimal transport on $\operatorname{RCD}$ spaces. The second fundamental
difference is in the perturbation argument pursued in section 4.

######  Proposition .

Let $(X,\mathsf{d},\mathfrak{m})$ be an $\operatorname{RCD}(K,N)$ metric
measure space for some $K\leq 0$ and $1\leq N<\infty$ and let
$(Y,\mathsf{d}_{Y})$ be a ${\sf CAT}(0)$ space. Let $\Omega\subset X$ be an
open domain and let $u:\Omega\to Y$ be a harmonic map. Let $f_{t}(x,\lambda)$
be as in (5.1). Then for any $p\in\Omega^{\prime}$ there exist a neighbourhood
$U_{p}=B_{R_{p}}(p)$ of $p$ and a time $t_{p}>0$ such that, for any
$0<t<t_{p}$ and any $\lambda\in[0,1]$, the function $U\ni x\mapsto
f_{t}(x,\lambda)$ is a super-solution of the Poisson equation

$\Delta
f_{t}(\cdot,\lambda)=-e^{-2K\lambda}\frac{K}{t}L_{t,\lambda}^{2}\,,\quad\text{on
}U_{p}\,.$

###### Proof.

The proof will be divided into three steps. In Step 1 we set up the
contradiction argument via an auxiliary function. In Step 2 we perturb the
auxiliary function to achieve a strict minimum at a _sufficiently regular_
point. In Step 3 we reach a contradiction with a second variation argument.

Step 1. Following the proof in [99], we notice that it is sufficient to prove
that $U\ni x\mapsto f_{t}(x,\lambda)$ is a super-solution of the Poisson
equation

$\Delta
f_{t}(\cdot,\lambda)=-e^{-2K\lambda}\frac{K}{t}L_{t,\lambda}^{2}+\theta\quad\text{
on $U_{p}$, for any $\theta>0$. }$

Let us suppose by contradiction that the claim above is not true for some
$t>0$, $\lambda\in[0,1]$ and $\theta_{0}>0$. Then, by subsection 2.1, there
exists an open domain $B\subset U_{p}$ such that, denoting by $v\in
W^{1,2}(B)$ the solution of the Poisson problem

$\Delta
v=-e^{-2K\lambda}\frac{K}{t}L_{t,\lambda}^{2}+\theta_{0}\,,\quad\text{on
$B$}\,,$ (5.19)

with $v=f_{t}(\cdot,\lambda)$ on $\partial B$ (i.e. $v-f_{t}(\cdot,\lambda)\in
W^{1,2}_{0}(B)$), it holds that

$\min_{x\in B}\left\\{f_{t}(x,\lambda)-v(x)\right\\}<0=\min_{x\in\partial
B}\left\\{f_{t}(x,\lambda)-v(x)\right\\}\,.$

In particular, $f_{t}(\cdot,\lambda)-v$ achieves a minimum in the interior of
$B$. Let $\bar{x}\in B$ be any such a minimum point. Define the function
$H:B\times U\to\mathbb{R}$ by

$H(x,y):=\frac{e^{-2K\lambda}\mathsf{d}^{2}(x,y)}{2t}-\mathsf{d}_{Y}(u(x),u(y))-v(x)\,.$

Let $\bar{y}\in S_{t}(\bar{x},\lambda)\Subset U$ be such that

$\mathsf{d}(\bar{x},\bar{y})=L_{t,\lambda}(x)\,.$

By the very definition of $S_{t}(\bar{x},\lambda)$, $H$ has a minimum at
$(\bar{x},\bar{y})$.

Step 2. We perturb $H$ to achieve a strict minimum at $(\bar{x},\bar{y})$,
with a controlled perturbation. To this aim, we consider

$0<\delta_{0}<C(K,N,{\rm diam}(U))\theta_{0}\,$ (5.20)

and define the function $H_{1}:B\times U\to\mathbb{R}$ by

$H_{1}(x,y):=H(x,y)+\delta_{0}\mathsf{d}^{2}(\bar{x},x)+\delta_{0}\mathsf{d}^{2}(\bar{y},y)\,.$

Since $(\bar{x},\bar{y})$ is a minimum for $H$, $(\bar{x},\bar{y})$ is the
unique strict minimum for $H_{1}$ in $B\times U$.

The next goal is to perturb again $H_{1}$ in order to make it achieve its
minimum at a point $(\tilde{x},\tilde{y})$ such that $\tilde{x}$ and
$\tilde{y}$ are _good points_ for section 3 and $\bar{x}$ is a Lebesgue point
for

$x\mapsto e^{-2K\lambda}\frac{K}{t}L_{t,\lambda}^{2}(x)\,.$

We notice that $\mathfrak{m}\otimes\mathfrak{m}$-a.e. point in $B\times U$
verifies these two properties.

We wish to construct perturbations with the tools developed in section 4.
Let us observe that $H_{1}$ has measure valued Laplacian on $B\times U$ with
positive part bounded from above by a constant

$C=C\left({\rm diam}U,{\rm diam}B,\lambda,t,\delta_{0},\left\lVert
L_{t,\lambda}\right\rVert_{L^{\infty}(B)}\right)\,.$

In order to verify this claim we consider the various terms appearing in the
definition of $H_{1}$ separately. The function

$(x,y)\mapsto\frac{e^{-2K\lambda}\mathsf{d}^{2}(x,y)}{2t}$

has measure valued Laplacian on $X\times X$ with positive part bounded above
by a constant $C(K,N,t,{\rm diam}U,{\rm diam}B)$ thanks to section 3. The
function

$(x,y)\mapsto\delta_{0}\mathsf{d}^{2}(\bar{x},x)+\delta_{0}\mathsf{d}^{2}(\bar{y},y)=:G(x,y)$
(5.21)

has measure valued Laplacian with positive part bounded by
$C(K,N,\delta_{0},{\rm diam}U,{\rm diam}B)$ thanks to the Laplacian comparison
and the tensorization of Sobolev spaces. The function

$(x,y)\mapsto-\mathsf{d}_{Y}(u(x),u(y))$

has non-positive measure valued Laplacian, thanks to section 3. Moreover,

$\Delta_{x,y}(-v)=-\Delta_{x}v=e^{-2K\lambda}\frac{K}{t}L_{t,\lambda}^{2}+\theta_{0}\,,$

by the very construction of $v$, see (5.19). Hence the function
$(x,y)\mapsto-v(x)$ has Laplacian bounded by $C(K,N,t,\lambda,{\rm diam}U,{\rm
diam}B)$, thanks to section 5 (ii).

For $\mathfrak{m}\otimes\mathfrak{m}$-a.e. $(z,z^{\prime})\in B\times U$, $z$
is a Lebesgue point of

$z\mapsto e^{-2K\lambda}\frac{K}{t}L_{t,\lambda}^{2}(z)\,$ (5.22)

and both $z$ and $z^{\prime}$ are such that, setting

$w_{q,P}(\cdot):=\mathsf{d}_{Y}^{2}(u(\cdot),u(q))-\mathsf{d}_{Y}^{2}(u(\cdot),P)+\mathsf{d}_{Y}^{2}(P,u(q))\,,$

it holds

$\limsup_{t\to 0}\frac{1}{t}P_{t}\left(w_{q,P}(\cdot)\right)(q)\leq 0\,,$

for every $P\in Y$, for $q=z$ and $q=z^{\prime}$. This is a consequence of
section 3.

Combining the above observations with section 4 we obtain that, for every
$\mu>0$ sufficiently small, there exist $a_{\mu}\in B$, $b_{\mu}\in U$ such
that the function $H_{a,b,\mu}:B\times U\to\mathbb{R}$ defined by

$H_{a,b,\mu}(x,y):=H_{1}(x,y)+\mu\mathsf{d}^{2}(a_{\mu},x)+\mu\mathsf{d}^{2}(b_{\mu},y)=:H_{1}(x,y)+G_{1,\mu}(x,y)\,$
(5.23)

achieves a strict minimum at a point $(\tilde{x}_{\mu},\tilde{y}_{\mu})\in
B\times U$ which verifies the good properties mentioned above. We choose any
of the triples $(a,b,\mu)$ such that these conditions are met and set
$H_{2,\mu}:=H_{a,b,\mu}$.

Step 3. Let us see how to reach a contradiction from the construction of the
previous two steps.
For the first part of this step, $\mu>0$ will be fixed and we will avoid the
subscript $\mu$ for the minimum point $(\tilde{x}_{\mu},\tilde{y}_{\mu})$ to
simplify the notation.
For any $s>0$, let us denote by $P_{s}\delta_{\tilde{x}}$ and
$P_{s}\delta_{\tilde{y}}$ the heat kernel measures at time $s$ centred at
$\tilde{x}$ and $\tilde{y}$ respectively. Moreover, we denote by $\Pi_{s}$ the
optimal transport plan for quadratic cost between $P_{s}\delta_{\tilde{x}}$
and $P_{s}\delta_{\tilde{y}}$. In particular, $\Pi_{s}$ is a probability
measure on $X\times X$ whose first and second marginals are
$P_{s}\delta_{\tilde{x}}$ and $P_{s}\delta_{\tilde{y}}$, respectively and

$\int_{X\times
X}\mathsf{d}^{2}(x,y)\mathop{}\\!\mathrm{d}\Pi_{s}(x,y)\leq\int_{X\times
X}\mathsf{d}^{2}(x,y)\mathop{}\\!\mathrm{d}\Pi(x,y)\,,$

for any probability measure $\Pi$ on $X\times X$ with the same marginals.
From the Wasserstein contractivity of the Heat Flow on
$\operatorname{RCD}(K,\infty)$ spaces, see [6], we get

$\int_{X\times X}\mathsf{d}^{2}(x,y)\mathop{}\\!\mathrm{d}\Pi_{s}(x,y)\leq
e^{-2Ks}\mathsf{d}^{2}(\tilde{x},\tilde{y})\,.$

From now on, when integrating with respect to heat kernel measures, or optimal
transport plans between heat kernel measures, continuous functions that are
not globally defined, we always understand that we are choosing global
extensions with controlled growth at infinity. The independence of the
particular choice is justified by [80, Lemma 2.53] and the first step of the
proof of section 5.

We set

$H_{2,\mu}(x,y):=\frac{e^{-2K\lambda}\mathsf{d}^{2}(x,y)}{2t}+F_{\mu}(x,y)$
(5.24)

and claim that

$\liminf_{s\to 0}\frac{\int_{X\times
X}H_{2,\mu}(x,y)\mathop{}\\!\mathrm{d}\Pi_{s}(x,y)-H_{2,\mu}(\tilde{x},\tilde{y})}{s}\geq
0\,,$

since $(\tilde{x},\tilde{y})$ is a minimum of $H_{2,\mu}$ on $B\times U$. This
is indeed a consequence of section 5.

On the other hand, let us estimate

$\displaystyle\limsup_{s\to 0}$ $\displaystyle\frac{\int_{X\times
X}H_{2,\mu}(x,y)\mathop{}\\!\mathrm{d}\Pi_{s}(x,y)-H_{2,\mu}(\tilde{x},\tilde{y})}{s}$
$\displaystyle\leq$ $\displaystyle\frac{e^{-2K\lambda}}{2t}\limsup_{s\to
0}\frac{\int_{X\times
X}\mathsf{d}^{2}(x,y)\mathop{}\\!\mathrm{d}\Pi_{s}(x,y)-\mathsf{d}^{2}(\tilde{x},\tilde{y})}{s}$
(5.25) $\displaystyle+$ $\displaystyle\limsup_{s\to 0}\frac{\int_{X\times
X}-\mathsf{d}_{Y}(u(x),u(y))\mathop{}\\!\mathrm{d}\Pi_{s}+\mathsf{d}_{Y}(u(\tilde{x}),u(\tilde{y}))}{s}$
(5.26) $\displaystyle+$ $\displaystyle\limsup_{s\to 0}\frac{\int_{X\times
X}\left(-v(x)\right)\mathop{}\\!\mathrm{d}\Pi_{s}(x,y)+v(\tilde{x})}{s}$
(5.27) $\displaystyle+$ $\displaystyle\limsup_{s\to 0}\frac{\int_{X\times
X}G(x,y)\mathop{}\\!\mathrm{d}\Pi_{s}(x,y)-G(\tilde{x},\tilde{y})}{s}$ (5.28)
$\displaystyle+$ $\displaystyle\limsup_{s\to 0}\frac{\int_{X\times
X}G_{1,\mu}(x,y)\mathop{}\\!\mathrm{d}\Pi_{s}(s,y)-G_{1,\mu}(\tilde{x},\tilde{y})}{s}\,,$
(5.29)

where the functions $G$ and $G_{1}$ have been introduced in (5.21) and (5.23)
respectively. We estimate each of the five terms separately.
We start observing that

$\frac{e^{-2K\lambda}}{2t}\limsup_{s\to 0}\frac{\int_{X\times
X}\mathsf{d}^{2}(x,y)\mathop{}\\!\mathrm{d}\Pi_{s}(x,y)-\mathsf{d}^{2}(\tilde{x},\tilde{y})}{s}\leq-K\frac{e^{-2K\lambda}}{t}\mathsf{d}^{2}(\tilde{x},\tilde{y})\,.$
(5.30)

Let us deal with (5.26). There are two possibilities. Either
$u(\tilde{x})=u(\tilde{y})$ in which case $\Pi_{s}$ is concentrated on the
diagonal of $X\times X$ and therefore it trivially holds

$\limsup_{s\to 0}\frac{\int_{X\times
X}-\mathsf{d}_{Y}(u(x),u(y))\mathop{}\\!\mathrm{d}\Pi_{s}+\mathsf{d}_{Y}(u(\tilde{x}),u(\tilde{y}))}{s}\leq
0\,.$

Otherwise $u(\tilde{x})\neq u(\tilde{y})$ in which case we can argue as
follows.

For any $(x,y)\in X\times X$, let us apply section 3 with points
$\\{u(x),u(\tilde{x}),u(\tilde{y}),u(y)\\}$. Then, denoting by $\tilde{z}$ the
midpoint of the segment $u(\tilde{x})u(\tilde{y})$, we obtain