# A note on Rubio de Francia’s extrapolation in tent spaces and applications

José María Martell  José María Martell
Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM
Consejo Superior de Investigaciones Cientificas
C/ Nicolás Cabrera, 13-15
E-28049 Madrid, Spain<EMAIL_ADDRESS>and  Pierre Portal  Pierre
Portal Mathematical Sciences Institute
Australian National University
Ngunnawal and Ngambri Country
Canberra ACT 0200, Australia<EMAIL_ADDRESS>
###### Abstract.

The Rubio de Francia extrapolation theorem is a very powerful result which
states that in order to show that certain operators satisfy weighted norm
inequalities with Muckenhoupt weights it suffices to see that the
corresponding inequalities hold for some fixed exponent, for instance $p=2$.
In this paper we extend this result and show that this extrapolation principle
allows one to obtain weighted estimates in tent spaces. From our extrapolation
result we automatically derive new estimates (and reprove some other)
concerning Calderón-Zygmund operators, operators associated with the Kato
conjecture, or fractional operators.

###### Key words and phrases:

Muckenhoupt weights, Rubio de Francia extrapolation, tent spaces, Hardy-
Littlewood maximal function, Calderón-Zygmund operators, singular integral
operators, Kato conjecture, fractional integrals.

###### 2010 Mathematics Subject Classification:

42B25, 42B20, 42B35, 42B37, 35J15

The first author acknowledges financial support from MCIN/AEI/
10.13039/501100011033 grants CEX2019-000904-S and PID2019-107914GB-I00. This
research was primarily conducted when the second author visited the first
author at the Instituto de Ciencias Matemáticas (ICMAT) in Madrid. The perfect
working environment provided by the ICMAT is very much appreciated.

###### Contents

  1. 1 Introduction
  2. 2 Applications
  3. 3 Notation
  4. 4 Proof of the main result
  5. 5 Off diagonal bounds and Weighted estimates

##  1\. Introduction

The celebrated extrapolation theorem of Rubio de Francia [18] gives that, if
an operator $T$ satisfies

$\|Tf\|_{L^{p_{0}}(w)}\lesssim\|f\|_{L^{p_{0}}(w)},\quad\text{for some
}p_{0}\in[1,\infty)\text{ and every }w\in A_{p_{0}},$

then

$\|Tf\|_{L^{p}(w)}\lesssim\|f\|_{L^{p}(w)},\quad\text{for every
}p\in(1,\infty)\text{ and every }w\in A_{p}.$

Here, $A_{p}$ denotes the class of Muckenhoupt weights in $\mathbb{R}^{n}$
with the underlying Lebesgue measure and $L^{p}(w)$ denotes the associated
weighted Lebesgue space with underlying measre $w(x)\,dx$. See Section 3 for
precise definitions.

The theory of Muckenhoupt weights has been extensively studied, and one of its
most basic features is the fact that $w\in A_{p}$ if and only if the Hardy-
Littlewood maximal function $M$ is bounded on $L^{p}(w)$ whenever
$p\in(1,\infty)$; and for $p=1$, $w\in A_{1}$ if and only if $M$ maps
continuously $L^{1}(w)$ into $L^{1,\infty}(w)$.

A modern treatment of extrapolation can be found in [12] where it is shown
that operators play no role in extrapolation, indeed one can reformulate it in
terms of pairs of functions. To elaborate on this, let $\mathcal{F}$ be some
family of pairs $(f,g)$ of non-negative measurable functions. Rubio de
Francia’s extrapolation theorem can then be rewritten as follows: if

(1.1) $\|f\|_{L^{p_{0}}(w)}\lesssim\|g\|_{L^{p_{0}}(w)},\quad\text{for some
}p_{0}\in[1,\infty),\text{ every }w\in A_{p_{0}},\text{ and every
}(f,g)\in\mathcal{F},$

then

(1.2) $\|f\|_{L^{p}(w)}\lesssim\|g\|_{L^{p}(w)},\quad\text{for every
}p\in(1,\infty),\text{ every }w\in A_{p},\text{ and every
}(f,g)\in\mathcal{F}.$

This formulation is very useful and has profound consequences. For example,
suppose that (1.1) holds (hence so does (1.2)) and we wish to show the
following family of vector-valued inequalities:

(1.3)
$\Big{\|}\Big{(}\sum_{j}f_{j}^{r}\Big{)}^{\frac{1}{r}}\Big{\|}_{L^{q}(w)}\lesssim\Big{\|}\Big{(}\sum_{j}g_{j}^{r}\Big{)}^{\frac{1}{r}}\Big{\|}_{L^{q}(w)},\quad\\{(f_{j},g_{j})\\}_{j}\subset\mathcal{F},$

with $1<q,r<\infty$ and $w\in A_{q}$. Fixing $r\in(1,\infty)$, there is some
choice of $q$ for which this estimate follows easily. Indeed, if $q=r$, we can
interchange the sum and the integral to obtain that for any $w\in A_{r}$,
(1.2) (applied to each $(f_{j},g_{j})$ with $p=r$ and $w\in A_{r}$) yields

(1.4)
$\Big{\|}\Big{(}\sum_{j}f_{j}^{r}\Big{)}^{\frac{1}{r}}\Big{\|}_{L^{r}(w)}^{r}=\sum_{j}\|f_{j}\|_{L^{r}(w)}^{r}\lesssim\sum_{j}\|g_{j}\|_{L^{r}(w)}^{r}=\Big{\|}\Big{(}\sum_{j}g_{j}^{r}\Big{)}^{\frac{1}{r}}\Big{\|}_{L^{r}(w)}^{r}$

Knowing that the desired family of estimate holds with some particular
exponent, $q=r$, we want to invoke Rubio de Francia’s extrapolation theorem
with a new family $\mathcal{F}_{r}$ consisting of the pairs $(F,G)$, where

$F=\Big{(}\sum_{j}f_{j}^{r}\Big{)}^{\frac{1}{r}},\qquad
G=\Big{(}\sum_{j}g_{j}^{r}\Big{)}^{\frac{1}{r}},$

and $\\{(f_{j},g_{j})\\}_{j}\subset\mathcal{F}$. In this way, (1.4) can be
written as

(1.5) $\|F\|_{L^{r}(w)}\lesssim\|G\|_{L^{r}(w)},\quad\text{for every }w\in
A_{r}\text{ and }(F,G)\in\mathcal{F}_{r}.$

Therefore, we can extrapolate and show that

$\|F\|_{L^{q}(w)}\lesssim\|G\|_{L^{q}(w)},\quad\text{for every
}q\in(1,\infty),\text{ every }w\in A_{q}\text{ and }(F,G)\in\mathcal{F}_{r}.$

That is, (1.3) holds as desired.

The previous application of Rubio de Francia’s extrapolation theorem shows
that, for one reason or another, sometimes there is a natural choice of an
exponent for which the desired estimate turns out to be simpler. The goal of
this paper is to show that this applies to the case of weighted tent spaces
which are introduced next. Given $0<r,p<\infty$ and $w\in A_{\infty}$ we say
that $F\in L^{r}_{\mathrm{loc}}(\mathbb{R}^{n+1}_{+})$ belongs to
$\mathscr{T}_{r}^{q}(w)$ provided

$\|F\|_{\mathscr{T}_{r}^{p}(w)}:=\|\mathcal{A}_{r}F\|_{L^{p}(w)}:=\Big{\|}\Big{(}\int\\!\\!\\!\int_{|\,\cdot\,-y|<t}|F(y,t)|^{r}\,\frac{dy\,dt}{t^{n+1}}\Big{)}^{\frac{1}{r}}\Big{\|}_{L^{p}(w)}<\infty.$

Much as in the case of the vector-valued inequalities estimates considered
above, the choice $p=r$ allows us to use Fubini’s theorem to rearrange the
different integrals. This is the basic idea to obtain the main result of this
paper which is formulated next. We also prove results in the weighted Lorentz
tent spaces $\mathscr{T}_{r}^{q,s}(w)$, with $0<r,p,s<\infty$ and $w\in
A_{\infty}$, defined using the quasi-norm

$\|F\|_{\mathscr{T}_{r}^{p,s}(w)}:=\|\mathcal{A}_{r}F\|_{L^{p,s}(w)}=\Big{\|}\Big{(}\int\\!\\!\\!\int_{|\,\cdot\,-y|<t}|F(y,t)|^{r}\,\frac{dy\,dt}{t^{n+1}}\Big{)}^{\frac{1}{r}}\Big{\|}_{L^{p,s}(w)}.$

######  Theorem 1.6.

Let $\mathcal{F}$ be a family of pairs of measurable functions $(F,G)$, with
$F,G:\mathbb{R}^{n+1}_{+}\to[0,\infty]$, and so that $F(\cdot,t)$,
$G(\cdot,t)$ are measurable for each $t\in(0,\infty)$. For any
$t\in(0,\infty)$ introduce $\mathcal{F}_{t}$, the family of pairs
$(F(\cdot,t),G(\cdot,t))$ with $(F,G)\in\mathcal{F}$. Let
$\varphi:[1,\infty)\to[1,\infty)$ be a non-decreasing function.

  * (i)

Assume that there exists $p_{0}\in[1,\infty)$ such that for every $w\in
A_{p_{0}}$ there holds

(1.7)
$\|F(\cdot,t)\|_{L^{p_{0}}(w)}\leq\varphi([w]_{A_{p_{0}}})\,\|G(\cdot,t)\|_{L^{p_{0}}(w)},\qquad(F(\cdot,t),G(\cdot,t))\in\mathcal{F}_{t},\
t\in(0,\infty).$

For every $r,p\in(1,\infty)$ there exists a non-decreasing function
$\psi:[1,\infty)\to[1,\infty)$ such that

(1.8)
$\|F\|_{\mathscr{T}_{r}^{p}(w)}\leq\psi([w]_{A_{p}})\,\|G\|_{\mathscr{T}_{r}^{p}(w)},\qquad(F,G)\in\mathcal{F},$

for every $w\in A_{p}$.

  * (ii)

Assume that there exists $p_{0}\in(0,\infty)$ such that for every $w\in
A_{\infty}$ there holds

$\|F(\cdot,t)\|_{L^{p_{0}}(w)}\leq\varphi([w]_{A_{\infty}})\,\|G(\cdot,t)\|_{L^{p_{0}}(w)},\qquad(F(\cdot,t),G(\cdot,t))\in\mathcal{F}_{t},\
t\in(0,\infty).$

For every $r,p,s\in(0,\infty)$ there exists a non-decreasing function
$\psi:[1,\infty)\to[1,\infty)$ such that

$\|F\|_{\mathscr{T}_{r}^{p}(w)}\leq\psi([w]_{A_{\infty}})\,\|G\|_{\mathscr{T}_{r}^{p}(w)},\qquad(F,G)\in\mathcal{F},$

and

$\|F\|_{\mathscr{T}_{r}^{p,s}(w)}\leq\psi([w]_{A_{\infty}})\,\|G\|_{\mathscr{T}_{r}^{p,s}(w)},\qquad(F,G)\in\mathcal{F},$

for every $w\in A_{\infty}$.

  * (iii)

Assume that there exist $0<p_{-}<p_{+}<\infty$, and $p_{0}\in[p_{-},p_{+}]$,
such that for every $w\in A_{\frac{p_{0}}{p_{-}}}\cap
RH_{(\frac{p_{+}}{p_{0}})^{\prime}}$ there holds

$\|F(\cdot,t)\|_{L^{p_{0}}(w)}\leq\varphi([w]_{A_{\frac{p_{0}}{p_{-}}}}+[w]_{RH_{(\frac{p_{+}}{p_{0}})^{\prime}}})\,\|G(\cdot,t)\|_{L^{p_{0}}(w)},\qquad(F(\cdot,t),G(\cdot,t))\in\mathcal{F}_{t},\
t\in(0,\infty).$

For every $r,p\in(p_{-},p_{+})$ there exists a non-decreasing function
$\psi:[1,\infty)\to[1,\infty)$ such that

$\|F\|_{\mathscr{T}_{r}^{p}(w)}\leq\psi([w]_{A_{\frac{p}{p_{-}}}}+[w]_{RH_{(\frac{p_{+}}{p})^{\prime}}})\,\|G\|_{\mathscr{T}_{r}^{p}(w)},\qquad(F,G)\in\mathcal{F},$

for every $w\in A_{\frac{p}{p_{-}}}\cap RH_{(\frac{p_{+}}{p})^{\prime}}$.

  * (iv)

Assume that there exist $1\leq p_{0}\leq q_{0}<\infty$, such that for every
$w\in A_{p_{0},q_{0}}$ there holds

$\|F(\cdot,t)\|_{L^{q_{0}}(w^{q_{0}})}\leq\varphi([w]_{A_{p_{0},q_{0}}})\,\|G(\cdot,t)\|_{L^{p_{0}}(w^{p_{0}})},\qquad(F(\cdot,t),G(\cdot,t))\in\mathcal{F}_{t},\
t\in(0,\infty).$

For every $p$ and $q$ such that $1<p\leq q<\infty$ and
$\frac{1}{p}-\frac{1}{q}=\frac{1}{p_{0}}-\frac{1}{q_{0}}$, and for every
$r\in(1,\infty)$ there exists a non-decreasing function
$\psi:[1,\infty)\to[1,\infty)$ such that

$\|F\|_{\mathscr{T}_{r}^{q}(w^{q})}\leq\psi([w]_{A_{p,q}})\,\|G\|_{\mathscr{T}_{r}^{p}(w^{p})},\qquad(F,G)\in\mathcal{F},$

for every $w\in A_{p,q}$.

In many applications, the function $F$ is of the form
$F(t,\cdot)=T_{t}(G(t,.))$ for some family of sublinear operators
$(T_{t})_{t>0}$. Even when, for all $t>0$, $T_{t}=T$ for a fixed operator $T$
bounded on $L^{p}(\mathbb{R}^{n})$, the estimate

(1.9)
$\Big{\|}\Big{(}\int\\!\\!\\!\int_{|\,\cdot\,-y|<t}|T(G(t,\cdot))(y)|^{r}\,\frac{dy\,dt}{t^{n+1}}\Big{)}^{\frac{1}{r}}\Big{\|}_{L^{p}}\lesssim\Big{\|}\Big{(}\int\\!\\!\\!\int_{|\,\cdot\,-y|<t}|G(y,t)|^{r}\,\frac{dy\,dt}{t^{n+1}}\Big{)}^{\frac{1}{r}}\Big{\|}_{L^{p}},$

is non-trivial. This is somewhat surprising, given that the analogous estimate
on the vertical square function space
$L^{p}(\mathbb{R}^{n};L^{2}(\mathbb{R}_{+},\frac{dt}{t}))$, rather than the
conical square function space $\mathscr{T}_{2}^{p}$, holds for simple reasons
(e.g., by the classical result of Marcinkiewicz and Zygmund [14]). To prove
(1.9), or its more general version with the family $(T_{t})_{t>0}$, one can
use a (different) extrapolation method based on off-diagonal estimates of
sufficiently high order decay, systemically developed in [2] (see Section 5).
Unfortunately, some natural operators such as the Riesz transforms do not have
enough off-diagonal decay and one can use weighted extrapolation instead, as
first demonstrated in [6].

Theorem 1.6 improves the approach developed in [6] by proving a general result
that extrapolates inequalities, rather than just operators, in the spirit of
[12]. This adds flexibility, and allows us to add weights to the tent space
norms, simplify proofs, and remove some unnecessary restrictions (see the
applications to fractional operators in Section 2). We also prove, in Section
5, an extrapolation result in weighted tent spaces under an off-diagonal
estimate assumption, and clarify the relationship between extrapolation
results using off-diagonal assumptions and those using weighted estimates as a
hypothesis.

Our approach is based on Lemma 4.2, which describes how the averaging
operators used to define tent spaces interact with weights. This is more
flexible than the approach used in [6], which is based on an understanding of
how the averaging operators interact with the operators one wants to
extrapolate (see [6, Lemma 4.1, Lemma 5.4]).

Given Lemma 4.2, our assumptions then translate into an estimate in some
weighted Lebesgue spaces for some particular exponent and for the pairs
$(\mathcal{A}_{r}F,\mathcal{A}_{r}G)$, with $(F,G)\in\mathcal{F}$. For the
sake of specificity let us sketch the proof of (i). Fix $r\in(1,\infty)$. We
will show that Fubini’s theorem implies that for every $w\in A_{r}$ one has

$\|\mathcal{A}_{r}F\|_{L^{r}(w)}\lesssim\|\mathcal{A}_{r}G\|_{L^{r}(w)},\qquad(F,G)\in\mathcal{F}.$

If we then define the family $\widetilde{\mathcal{F}}_{r}$ consisting of the
pairs $(\mathcal{A}_{r}F,\mathcal{A}_{r}G)$ with $(F,G)\in\mathcal{F}$, we can
extrapolate from the exponent $r$ to obtain estimates for
$\widetilde{\mathcal{F}}_{r}$ in $L^{p}(w)$ for every $p\in(1,\infty)$ and for
all $w\in A_{p}$. These are, in turn, the desired estimates for the pairs of
the family $\mathcal{F}$ in $\mathscr{T}_{r}^{p}(w)$.

As just shown for (i), one can relate the initial estimates with some weighted
norm inequality for the pairs $(\mathcal{A}_{r}F,\mathcal{A}_{r}G)$ with
$(F,G)\in\mathcal{F}$. This means that we can invoke some other known
extrapolation results and obtain other estimates. For instance, we can
consider tent spaces associated to Banach functions space and in the context
of (i) show that

$\|\mathcal{A}_{r}F\|_{\mathbb{X}(w)}\lesssim\|\mathcal{A}_{r}G\|_{\mathbb{X}(w)}$

with $\mathbb{X}$ being a rearrangement invariant Banach function space with
Boyd indices $1<p_{\mathbb{X}}\leq q_{\mathbb{X}}<\infty$ and $w\in
A_{p_{\mathbb{X}}}$ (see [12, Theorem 4.6]). Examples to which this applies
are $L^{p,q}$, $1<p<\infty$, $1\leq q\leq\infty$; or $L^{p}(\log L)^{\alpha}$
with $1<p<\infty$ and $\alpha\in\mathbb{R}$. In both cases
$p_{\mathbb{X}}=q_{\mathbb{X}}=p$ and hence the weights should be in $A_{p}$.
Other examples are $\mathbb{X}=L^{p}\cap L^{q}$ or $L^{p}+L^{q}$ with
$1<p<q<\infty$, where $p_{\mathbb{X}}=p$ and $q_{\mathbb{X}}=q$, and the
weights are assumed to be in $A_{p}$.

More generally we can introduce some different weights inside the previous
norms and obtain

$\|(\mathcal{A}_{r}F)\,u\|_{\mathbb{X}(w)}\lesssim\|(\mathcal{A}_{r}G)\,u\|_{\mathbb{X}(w)}$

with $\mathbb{X}$ being a rearrangement invariant Banach function space with
Boyd indices $1<p_{\mathbb{X}}\leq q_{\mathbb{X}}<\infty$,
$u^{p_{\mathbb{X}}}\,w\in A_{p_{\mathbb{X}}}$, $u^{q_{\mathbb{X}}}\,w\in
A_{q_{\mathbb{X}}}$, and $w\in A_{\infty}$ (see [8, Theorem 3.1, Example
2.46]). For instance, if $\mathbb{X}=L^{p,q}$, $1<p<\infty$, $1\leq
q\leq\infty$; or $L^{p}(\log L)^{\alpha}$ with $1<p<\infty$ and
$\alpha\in\mathbb{R}$ the conditions on the weights reduce to $u^{p}\,w\in
A_{p}$ and $w\in A_{\infty}$. As before, for $\mathbb{X}=L^{p}\cap L^{q}$ or
$L^{p}+L^{q}$ with $1<p<q<\infty$, the weights should satisfy $u^{p}\,w\in
A_{p}$, $u^{q}\,w\in A_{q}$, and $w\in A_{\infty}$.

We can also consider Banach function spaces which are not necessarily
rearrangement invariant. Indeed, if $\mathbb{X}$ is Banach function space so
that the Hardy-Littlewood maximal function is bounded both on $\mathbb{X}$ and
its associate space $\mathbb{X}^{\prime}$ (cf. [15, Theorem 8.2], [8, Theorem
3.1]), then we also get that
$\|\mathcal{A}_{r}F\|_{\mathbb{X}}\lesssim\|\mathcal{A}_{r}G\|_{\mathbb{X}}$.
That is the case of variable Lebesgue spaces $\mathbb{X}=L^{p(\cdot)}$ where
the variable exponent $p(\cdot)$ satisfies the so-called log-Hölder condition
(see [8] and the references therein). One could also obtain estimates of the
form
$\|(\mathcal{A}_{r}F)\,u\|_{\mathbb{X}}\lesssim\|(\mathcal{A}_{r}G)\,u\|_{\mathbb{X}}$
provided the Hardy-Littlewood maximal function and some appropriate dual
operator satisfies similar inequalities (see [8, Theorem 3.1]). As a
consequence,$\|(\mathcal{A}_{r}F)\,u\|_{L^{p(\cdot)}}\lesssim\|(\mathcal{A}_{r}G)\,u\|_{L^{p(\cdot)}}$
holds for variable exponents $p(\cdot)$ satisfying the log-Hölder condition
and for all $u\in A_{p(\cdot)}$ (see [8, Example 2.39]).

More general weighted Banach functions spaces are allowed, see [8, Theorem
3.1], or even estimates in modular spaces, see [12, Section 4.3] or [8,
Theorem 4.1]. The precise statements are left to the interested reader.

The plan of the paper is as follows. We present some applications of the main
result in the next section: Corollary 2.3 is a general result which is applied
to Calderón-Zygmund operators, operators associated with the Kato conjecture,
and fractional operators. Notation is fixed in Section 3, before we turn to
the proofs in Section 4. Finally, in Section 5, we consider the interaction
between extrapolation results on tent spaces based on off-diagonal bounds, and
those based on weighted estimates.

##  2\. Applications

We denote by $M$ the (uncentered) Hardy-Littlewood maximal function and by
$\mathcal{M}$ its corresponding extension to functions defined in
$\mathbb{R}^{n+1}_{+}$. That is, if $F:\mathbb{R}^{n+1}_{+}\to\mathbb{R}$ with
$F(\cdot,t)\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})$ we set

$\mathcal{M}(F)(x,t):=M(F(\cdot,t))(x),\qquad(x,t)\in\mathbb{R}^{n+1}_{+}.$

Analogously, for an operator $T$ on $\mathbb{R}^{n}$ defined on some family of
measurable functions we define its extension

(2.1)
$\mathcal{T}(F)(x,t):=T(F(\cdot,t))(x),\qquad(x,t)\in\mathbb{R}^{n+1}_{+},$

provided $F(\cdot,t)$ belongs to the domain of $T$.

The first consequence of Theorem 1.6 is that we can extend [6, Theorem 2.1] to
consider the boundedness of the Hardy-Littlewood maximal function in weighted
tent spaces.

######  Corollary 2.2.

Given $1<r<\infty$, $\mathcal{M}$ is bounded on $\mathscr{T}_{r}^{p}(w)$ for
all $1<p<\infty$ and all $w\in A_{p}$, and bounded from
$\mathscr{T}_{r}^{1}(w)$ to $\mathscr{T}_{r}^{1,\infty}(w)$ for all $w\in
A_{1}$.

The proof of the case $1<p<\infty$ is particularly easy. Let $\mathcal{F}$ be
the family of pairs $(\mathcal{M}(G),|G|)$, where
$G:\mathbb{R}^{n+1}_{+}\to\mathbb{R}$ with $G(\cdot,t)\in
L^{1}_{\mathrm{loc}}(\mathbb{R})$. Note that for every $t>0$ and for every
$w\in A_{2}$ by [7]

$\|\mathcal{M}(G)(\cdot,t)\|_{L^{2}(w)}=\|M(G(\cdot,t))\|_{L^{2}(w)}\lesssim[w]_{A_{2}}\,\|G(\cdot,t)\|_{L^{2}(w)}.$

We can then invoke Theorem 1.6 part (i) with $p_{0}=2$ to obtain for every
$p,r\in(1,\infty)$ and every $w\in A_{p}$

$\|\mathcal{M}(G)\|_{\mathscr{T}_{r}^{p}(w)}\lesssim\|G\|_{\mathscr{T}_{r}^{p}(w)}.$

The case $p=1$ does not follow by extrapolation and its proof is postponed
until the end of Section 4.

Using the very same ideas, Theorem 1.6 and Corollary 2.2 have the following
immediate consequence. The proof is easy and left to the interested reader.

######  Corollary 2.3.

  * (a)

Assume that $T$ is an operator satisfying
$\|Tf\|_{L^{p_{0}}(w)}\lesssim\|f\|_{L^{p_{0}}(w)},$ for some
$p_{0}\in(1,\infty)$ and for all $w\in A_{p_{0}}$ (equivalently, for all
$p\in(1,\infty)$ and all $w\in A_{p}$). Then, $\mathcal{T}$ is bounded in
$\mathscr{T}_{r}^{p}(w)$ for all $p,r\in(1,\infty)$ and all $w\in A_{p}$, that
is,

$\|\mathcal{T}(F)\|_{\mathscr{T}_{r}^{p}(w)}\lesssim\|F\|_{\mathscr{T}_{r}^{p}(w)}.$

  * (b)

Assume that $T$ is an operator satisfying
$\|Tf\|_{L^{p_{0}}(w)}\lesssim\|Mf\|_{L^{p_{0}}(w)},$ for some
$p_{0}\in(0,\infty)$ and for all $w\in A_{\infty}$ (equivalently, for all
$p\in(0,\infty)$ and all $w\in A_{\infty}$). Then, for all
$p,r,s\in(0,\infty)$ and all $w\in A_{\infty}$, there hold

$\|\mathcal{T}(F)\|_{\mathscr{T}_{r}^{p}(w)}\lesssim\|\mathcal{M}(F)\|_{\mathscr{T}_{r}^{p}(w)},\qquad\|\mathcal{T}(F)\|_{\mathscr{T}_{r}^{p,s}(w)}\lesssim\|\mathcal{M}(F)\|_{\mathscr{T}_{r}^{p,s}(w)}.$

As a consequence, $\mathcal{T}$ is bounded in $\mathscr{T}_{r}^{p}(w)$ for all
$p,r\in(1,\infty)$ and all $w\in A_{p}$, that is,

$\|\mathcal{T}(F)\|_{\mathscr{T}_{r}^{p}(w)}\lesssim\|F\|_{\mathscr{T}_{r}^{p}(w)};$

and $\mathcal{T}$ is bounded from $\mathscr{T}_{r}^{1}(w)$ to
$\mathscr{T}_{r}^{1,\infty}(w)$ for all $r\in(1,\infty)$ and all $w\in A_{1}$,
that is,

$\|\mathcal{T}(F)\|_{\mathscr{T}_{r}^{1,\infty}(w)}\lesssim\|F\|_{\mathscr{T}_{r}^{1}(w)}.$

  * (c)

Assume that there exist $0<p_{-}<p_{+}<\infty$ and that $T$ is an operator
satisfying $\|Tf\|_{L^{p_{0}}(w)}\lesssim\|f\|_{L^{p_{0}}(w)},$ for some
$p_{0}\in(p_{-},p_{+})$ and for all $w\in A_{{p_{0}}/{p_{-}}}\cap
RH_{({p_{+}}/{p_{0}})^{\prime}}$ (equivalently, for all $p\in(p_{-},p_{+})$
and all $w\in A_{{p}/{p_{-}}}\cap RH_{({p_{+}}/{p})^{\prime}}$). Then,
$\mathcal{T}$ is bounded in $\mathscr{T}_{r}^{p}(w)$ for all
$p,r\in(p_{-},p_{+})$ and all $w\in A_{{p}/{p_{-}}}\cap
RH_{({p_{+}}/{p})^{\prime}}$, that is,

$\|\mathcal{T}(F)\|_{\mathscr{T}_{r}^{p}(w)}\lesssim\|F\|_{\mathscr{T}_{r}^{p}(w)}.$

  * (d)

Assume that there exist $1<p_{0}\leq q_{0}<\infty$ and that $T$ is an operator
satisfying
$\|Tf\|_{L^{q_{0}}(w^{q_{0}})}\lesssim\|f\|_{L^{p_{0}}(w^{p_{0}})},$ for every
$w\in A_{p_{0},q_{0}}$ (equivalently, for all $1<p\leq q<\infty$ with
$\frac{1}{p}-\frac{1}{q}=\frac{1}{p_{0}}-\frac{1}{q_{0}}$ and all $w\in
A_{p,q}$). Then, $\mathcal{T}$ is bounded from $\mathscr{T}_{r}^{p}(w^{p})$ to
$\mathscr{T}_{r}^{q}(w^{q})$ for all $p$ and $q$ such that $1<p\leq q<\infty$
and $\frac{1}{p}-\frac{1}{q}=\frac{1}{p_{0}}-\frac{1}{q_{0}}$, all
$r\in(1,\infty)$, and all $w\in A_{p,q}$, that is,

$\|\mathcal{T}(F)\|_{\mathscr{T}_{r}^{q}(w^{q})}\lesssim\|F\|_{\mathscr{T}_{r}^{p}(w^{p})}.$

In the previous result all the involved boundedness should hold with a
constant that depends on the characteristic of the weights in terms on a non-
decreasing function.

Let us now present some relevant examples of operators to which the previous
result applies:

### Calderón-Zygmund operators and Singular Integral operators

Any Calderón-Zygmund operator $T$, like the Hilbert or the Riesz transform, is
bounded on $L^{p}(w)$ for every $p\in(1,\infty)$ and $w\in A_{p}$. It also
satisfies Coifmann-Fefferman inequalities (cf. [10, 11]), that is, $T$ is
controlled by $M$ in $L^{p}(w)$ for every $p\in(0,\infty)$ and $w\in
A_{\infty}$. As such we can apply Corollary 2.3 parts (a) and (b) to obtain
that $\mathcal{T}$ is bounded in $\mathscr{T}_{r}^{p}(w)$ for every
$1<p,r<\infty$ and every $w\in A_{p}$ and is also bounded from
$\mathscr{T}_{r}^{1}(w)$ to $\mathscr{T}_{r}^{1,\infty}(w)$ for every
$1<r<\infty$ and every $w\in A_{1}$, this extends [6, Theorem 2.2]. Moreover,
$\mathcal{T}$ can be controlled by $\mathcal{M}$ in $\mathscr{T}_{r}^{p}(w)$,
and even in $\mathscr{T}_{r}^{p,s}(w)$, for every $0<p,r,s<\infty$ and $w\in
A_{\infty}$. This can be generalized to other singular integral operators. For
instance we can consider commutators $[T,b]$ with $T$ being a Calderón-Zygmund
operator and $b\in\mathrm{BMO}$. It is well-known that these commutators are
bounded in $L^{p}(w)$ for $1<p<\infty$ and $w\in A_{p}$ and they can be
controlled by $M^{2}=M\circ M$ in $L^{p}(w)$ for every $p\in(0,\infty)$ and
$w\in A_{\infty}$. We can therefore obtain that these estimates readily extend
to weighed tent spaces, further details are left to the interested reader.

### Operators associated with the Kato conjecture

Let $A$ be an $n\times n$ matrix of complex and $L^{\infty}$-valued
coefficients defined on $\mathbb{R}^{n}$. We assume that $A$ satisfies the
following ellipticity condition: there exist $0<\lambda\leq\Lambda<\infty$
such that

$\lambda|\xi|^{2}\leq\mathrm{Re}\,A(x)\xi\cdot\bar{\xi}\qquad\text{and}\qquad|A(x)\xi\cdot\bar{\eta}|\leq\Lambda|\xi||\eta|,$

for all $\xi,\eta\in\mathbb{C}^{n}$ and almost every $x\in\mathbb{R}^{n}$. We
have used the notation $\xi\cdot\eta=\sum_{j=1}^{n}\xi_{j}\eta_{j}$, and
therefore $\xi\cdot\bar{\eta}$ is the usual inner product in $\mathbb{C}^{n}$.
Note that then
$A(x)\xi\cdot\bar{\eta}=\sum_{j,k}a_{j,k}(x)\xi_{k}\bar{\eta}_{j}$. Associated
with this matrix we define the second order divergence form operator
$Lu=-\mathop{\operatorname{div}}\nolimits(A\nabla u)$, which is understood in
the standard weak sense as a maximal-accretive operator on the space
$L^{2}(\mathbb{R}^{n})$ with domain $D(L)$ by means of a sesquilinear form.

Associated with this operator we can consider the functional calculus
$\varphi(L)$ where $\varphi$ is holomorphic and bounded in an appropriate
sector, the Riesz transform $\nabla L^{-1/2}$, and some square functions. The
$L^{p}$ theory for these operators was developed in the monograph [1] and
weighted norm inequalities were obtained in [5] using a generalized Calderón-
Zygmund theory from [3]. As in [1] and [4], we denote by
$(p_{-}(L),p_{+}(L))$, respectively $(q_{-}(L),q_{+}(L))$, the maximal open
interval on which the heat semigroup $(e^{-tL})_{t>0}$, respectively its
gradient $(\sqrt{t}\nabla e^{-tL})_{t>0}$, is uniformly bounded on
$L^{p}(\mathbb{R}^{n})$. It is obtained in [1] that $p_{-}(L)=q_{-}(L)$ and
$2<q_{+}(L)\leq p_{+}(L)$.

Recall the weighted norm inequalities from [5]:

$\displaystyle\|\varphi(L)f\|_{L^{p}(w)}\lesssim\|\varphi\|_{\infty}\|f\|_{L^{p}(w)},\quad\forall\,p\in(p_{-}(L),p_{+}(L)),\,\,w\in
A_{p/p_{-}(L)}\cap RH_{(p_{+}(L)/p)^{\prime}};$ $\displaystyle\|\nabla
L^{-1/2}f\|_{L^{p}(w)}\lesssim\|f\|_{L^{p}(w)},\quad\forall\,p\in(q_{-}(L),q_{+}(L)),\,\,w\in
A_{p/q_{-}(L)}\cap RH_{(q_{+}(L)/p)^{\prime}};$

and

$\displaystyle\|L^{1/2}f\|_{L^{p}(w)}\lesssim\|\nabla
f\|_{L^{p}(w)},\quad\forall\,p\in(p_{-}(L),p_{+}(L)),\,\,w\in
A_{p/p_{-}(L)}\cap RH_{(p_{+}(L)/p)^{\prime}}.$

Using these estimates, and invoking Corollary 2.3 part (c) we can easily
obtain that $\varphi(L)$ can be extended as a bounded operator in
$\mathscr{T}_{r}^{p}(w)$ for every $p,r\in(p_{-}(L),p_{+}(L))$ and $w\in
A_{p/p_{-}(L)}\cap RH_{(p_{+}(L)/p)^{\prime}}$. In the case of the Riesz
transform $\nabla L^{-1/2}$ we get estimates in $\mathscr{T}_{r}^{p}(w)$ for
every $p,r\in(q_{-}(L),q_{+}(L))$ and $w\in A_{p/q_{-}(L)}\cap
RH_{(q_{+}(L)/p)^{\prime}}$. The latter provides a much easier proof of [6,
Theorem 2.4], which only considers the unweighted situation.

Finally for the reverse inequalities for square roots we get

$\displaystyle\|L^{1/2}F\|_{\mathscr{T}_{r}^{p}(w)}\lesssim\|\nabla_{x}F\|_{\mathscr{T}_{r}^{p}(w)},\quad\forall\,p,r\in(p_{-}(L),p_{+}(L)),\,\,w\in
A_{p/p_{-}(L)}\cap RH_{(p_{+}(L)/p)^{\prime}}.$

In particular we get the following Kato type estimates in weighted tent
spaces:

$\displaystyle\|L^{1/2}F\|_{\mathscr{T}_{r}^{p}(w)}\approx\|\nabla_{x}F\|_{\mathscr{T}_{r}^{p}(w)},\quad\forall\,p,r\in(q_{-}(L),q_{+}(L)),\,\,w\in
A_{p/q_{-}(L)}\cap RH_{(q_{+}(L)/p)^{\prime}}.$

In these estimates it is understood that the operators in question are acting
only on the space variable (much as in (2.1)), but this time, abusing the
notation, we have not used a different symbol to denote this extension.

### Fractional operators

Let $I_{\alpha}$ denote the fractional integral (aka Riesz potential) of order
$\alpha\in(0,n)$ and let $M_{\alpha}$ be the associated maximal fractional
integral operator. We write $\mathcal{I}_{\alpha}$ and $\mathcal{M}_{\alpha}$
for the associated extensions to $\mathbb{R}^{n+1}_{+}$ (cf. (2.1)). In [17,
Theorem 1] it was shown that $I_{\alpha}$ is controlled by $M_{\alpha}$ in
$L^{p}(w)$ for all $p\in(0,\infty)$ and $w\in A_{\infty}$. By Corollary 2.3
part $(b)$ we readily obtain that $\mathcal{I}_{\alpha}$ is controlled by
$\mathcal{M}_{\alpha}$ in the weighted tent spaces $\mathscr{T}_{r}^{p}(w)$,
for every $0<p,r<\infty$ and $w\in A_{\infty}$. On the other hand, [17] also
shows that $I_{\alpha}$ and $M_{\alpha}$ are bounded from $L^{p}(w^{p})$ to
$L^{q}(w^{q})$ for every $p\in(1,n/\alpha)$, $q$ so that
$\frac{1}{p}-\frac{1}{q}=\frac{\alpha}{n}$, and $w\in A_{p,q}$. Invoking
Corollary 2.3 part (d) we conclude that $\mathcal{I}_{\alpha}$ and
$\mathcal{M}_{\alpha}$ are bounded in the weighted tent spaces
$\mathscr{T}_{r}^{p}(w)$ for every $p\in(1,n/\alpha)$, $q$ so that
$\frac{1}{p}-\frac{1}{q}=\frac{\alpha}{n}$, $r\in(1,\infty)$, and $w\in
A_{p,q}$. This extends [6, Theorem 2.3] which only considers the unweighted
case and where it was additionally assumed that $r>n/(n-\alpha)$.

##  3\. Notation

Throughout this paper, we work in $\mathbb{R}^{n}$, $n\geq 1$, endowed with
the Lebesgue measure which is denoted by $dx$. The characteristic function of
a measurable set $E$ is denoted by $\mathbf{1}_{E}$, and we use $B$ to denote
Euclidean balls. The Hardy-Littlewood maximal operator $M$ is defined for each
locally integrable function $f$ by

(3.1) $Mf(x):=\sup_{x\in
B}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B}|f(y)|\,dy,\qquad
x\in\mathbb{R}^{n},$

where the sup runs over all Euclidean balls $B$ containing $x$. Here and
elsewhere barred integrals are used to denote averages.

A measurable function $w$ on $\mathbb{R}^{n}$ is called a weight if
$0<w(x)<\infty$ for a.e. $x\in\mathbb{R}^{n}$. For every $p\in(0,\infty)$ and
every weight $w$, we define the associated weighted Lebesgue space
$L^{p}(w):=L^{p}(w(x)\,dx)$ and the same applies to weighted Lorentz spaces
$L^{p,s}(w):=L^{p,s}(w(x)\,dx)$ for $p,s\in(0,\infty)$.

Recall that we define tent spaces as follows: given $0<r,p<\infty$ and a
weight $w$ we say that $F\in L^{r}_{\mathrm{loc}}(\mathbb{R}^{n+1}_{+})$
belongs to $\mathscr{T}_{r}^{q}(w)$ provided

$\|F\|_{\mathscr{T}_{r}^{p}(w)}:=\|\mathcal{A}_{r}F\|_{L^{p}(w)}:=\Big{\|}\Big{(}\int\\!\\!\\!\int_{|\,\cdot\,-y|<t}|F(y,t)|^{r}\,\frac{dy\,dt}{t^{n+1}}\Big{)}^{\frac{1}{r}}\Big{\|}_{L^{p}(w)}<\infty.$

Weighted Lorentz tents spaces are defined similarly.

Given $p\in(1,\infty)$, we define the class $A_{p}$ as the collection of all
weights $w$ satisfying

$[w]_{A_{p}}:=\sup_{B}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B}w(x)\,dx\right)\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B}w(x)^{1-p^{\prime}}\,dx\right)^{p-1}<\infty,$

where $p^{\prime}$ is the Hölder conjugate exponent of $p$, i.e.,
$\frac{1}{p}+\frac{1}{p^{\prime}}=1$. As for the case $p=1$, we say that $w\in
A_{1}$ if

$[w]_{A_{1}}:=\sup_{B}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B}w(y)\,dy\right)\|w^{-1}\mathbf{1}_{B}\|_{L^{\infty}(\mathbb{R}^{n})}<\infty.$

Finally, we define

$A_{\infty}:=\bigcup_{p\geq 1}A_{p},$

and $[w]_{A_{\infty}}=\inf_{p:w\in A_{p}}[w]_{A_{p}}$.

Given $1<p\leq\infty$, $1\leq q<\infty$, we define the class $A_{p,q}$ as the
collection of all weights $w$ satisfying

(3.2)
$[w]_{A_{p,q}}:=\sup_{B}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B}w(x)^{q}\,dx\right)\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B}w(x)^{-p^{\prime}}\,dx\right)^{\frac{q}{p^{\prime}}}<\infty.$

The reverse Hölder classes are defined in the following way: we say that $w\in
RH_{s}$ for $s\in(1,\infty)$ if

$[w]_{RH_{s}}:=\sup_{B}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B}w(x)^{s}\,dx\right)^{\frac{1}{s}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B}w(x)\,dx\right)^{-1}<\infty.$

Regarding the endpoint $s=\infty$, $w\in RH_{\infty}$ means that

$[w]_{RH_{\infty}}:=\sup_{B}\|w\mathbf{1}_{B}\|_{L^{\infty}(\mathbb{R}^{n})}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B}w(x)\,dx\right)^{-1}<\infty.$

##  4\. Proof of the main result

We begin with some auxiliary results.

######  Lemma 4.1.

For every $0\leq h\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})$,
$x\in\mathbb{R}^{n}$, and $s,t\in(0,\infty)$ there holds

$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,s)}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(y,t)}h(z)\,dz\Big{)}\,dy\leq
2^{n}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,s+t)}h(y)\,dy.$

###### Proof.

Note first that if $y\in B(x,s)$ then $B(y,t)\subset B(x,s+t)$. Assume first
that $s\leq t$. Then,

$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,s)}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(y,t)}h(z)\,dz\Big{)}\,dy\leq\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,s)}\Big{(}\frac{|B(x,s+t)|}{|B(y,t)|}\,\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,s+t)}h(z)\,dz\Big{)}\,dy\\\
=\frac{(s+t)^{n}}{t^{n}}\,\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,s+t)}h(z)\,dz\leq
2^{n}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,s+t)}h(y)\,dy.$

On the other hand, if $t\leq s$, we observe that Fubini’s theorem implies

$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,s)}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(y,t)}h(z)\,dz\Big{)}\,dy=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,s)}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(y,t)}h(z)\,\mathbf{1}_{B(x,s+t)}\,dz\Big{)}\,dy\\\
\leq\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,s+t)}h(z)\,\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(z,t)}\frac{|B(x,s+t)|}{|B(x,s)|}\,dy\Big{)}\,dz=\frac{(s+t)^{n}}{s^{n}}\,\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,s+t)}h(z)\,dz\\\
\leq
2^{n}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,s+t)}h(y)\,dy.$

This completes the proof. ∎

######  Lemma 4.2.

Let $1\leq p,r<\infty$ and $w\in A_{p}$. Define

$W_{t}(x):=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,t)}w(y)\,dy,\qquad
x\in\mathbb{R}^{n},\ t>0.$

Then, $W_{t}\in A_{p}$ with $[W_{t}]_{A_{p}}\leq 2^{n\,p}\,[w]_{A_{p}}$, and
for every $F\in L^{r}_{\mathrm{loc}}(\mathbb{R}^{n+1})$

$\|\mathcal{A}_{r}F\|_{L^{r}(w)}^{r}=v_{n}\int_{0}^{\infty}\|F(\cdot,t)\|_{L^{r}(W_{t})}^{r}\,\frac{dt}{t},$

where $v_{n}=|B(0,1)|$.

###### Proof.

Consider first the case $p=1$. Given $B=B(x,s)$, by Lemma 4.1 we obtain

$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B}W_{t}(y)\,dy=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,s)}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(y,t)}w(z)\,dz\Big{)}\,dy\leq
2^{n}\,\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,s+t)}w(y)\,dy\\\
\leq 2^{n}\,[w]_{A_{1}}\mathop{\mathrm{missing}}{\,ess\,inf\,}_{B(x,s+t)}w\leq
2^{n}\,[w]_{A_{1}}\mathop{\mathrm{missing}}{\,ess\,inf\,}_{B(x,s)}w.$

This readily implies that $W_{t}\in A_{1}$ with $[W_{t}]_{A_{1}}\leq
2^{n}\,[w]_{A_{1}}$.

Assume next that $1<p<\infty$. Using that $\eta(t)=t^{1-p^{\prime}}$, $t>0$,
is convex; Jensen’s inequality; and Lemma 4.1 we arrive at

$\displaystyle\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B}W_{t}(y)\,dy\Big{)}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B}W_{t}(y)^{1-p^{\prime}}\,dy\Big{)}^{p-1}$
$\displaystyle\qquad\qquad=\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,s)}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(y,t)}w(z)\,dz\Big{)}\,dy\Big{)}\,\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,s)}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(y,t)}w(z)\,dz\Big{)}^{1-p^{\prime}}\,dy\Big{)}^{p-1}$
$\displaystyle\qquad\qquad\leq\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,s)}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(y,t)}w(z)\,dz\Big{)}\,dy\Big{)}\,\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,s)}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(y,t)}w(z)^{1-p^{\prime}}\,dz\Big{)}\,dy\Big{)}^{p-1}$
$\displaystyle\qquad\qquad\leq
2^{n\,p}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,s+t)}w(y)\,dy\Big{)}\,\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,s+t)}w(y)^{1-p^{\prime}}\,dy\Big{)}^{p-1}$
$\displaystyle\qquad\qquad\leq 2^{n\,p}[w]_{A_{p}}.$

From here we can immediately see that $W_{t}\in A_{p}$ with
$[W_{t}]_{A_{p}}\leq 2^{n\,p}\,[w]_{A_{p}}$.

Finally, Fubini’s theorem readily gives that

$\|\mathcal{A}_{r}F\|_{L^{r}(w)}^{r}=\int_{\mathbb{R}^{n+1}}\int\\!\\!\\!\int_{|x-y|<t}|F(y,t)|^{r}\,\frac{dy\,dt}{t^{n+1}}\,w(x)\,dx=\int_{0}^{\infty}\\!\\!\\!\int_{\mathbb{R}^{n}}|F(y,t)|^{r}\,\Big{(}\frac{1}{t^{n}}\int_{B(y,t)}w(x)\,dx\Big{)}\,dy\,\frac{dt}{t}\\\
=v_{n}\,\int_{0}^{\infty}\\!\\!\\!\int_{\mathbb{R}^{n}}|F(y,t)|^{r}\,W_{t}(y)\,dy\,\frac{dt}{t}=v_{n}\int_{0}^{\infty}\|F(\cdot,t)\|_{L^{r}(W_{t})}^{r}\,\frac{dt}{t},$

and this completes the proof. ∎

We are now ready to prove Theorem 1.6.

###### Proof of Theorem 1.6.

We start with (i). Fix $r\in(1,\infty)$ and let $w\in A_{r}$ be arbitrary. By
(1.7) and [12, Theorem 3.9] it follows that (1.7) holds with $r$ in place of
$p_{0}$ and with some non-decreasing function $\widetilde{\varphi}$ replacing
$\varphi$. This, together with Lemma 4.2, gives

$\|\mathcal{A}_{r}F\|_{L^{r}(w)}^{r}=v_{n}\int_{0}^{\infty}\|F(\cdot,t)\|_{L^{r}(W_{t})}^{r}\,\frac{dt}{t}\leq
v_{n}\int_{0}^{\infty}\widetilde{\varphi}([W_{t}]_{A_{r}})\|G(\cdot,t)\|_{L^{r}(W_{t})}^{r}\,\frac{dt}{t}\\\
\leq
v_{n}\widetilde{\varphi}(2^{n}[w]_{A_{r}})\,\int_{0}^{\infty}\|G(\cdot,t)\|_{L^{r}(W_{t})}^{r}\,\frac{dt}{t}=\widetilde{\varphi}(2^{n}[w]_{A_{r}})\,\|\mathcal{A}_{r}G\|_{L^{r}(w)}^{r}.$

Since $w\in A_{r}$ is arbitrary we can extrapolate invoking again [12, Theorem
3.9] —with the family of pairs $(\mathcal{A}_{r}F,\mathcal{A}_{r}G)$, where
$(F,G)\in\mathcal{F}$— to conclude that (1.8) holds.

The proof of (ii) and (iii) can be carried out similarly invoking [12,
Corollary 3.15 and Corllary 3.17] and [9, Theorem 2.1], and [12, Theorem 3.31]
respectively (see also [3, Theorem 4.9]). Further details are left to the
reader.

Proving (iv) requires some extra arguments that are inspired by [12, p. 49].
Fix $r\in(1,\infty)$ and pick $p_{1}$ and $q_{1}$ such that $1<p_{1}\leq r\leq
q_{1}<\infty$ and
$\frac{1}{p_{1}}-\frac{1}{q_{1}}=\frac{1}{p_{0}}-\frac{1}{q_{0}}$ (this is
possible for all values of $r\in(1,\infty)$). Take an arbitrary $w\in
A_{p_{1},q_{1}}$ and $(F,G)\in\mathcal{F}$. Note that by Minkowski’s
inequality with exponent $q_{1}/r\geq 1$ and Jensen’s inequality we arrive at

(4.3)
$\|\mathcal{A}_{r}F\|_{L^{q_{1}}(w^{q_{1}})}^{r}=v_{n}\,\Big{\|}\int_{0}^{\infty}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(\cdot,t)}|F(y,t)|^{r}\,dy\Big{)}\,\frac{dt}{t}\Big{\|}_{L^{\frac{q_{1}}{r}}(w^{q_{1}})}\\\
\leq
v_{n}\,\int_{0}^{\infty}\Big{\|}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(\cdot,t)}|F(y,t)|^{r}\,dy\Big{\|}_{L^{\frac{q_{1}}{r}}(w^{q_{1}})}\,\frac{dt}{t}\leq
v_{n}\,\int_{0}^{\infty}\Big{\|}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(\cdot,t)}|F(y,t)|^{q_{1}}\,dy\Big{)}^{\frac{r}{q_{1}}}\Big{\|}_{L^{\frac{q_{1}}{r}}(w^{q_{1}})}\,\frac{dt}{t}.$

Define

$\widetilde{W}_{t}(x):=\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,t)}w(y)^{q_{1}}\,dy\Big{)}^{\frac{1}{q_{1}}},\qquad
x\in\mathbb{R}^{n},\ t>0.$

Using that $w\in A_{p_{1},q_{1}}$ if and only if $w^{q_{1}}\in
A_{1+q_{1}/p_{1}^{\prime}}$, and Lemma 4.2 we readily obtain that
$\widetilde{W}_{t}^{q_{1}}\in A_{1+q_{1}/p_{1}^{\prime}}$ with
$[\widetilde{W}_{t}^{q_{1}}]_{A_{1+q_{1}/p_{1}^{\prime}}}\leq
2^{n\,(1+q_{1}/p_{1}^{\prime})}\,[w^{q_{1}}]_{A_{1+q_{1}/p_{1}^{\prime}}}$.
Equivalently, $\widetilde{W}_{t}\in A_{p_{1},q_{1}}$ with
$[\widetilde{W}_{t}]_{A_{p_{1},q_{1}}}\leq
2^{n\,(1/q_{1}+1/p_{1}^{\prime})}\,[w]_{A_{p_{1},q_{1}}}$. On the other hand,
using that $w\in A_{p_{1},q_{1}}$; that
$\eta(t)=t^{-{p_{1}}/{p_{1}^{\prime}}}$, $t>0$, is convex; and Jensen’s
inequality we obtain

$\widetilde{W}_{t}(x)^{p_{1}}=\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,t)}w(y)^{q_{1}}\,dy\Big{)}^{\frac{{}^{p_{1}}}{q_{1}}}\leq[w]_{A_{p_{1},q_{1}}}^{p_{1}}\,\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,t)}w(y)^{-p_{1}^{\prime}}\,dy\Big{)}^{-\frac{{p_{1}}}{p_{1}^{\prime}}}\leq[w]_{A_{p_{1},q_{1}}}^{p_{1}}\,\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,t)}w(y)^{p_{1}}\,dy.$

All this give for any $t\in(0,\infty)$ by Fubini’s inequality and our
hypothesis

$\displaystyle\Big{\|}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(\cdot,t)}|F(y,t)|^{q_{1}}\,dy\Big{)}^{\frac{r}{q_{1}}}\Big{\|}_{L^{\frac{q_{1}}{r}}(w^{q_{1}})}^{\frac{1}{r}}$
$\displaystyle=\Big{(}\int_{\mathbb{R}^{n}}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,t)}|F(y,t)|^{q_{1}}\,dy\Big{)}\,w(x)^{q_{1}}\,dx\Big{)}^{\frac{1}{q_{1}}}$
$\displaystyle=\Big{(}\int_{\mathbb{R}^{n}}|F(y,t)|^{q_{1}}\,\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(y,t)}w(x)^{q_{1}}\,dx\Big{)}\,dy\Big{)}^{\frac{1}{q_{1}}}$
$\displaystyle=\|F(\cdot,t)\|_{L^{q_{1}}(\widetilde{W}_{t}^{q_{1}})}$
$\displaystyle\lesssim\|G(\cdot,t)\|_{L^{p_{1}}(\widetilde{W}_{t}^{p_{1}})}$
$\displaystyle\lesssim\Big{(}\int_{\mathbb{R}^{n}}|G(y,t)|^{p_{1}}\,\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(y,t)}w(x)^{p_{1}}\,dx\Big{)}\,dy\Big{)}^{\frac{1}{p_{1}}}$
$\displaystyle=\Big{(}\int_{\mathbb{R}^{n}}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,t)}|G(y,t)|^{p_{1}}\,dy\Big{)}\,w(x)^{p_{1}}\,dx\Big{)}^{\frac{1}{p_{1}}}.$

Plugging this into (4.3), using Minkowski’s inequality with exponent
$r/p_{1}\geq 1$ and Jensen’s inequality we conclude that

$\|\mathcal{A}_{r}F\|_{L^{q_{1}}(w^{q_{1}})}^{p_{1}}\lesssim\Big{(}\int_{0}^{\infty}\Big{(}\int_{\mathbb{R}^{n}}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,t)}|G(y,t)|^{p_{1}}\,dy\Big{)}\,w(x)^{p_{1}}\,dx\Big{)}^{\frac{r}{p_{1}}}\,\frac{dt}{t}\Big{)}^{\frac{p_{1}}{r}}\\\
\leq\int_{\mathbb{R}^{n}}\Big{(}\int_{0}^{\infty}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,t)}|G(y,t)|^{p_{1}}\,dy\Big{)}^{\frac{r}{p_{1}}}\,\frac{dt}{t}\,\Big{)}^{\frac{p_{1}}{r}}\,w(x)^{p_{1}}\,dx\\\
\leq\int_{\mathbb{R}^{n}}\Big{(}\int_{0}^{\infty}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,t)}|G(y,t)|^{r}\,dy\Big{)}\,\frac{dt}{t}\,\Big{)}^{\frac{p_{1}}{r}}\,w(x)^{p_{1}}\,dx=v_{n}^{-\frac{p_{1}}{r}}\|\mathcal{A}_{r}G\|_{L^{p_{1}}(w^{p_{1}})}^{p_{1}}.$

All these show that
$\|\mathcal{A}_{r}F\|_{L^{q_{1}}(w^{q_{1}})}\lesssim\|\mathcal{A}_{r}G\|_{L^{p_{1}}(w^{p_{1}})}$
for every $(F,G)\in\mathcal{F}$ and for every $w\in A_{p_{1},q_{1}}$. At this
point we invoke [12, Theorem 3.23] —with the family of pairs
$(\mathcal{A}_{r}F,\mathcal{A}_{r}G)$, where $(F,G)\in\mathcal{F}$— to obtain
the desired estimate. ∎

###### Proof of Corollary 2.2.

As observed above the case $1<p<\infty$ is particularly easy. Indeed, let
$\mathcal{F}$ be the family of pairs $(\mathcal{M}(G),|G|)$, where
$G:\mathbb{R}^{n+1}_{+}\to\mathbb{R}$ with $G(\cdot,t)\in
L^{1}_{\mathrm{loc}}(\mathbb{R})$. Note that for every $t>0$ and for every
$w\in A_{2}$ by [7]

$\|\mathcal{M}(G)(\cdot,t)\|_{L^{2}(w)}=\|M(G(\cdot,t))\|_{L^{2}(w)}\lesssim[w]_{A_{2}}\,\|G(\cdot,t)\|_{L^{2}(w)}.$

We can then invoke Theorem 1.6 part (i) with $p_{0}=2$ to obtain for every
$p,r\in(1,\infty)$ and every $w\in A_{p}$

$\|\mathcal{M}(G)\|_{\mathscr{T}_{r}^{p}(w)}\lesssim\|G\|_{\mathscr{T}_{r}^{p}(w)}.$

We are left with considering the case $p=1$. We invoke [6, Lemma 4.1] which
shows that for every $r\in(1,\infty)$, $t>0$ and $x\in\mathbb{R}^{n}$ the
following estimates holds

$\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,t)}Mf(x)^{r}\,dx\Big{)}^{\frac{1}{r}}\lesssim\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,2t)}|f(x)|^{r}\,dx\Big{)}^{\frac{1}{r}}+M\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(\cdot,t)}|f(z)|\,dz\Big{)}(x).$

This implies that

$\mathcal{A}_{r}(\mathcal{M}(G))(x)\lesssim\mathcal{A}_{r}^{2}G(x)+\Big{(}\int_{0}^{\infty}\mathcal{M}(\widetilde{G})(x)^{r}\,\frac{dt}{t}\Big{)}^{\frac{1}{r}}$

with
$\widetilde{G}(x,t)=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,t)}|G(z,t)|\,dz$,
and where

$\mathcal{A}_{r}^{2}(G)(x)=\Big{(}\int\\!\\!\\!\int_{|x-y|<2\,t}|F(y,t)|^{r}\,\frac{dy\,dt}{t^{n+1}}\Big{)}^{\frac{1}{r}}.$

Tchebychev’s inequality and the change of angle formula (see for instance [16,
Section 3.1]) imply

$\|\mathcal{A}_{r}^{2}G\|_{L^{1,\infty}(w)}\leq\|\mathcal{A}_{r}^{2}G\|_{L^{1}(w)}\lesssim\|\mathcal{A}_{r}G\|_{L^{1}(w)}=\|G\|_{\mathscr{T}_{r}^{1}(w)}.$

On the other hand the weighted Fefferman-Stein vector-valued weak-type
estimates for the Hardy-Littlewood maximal function (see [19]) and Jensen’s
inequality yield

$\Big{\|}\Big{(}\int_{0}^{\infty}\mathcal{M}(\widetilde{G})(\cdot)^{r}\,\frac{dt}{t}\Big{)}^{\frac{1}{r}}\Big{\|}_{L^{1,\infty}(w)}\lesssim\Big{\|}\Big{(}\int_{0}^{\infty}\widetilde{G}(\cdot,t)^{r}\,\frac{dt}{t}\Big{)}^{\frac{1}{r}}\Big{\|}_{L^{1}(w)}\\\
=\Big{\|}\Big{(}\int_{0}^{\infty}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(\cdot,t)}|G(z,t)|\,dz\Big{)}^{r}\,\frac{dt}{t}\Big{)}^{\frac{1}{r}}\Big{\|}_{L^{1}(w)}\leq\Big{\|}\Big{(}\int_{0}^{\infty}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(\cdot,t)}|G(z,t)|^{r}\,dz\,\frac{dt}{t}\Big{)}^{\frac{1}{r}}\Big{\|}_{L^{1}(w)}\\\
=v_{n}^{-1}\|\mathcal{A}_{r}G\|_{L^{1}(w)}=v_{n}^{-1}\|G\|_{\mathscr{T}_{r}^{1}(w)},$

where $v_{n}=|B(0,1)|$. All these imply

$\|\mathcal{M}(G)\|_{\mathscr{T}_{r}^{1,\infty}(w)}=\|\mathcal{A}_{r}(\mathcal{M}(G))\|_{L^{1,\infty}(w)}\lesssim\|\mathcal{A}_{r}^{2}(G)\|_{L^{1,\infty}(w)}+\Big{\|}\Big{(}\int_{0}^{\infty}\mathcal{M}(\widetilde{G})(\cdot)^{r}\,\frac{dt}{t}\Big{)}^{\frac{1}{r}}\Big{\|}_{L^{1,\infty}(w)}\\\
\lesssim\|G\|_{\mathscr{T}_{r}^{1}(w)}.$

This completes the proof. ∎

##  5\. Off diagonal bounds and Weighted estimates

As we have seen along the paper, Rubio de Francia’s extrapolation can be used
to obtain estimates in tent spaces once some family of weighted estimates are
known. Nonetheless, there are other extrapolation techniques, based on off-
diagonal estimates, which can be used to derive estimates in the scale of tent
spaces.

Let us first introduce some notation. Given a family of bounded linear
operators $(T_{t})_{t>0}$, we recall that for each $t>0$, we write
$\mathcal{T}_{t}$ for the extension of $T_{t}$ to functions defined in
$\mathbb{R}^{n+1}_{+}$. We then set for each
$F:\mathbb{R}^{n+1}_{+}\to\mathbb{R}$

$\mathbf{T}(F)(x,t):=\mathcal{T}_{t}(F)(x)=T_{t}(F(\cdot,t))(x),\qquad(x,t)\in\mathbb{R}^{n+1}_{+}.$

We next recall the definition of off-diagonal decay:

######  Definition 5.1.

Let $1<r<\infty$. A family of bounded linear operators $(T_{t})_{t>0}$ on
$L^{r}(\mathbb{R}^{n})$ is said to have $L^{r}-L^{r}$ off-diagonal decay of
order $M>0$ if

$\|\mathbf{1}_{E}T_{t}(\mathbf{1}_{F}f)\|_{r}\lesssim\Big{(}1+\frac{d(E,F)}{t}\Big{)}^{-M}\|\mathbf{1}_{F}f\|_{r},$

for all Borel sets $E,F\subset\mathbb{R}^{n}$, all $t>0$, and all $f\in
L^{r}(\mathbb{R}^{n})$.

In [13, Theorem 5.2] it is shown that if $(T_{t})_{t>0}$ has $L^{2}-L^{2}$
off-diagonal decay of order $M>\frac{n}{\min(p,2)}$, then $\mathbf{T}$ extends
to a bounded operator on $\mathscr{T}_{2}^{p}$. This result is perhaps the
simplest example of an extrapolation result based on off-diagonal bounds. We
observe that if $T_{t}\equiv T$ for a given Calderón-Zygmund operator $T$,
off-diagonal decay is, in general, only of order $\frac{n}{2}$, and [13,
Theorem 5.2] does not apply. However, Corollary 2.3 still gives that
$\mathbf{T}$ extends to a bounded operator on $\mathscr{T}_{2}^{p}(w)$ for all
$w\in A_{p}$.

In many applications, one deals with families of operators $(T_{t})_{t>0}$
that have $L^{2}-L^{2}$ off-diagonal decay of arbitrary high order (e.g., when
the decay is exponential). Our next result shows that, for such families (the
precise order needed here being $M>n$), $\mathbf{T}$ extends to a bounded
operator on $\mathscr{T}_{2}^{p}(w)$ for all $p\in(1,\infty)$ and all $w\in
A_{p}$.

######  Proposition 5.2.

Let $(T_{t})_{t>0}$ be a family of bounded linear operators on
$L^{r}(\mathbb{R}^{n})$ with $L^{r}-L^{r}$ off-diagonal decay of order $M$ for
some $r\geq 1$ and some $M>n/r$. For every $p$ satisfying $n/M<p<\infty$ and
every $w\in A_{p\,M/n}$, $\mathbf{T}$ extends to a bounded operator on
$\mathscr{T}_{r}^{p}(w)$.

The role of $L^{2}-L^{2}$ off-diagonal and weighted assumptions in
extrapolating tent space estimates can thus be summarized as follows:

  1. (i)

Regardless of the order of off-diagonal decay if weighted bounds are
available, then Rubio de Francia’s extrapolation gives that $\mathbf{T}$
extends to a bounded operator on $\mathscr{T}_{2}^{p}(w)$ for all
$p\in(1,\infty)$ and all $w\in A_{p}$ (Corollary 2.3).

  2. (ii)

If the order of off-diagonal decay is greater than $\frac{n}{\min(p,2)}$, then
extrapolation based on off-diagonal bounds gives that $\mathbf{T}$ extends to
a bounded operator on $\mathscr{T}_{2}^{p}$ ([13, Theorem 5.2]). This follows
as well from Proposition 5.2 by taking $M=\frac{n\,(1+\epsilon)}{\min(p,2)}$
with $0<\epsilon\ll 1$, $r=2$, and $w\equiv 1$.

  3. (iii)

If the order of off-diagonal decay is greater than or equal to $n$ (that is
the case if the decay is exponential), then Proposition 5.2 with $r=2$ and
$M=n$ gives that $\mathbf{T}$ extends to a bounded operator on
$\mathscr{T}_{2}^{p}(w)$ for all $p\in(1,\infty)$ and all $w\in A_{p}$,
without assuming a priori weighted estimates.

###### Proof of Proposition 5.2.

Fix $n/M<p<\infty$ and $w_{0}\in A_{p\,M/n}$. Using the openness of the
Muckenhoupt classes of weights and that $M>n/r$ we can find $\eta$ with
$\frac{n}{\min\\{p,r\\}\,M}<\eta<1$ so that $w_{0}\in A_{p\,M\,\eta/n}$. We
can then pick $p_{0}$ such that $\max\\{r/p,1\\}<p_{0}<r\,M\,\eta/n$ and
$w_{0}\in A_{p\,p_{0}/r}$.

Keeping these choices in mind we are going to use Rubio de Francia’s
extrapolation to obtain our desired estimate. For starters, localizing $f$ on
dyadic annuli of the form $C_{1}(x,t)=B(x,4\,t)$ and
$C_{j}(x,t)=B(x,2^{j+1}\,t)\setminus B(x,2^{j}\,t)$ for $j\geq 2$, and using
the off-diagonal decay assumption, we first note that

$\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,t)}|T_{t}f(y)|^{r}\,dy\Big{)}^{\frac{1}{r}}\lesssim\sum_{j=1}^{\infty}2^{-j(M-\frac{n}{r})}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,2^{j}\,t)}|f(y)|^{r}\,dy\Big{)}^{\frac{1}{r}},\qquad\text{for
every }x\in\mathbb{R}^{n},\ t>0.$

Take next an arbitrary $w\in A_{p_{0}}$ and define $W_{t}$ as in Lemma 4.2.
The fact that $w\in A_{p_{0}}$ easily implies that

$w(\lambda B)\leq[w]_{A_{p_{0}}}\,\lambda^{np_{0}}\,w(B),\qquad\text{for every
ball $B$ and for every }\lambda>1.$

This readily implies that
$W_{\lambda\,t}(x)\leq[w]_{A_{p_{0}}}\,\lambda^{n(p_{0}-1)}W_{t}(x)$ for every
$x\in\mathbb{R}^{n}$, $t>0$, and $\lambda>1$. This together with our
assumptions give for any fixed $t>0$:

$\displaystyle\|\mathbf{T}(F)(\cdot,t)\|_{L^{r}(W_{t})}$
$\displaystyle=\Big{(}\int_{\mathbb{R}^{n}}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,t)}|T_{t}(F(\cdot,t))(y)|^{r}\,dy\Big{)}\,w(x)\,dx\Big{)}^{\frac{1}{r}}$
$\displaystyle\leq\Big{(}\int_{\mathbb{R}^{n}}\Big{(}\sum_{j=1}^{\infty}2^{-j(M-\frac{n}{r})}\,\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,2^{j}\,t)}|F(y,t)|^{r}\,dy\Big{)}^{\frac{1}{r}}\Big{)}^{r}\,w(x)\,dx\Big{)}^{\frac{1}{r}}$
$\displaystyle\leq\sum_{j=1}^{\infty}2^{-j(M-\frac{n}{r})}\,\Big{(}\int_{\mathbb{R}^{n}}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,2^{j}\,t)}|F(y,t)|^{r}\,dy\Big{)}\,w(x)\,dx\Big{)}^{\frac{1}{r}}$
$\displaystyle=\sum_{j=1}^{\infty}2^{-j(M-\frac{n}{r})}\,\Big{(}\int_{\mathbb{R}^{n}}|F(y,t)|^{r}\,W_{2^{j}\,t}(y)\,dy\Big{)}^{\frac{1}{r}}$
$\displaystyle\leq[w]_{A_{p_{0}}}\,\sum_{j=1}^{\infty}2^{-j(M-\frac{p_{0}\,n}{r})}\,\Big{(}\int_{\mathbb{R}^{n}}|F(y,t)|^{r}\,W_{t}(y)\,dx\Big{)}^{\frac{1}{r}}$
$\displaystyle\lesssim[w]_{A_{p_{0}}}\,\|F(\cdot,t)\|_{L^{r}(W_{t})},$

where we have used the fact that $p_{0}\,n/r<M\,\eta<M$. This and Lemma 4.2
allows us to arrive at

$\int_{\mathbb{R}^{n}}\mathcal{A}_{r}(\mathbf{T}(F))(x)^{r}\,w(x)\,dx=v_{n}\int_{0}^{\infty}\|\mathbf{T}(F)(\cdot,t)\|_{L^{r}(W_{t})}^{r}\,\frac{dt}{t}\\\
\lesssim
v_{n}\,[w]_{A_{p_{0}}}\,\int_{0}^{\infty}\|F(\cdot,t)\|_{L^{r}(W_{t})}^{r}\,\frac{dt}{t}=[w]_{A_{p_{0}}}\,\int_{\mathbb{R}^{n}}\mathcal{A}_{r}(F)(x)^{r}\,w(x)\,dx.$

We can rewrite this as

$\int_{\mathbb{R}^{n}}\big{(}\mathcal{A}_{r}(\mathbf{T}(F))(x)^{\frac{r}{p_{0}}}\big{)}^{p_{0}}\,w(x)\,dx\leq
M_{0}^{r}\,[w]_{A_{p_{0}}}\,\int_{\mathbb{R}^{n}}\big{(}\mathcal{A}_{r}(F)(x)^{\frac{r}{p_{0}}}\big{)}^{p_{0}}\,w(x)\,dx.$

Since $w\in A_{p_{0}}$ is arbitrary we can invoke Rubio de Francia
extrapolation theorem [12, Theorem 3.9] with the family of pairs
$\big{(}\mathcal{A}_{r}(\mathbf{T}(F))(x)^{\frac{r}{p_{0}}},\mathcal{A}_{r}(F)(x)^{\frac{r}{p_{0}}}\big{)}$,
and obtain

(5.3)
$\int_{\mathbb{R}^{n}}\big{(}\mathcal{A}_{r}(\mathbf{T}(F))(x)^{\frac{r}{p_{0}}}\big{)}^{q}\,v(x)\,dx\lesssim\int_{\mathbb{R}^{n}}\big{(}\mathcal{A}_{r}(F)(x)^{\frac{r}{p_{0}}}\big{)}^{q}\,v(x)\,dx,$

for all $1<q<\infty$ and all $v\in A_{q}$.

To obtain the desired estimate we select $q:=p\,p_{0}/r$ and $v=w_{0}$. Note
that our previous choices guarantee that $1<q<\infty$ and $v\in A_{q}$. Hence
(5.3) holds and the proof is complete. ∎

######  Remark 5.4.

A remaining open question is to determine the optimal level of off-diagonal
decay required to extrapolate tent space estimates. Noting that
$\mathscr{T}_{2}^{2}$ estimates do not require any off-diagonal decay
(provided that the family $(T_{t})_{t>0}$ is uniformly bounded on $L^{2}$),
one would thus expect the required level of off-diagonal decay for a
$\mathscr{T}_{p}^{2}$ estimate to improve as $p$ approaches $2$.
Unfortunately, our current methods, whether based on the optimal change of
angle in tent spaces in [13] or on the doubling property of Muckenhoupt
weights in Proposition 5.2, do not seem to pick up such an improvement as $p$
approaches $2$.

## References

  * [1] P. Auscher, _On necessary and sufficient conditions for $L^{p}$-estimates of Riesz transforms associated to elliptic operators on $\mathbb{R}^{n}$ and related estimates_, Mem. Am. Math. Soc. 186 (871), (2007).
  * [2] P. Auscher, C. Kriegler, S. Monniaux, and P. Portal, _Singular integral operators on tent spaces_ , J. Evol. Equ. 12 (2012) 741–765.
  * [3] P. Auscher and J.M. Martell, _Weighted norm inequalities, off-diagonal estimates and elliptic operators. I. General operator theory and weights_ , Adv. Math. 212 (2007), 225–276.
  * [4] P. Auscher and J.M. Martell, _Weighted norm inequalities, off-diagonal estimates and elliptic operators. II. Off-diagonal estimates on spaces of homogeneous type_ , J. Evol. Equ. 7 (2007), 265–316.
  * [5] P. Auscher and J.M. Martell, _Weighted norm inequalities, off-diagonal estimates and elliptic operators. III. Harmonic analysis of elliptic operators_ , J. Funct. Anal. 241 (2006), 703–746.
  * [6] P. Auscher and C. Prisuelos-Arribas, Tent space boundedness via extrapolation, Math. Z. 286 (2017), no. 3-4, 1575–1604.
  * [7] S. M. Buckley, Estimates for operator norms on weighted spaces and reverse Jensen inequalities, Trans. Amer. Math. Soc. 340 (1993), no. 1, 253–272.
  * [8] M. Cao, J.J. Marín, and J.M. Martell, Extrapolation on function and modular spaces, and applications, Adv. Math. 406 (2022), 108520.
  * [9] D. Cruz-Uribe, J. M. Martell, and C. Pérez, _Extrapolation from $A_{\infty}$ weights and applications_, J. Funct. Anal. 213 (2004), 412–439.
  * [10] R. Coifman, Distribution function inequalities for singular integrals, Proc. Nat. Acad. Sci. U.S.A. 69 (1972), 2838–2839.
  * [11] R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250.
  * [12] D. Cruz-Uribe, J.M. Martell, and C. Pérez, _Weights, extrapolation and the theory of Rubio de Francia_ , Operator Theory: Advances and Applications, Vol. 215, Birkhäuser/Springer Basel AG, Basel, 2011.
  * [13] T. Hytönen, J. van Neerven, and P. Portal, _Conical square function estimates in UMD Banach spaces and applications to $H^{\infty}$-functional calculi_, J. Anal. Math. 106 (2008) 317–351.
  * [14] J. Marcinkiewicz and A. Zygmund, _Quelques inégalités pour les opérations linéaires_ , Fund. Math. 32 (1939), 112–121.
  * [15] J.J. Marín, J.M. Martell, D. Mitrea, I. Mitrea, and M. Mitrea, Singular Integral Operators, Quantitative Flatness, and Boundary Problems, Progress in Mathematics, 344. Birkhäuser/ Springer, Cham, 2022.
  * [16] J.M. Martell and C.Prisuelos-Arribas, Weighted Hardy spaces associated with elliptic operators. Part I: Weighted norm inequalities for conical square functions Trans. Amer. Math. Soc. 369 (2017), no. 6, 4193–4233.
  * [17] B.Muckenhoupt and R. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192 (1974), 261–274.
  * [18] J.L. Rubio de Francia, Factorization and extrapolation of weights, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 2, 393–395.
  * [19] J.L. Rubio de Francia, F.J. Ruiz, and J.L. Torrea, Calderón-Zygmund theory for operator-valued kernels, Adv. Math. 62 (1986), no. 1, 7–48.


# Emerging Diversity in a Population of Evolving Intransitive Dice

Julius B. Kirkegaard  Kim Sneppen  Niels Bohr Institute, University of
Copenhagen, 2100 Copenhagen, Denmark

###### Abstract

Exploiting the mathematical curiosity of intransitive dice, we present a
simple theoretical model for co-evolution that captures scales ranging from
the genome of the individual to the system-wide emergence of species
diversity. We study a set of evolving agents that interact competitively in a
closed system, in which both the dynamics of mutations and competitive
advantage emerge directly from interpreting a genome as the sides of a die.
The model demonstrates sympatric speciation where new species evolve from
existing ones while in contact with the entire ecosystem. Allowing free
mutations both in the genomes and the mutation rates, we find, in contrast to
hierarchical models of fitness, the emergence of a metastable state of finite
mutation rate and diversity.

##  I Introduction

Evolution is the optimization scheme of the biological realm: with the correct
initial conditions and ample time, random mutations and natural selection are
sufficient to ensure the emergence of highly complex organisms. But the nature
of what is being optimized is context-dependent. Even for a fixed environment,
evolution does not necessarily have a single goal. If we were to rerun the
“experiment” of evolution, it is virtually guaranteed this would result in
species distinct from those that are alive today.

Evolution of specific biological features of a species depend on properties of
the other species in its environment ehrlich1964butterflies ;
woolhouse2002biological . Such co-evolution is believed to be central to
describe evolution on the large scale KaufffmanBook ; Bak1992 ; Sneppen1993 ;
Weitz2005 ; Xue2017  as implied by the Red Queen hypothesis of Van Valen
van1973new ; liow2011red . Further, species interactions are not necessarily
ranked, as observed among corals, plants and microbes jackson1975alleopathy ;
taylor1990complex ; cameron2009parasite ; kirkup2004antibiotic ; Kerr2002 .
Many studies have been devoted to understanding and evolving such intransitive
interactions, ranging from molecular scale autocatalytic network
eigen1978hypercycle ; jain1998autocatalytic ; segre2000composing  to
extensions of the competitive game of rock, paper and scissors Kerr2002 ;
Laird2006 ; Reichenbach2006 ; Reichenbach2007 ; Mathiesen2011 ; Mitarai2012 ;
Dobrinevski2012 ; Ulrich2014 ; szolnoki2014cyclic ; Levine2017 ; Soliveres2018
.

Non-hierarchical species dynamics can readily be studied for a set of species
whose interactions are fixed. Here, in contrast, we are interested in evolving
systems where intransitive interactions emerge ex nihilo. We suggest a minimal
model for such a system consisting of individuals that interact by rolling
dice.

Figure 1: The intransitive interaction of five 9-sided proper dice. Direction
of the arrows indicate domination. The graph formed from the interactions has
two Hamiltonian paths, one of which is indicated in the background. In other
words, out of the $4!=24$ orderings of the dice, there are two that form an
intransitive loop of dominance.

##  II Model

We define the characteristics of an individual in our system by its genome as
given by a list of $n$ integers. We only consider competitive interactions
between species and settle these by interpreting the genome of integers as the
sides of dice that are rolled. The outcome of a fight is stochastic, but
certain dice will tend to out-compete other dice. For instance, a fight
between $A=(3,3,3,3,3,6)$ and $B=(2,2,2,5,5,5)$ will typically be won by $A$
despite $\sum_{i}A_{i}=\sum_{i}B_{i}$. The probability of $A$ winning in the
present example is $n^{-2}\sum_{i}\sum_{j}[A_{i}>B_{j}]=7/12$. What makes this
particular interaction interesting in the context of competing species is the
fact that we can introduce a species $C$ such that both $B\succ C$ and $C\succ
A$, or succinctly: $A\succ B\succ C\succ A$. This is for instance the case for
$C=(1,4,4,4,4,4)$. This intransitive behavior of dice is well-known
Finkelstein2006 ; Conrey2016a , but its applicability and simplicity for
modeling co-evolution are unexplored.

A plethora of systems could be designed around the above interaction rule. We
choose to consider one of the simplest and study $k$ individuals that interact
in a well-mixed scenario. At each time step of our simulation, we let each
individual randomly attack another. Two individuals, $X$ and $Y$, are
considered to belong to the same species if $\sum_{i}|X_{i}-Y_{i}|<\delta$, in
which case they will not fight. Otherwise, the losing individual of the
competition will be replaced by a copy of the winner. On ties, a random winner
is chosen.

Figure 2: Probability of intransitive (Hamiltonian) $k$-loops in sets of $k$
proper dice as a function of the number of sides $n$ on the dice. Here, proper
dice are sorted, has $\sum_{i}X_{i}=n(n+1)/2$ and $X_{i}\leq n$. A set of $k$
dice $\\{X_{i}\\}$ has a $k$-loop if an ordering $\sigma$ exists such that
$X_{\sigma_{1}}\prec X_{\sigma_{2}}\prec\cdots\prec X_{\sigma_{k}}\prec
X_{\sigma_{1}}$. Each point is the result of averaging over $100,\\!000$ Monte
Carlo samples with uniform probability for each valid die. Figure 3:
Speciation. (a–b) show $1,\\!500$ individuals of the system with links between
them if they consider one another the same species. (a) has $n=9$ and (b)
$n=14$, and in both cases $\mu=0.1$. Nodes are colored, for visualization
purposes only, using the Leiden community detection algorithm Traag2019  on
the undirected, unweighted graph. (c–e) show species richness $S$ (solid
lines) and diversity $D$ (dashed) stabilizing in a simulations of varying
genomic complexity $n$ and mutation rate $\mu$ as indicated by the legend.
Curves are averages over simulations with initial conditions of all dice
equaling the standard die. All simulation were run with $\delta=3$.
$k=10,\\!000$.

A crucial novelty of our model is that the dice interpretation not only sets
the rules for interactions but also naturally provides a genome space in which
mutations may occur. In our model, at each time step, each individual mutates
with probability $\mu$. A mutation event is the random change ($\pm 1$) of one
of its genome digits. As dice that are permutations of one another have
identical competitive advantages, we restrict our genome space to that of
ordered dice. Thus we disallow mutations that break the sorted nature of a
genome. For instance, $A$ may mutate to $(2,3,3,3,3,6)$ but not to
$(3,2,3,3,3,6)$. In effect this accelerates the dynamics of our system, as
most neutral competitions are avoided that would otherwise have to be settled
by stochastic extinction. Further, naturally, it is universally better for a
species to mutate up in sum rather than down. We set a fitness ceiling by only
allowing mutations that keep $\sum_{i}X_{i}\leq n\,(n+1)/2$ (the sum of the
standard die) and any $X_{i}\leq n$; the latter of which excludes a large of
set of dice that are typically less competitive and allows efficient
enumeration of the set of allowed dice.

In total, we have thus created a mutation model where universal fitness can be
measured, but where, occasionally, universally unfavorable mutations are
preferred to adapt to coexistent competitors.

Instead of defining the interaction as a single roll of the dice, one might
consider competitions of $r$ rounds. As $r\rightarrow\infty$, any slight
competitive advantage will result in certain overall wins. In this way, $r$
can be chosen to control the ruggedness of the fitness landscape KaufffmanBook
. Here, we limit our attention to $r=1$.

##  III Probability of intransitivity

Before delving into the dynamics of the model, it is useful to have a feeling
for the prevalence of intransitivity in random dice. Consider all dice such
that $\sum X_{i}=n(n+1)/2$ and $1\leq X_{i}\leq n$. Fig. 2 shows the
probability that $k$ such dice contains at least one intransitive
(Hamiltonian) $k$-loop as function of the number of sides of the dice $n$. The
plot shows the fact that if you choose $k=3$ random dice (with $n<15$ sides),
the probability that these dice interact intransitively is less than $\sim
20\%$. With a larger set of dice, not only do the probability of $k$-loops
increase, but so does probability of smaller sub-cycles [see SI paperSI  for a
version of Fig. 2 for loops of any size]. In Fig. 2 we only consider a small
number of dice, but very long intransitive cycles can also be found. For
instance, in the set of $k=910$ allowed $9$-sided dice, the longest possible
cycle is at least $891$ (finding the precise length is NP-hard).

Thus, intransitivity is by no means rare, but (short) loops are not the norm
either. However, in the dynamics of our model, it is much more unlikely for a
species to go extinct in an intransitive loop than when species interact in a
dominant manner. Thus we expect one of two things to happen in the long run:
the system will be taken over by one species or will be inhabited by a number
of species that interact intransitively and show oscillations.

There is also a large heterogeneity in the advantages of the different dice.
For instance, for $n=6$, the die that has an advantage over most other dice is
$(1,3,3,4,5,5)$, beating on average $\sim 60\,\%$ of the other dice. The worst
is $(1,2,2,4,6,6)$, which is better than only $\sim 40\,\%$. In contrast, the
standard die $(1,2,3,4,5,6)$ has precisely a $50\,\%$ chance of beating any
other proper die. Similar conclusions can be made for all $n$.

We note that the precise statistics of Fig. 2 would be different if we
considered not only proper dice, as some dice have many more permutations of
their sides than others. The qualitative conclusions drawn would remain
similar, nonetheless.

##  IV Finite mutation rate upholds species diversity

Random, well-mixed ecosystems of many competing species interacting under
demographic noise are unstable may1972will , and competitive exclusion often
leads to a collapse to only a single or a few surviving species
hardin1960competitive ; levin1970community ; haerter2016food . This,
naturally, also applies to the present dynamical system. However, since the
genome space of our model is not hierarchically organized, a finite mutation
rate can induce a perpetual co-evolutionary arms race.

One complication in counting the number of species in a system is due to the
fact that the network of individuals is very unlikely to organize into fully
connected components. This is a complication that is not unique to our system,
but indeed any problem related to speciation tucker2017guide . For each pair
of individuals, our genome distance rule specifies if they belong to the same
species. Denote by $C_{i}$ the number of individuals that individual $i$ is
considered the same species as (including $i$ itself). An effective measure
for species richness is then given by $S=\sum_{i}C_{i}^{-1}$. In the case of
fully connected species, with no overlap between them, this measure coincides
with simply counting the number of species. Likewise, we can define a species
diversity measure that also accounts for evenness as
$D=k^{2}\left(\sum_{i}C_{i}\right)^{-1}$. This is equal to the number of
species only if each species occupy the same fraction of the entire system and
thus small species contribute only negligibly to its value.

Fig. 3 shows that both the mutation rate $\mu$ and the genome complexity $n$
(dice size) set the number of species that a system of a certain size can
maintain. At low mutation rates, the system is dominated by a small cloud of
individuals that form a quasispecies eigen1989molecular , since, in this case,
a single species can be locally dominant and no individuals can escape this
local optimum at the low mutation rate.

At higher mutation rates, however, intransitive interactions appear and
oscillatory dynamics of a high diversity system emerges. The systems have
intransitive loops of many lengths, but the dynamics are dominated by short
cycles ($\lesssim 5$ in the systems studied here). This is demonstrated and
studied in the SI by considering the mean-field Lotka–Volterra equations of
the system. In detail, the system behaves oscillatory with a frequency that
remain relatively constant for multiple oscillation periods. On long time
scales, however, stochastic events can change the dynamics, such as when a
species in an intransitive loop stochastically goes extinct or when an
individual suddenly mutates to dominate the existent intransitive interactions
initiating a “punctuated equilibrium” event causing a sudden shift in
oscillation frequency.

Fig. 3(a–b) visualizes species connectivity in the steady-state ecosystems
that evolve from the dynamics of the model. In these graphs, an edge is drawn
between two dice if they consider each other to belong to the same species.
Despite starting with a single species, the system can evolve to one that has
many species that are genomically disconnected. While no single measure can
capture the complexity of these inter-species connections, running community
detection algorithms on these graphs tend to find a number of clusters in the
same order of magnitude as our richness $S$ and diversity $D$ measures. We
note that for comparisons with other system sizes $k$, the values of the
mutation rates should be rescaled accordingly, as the number of attempted
mutation events per step is $\sim\mu k$. Thus systems with a higher number of
individuals can support high diversity in spite of having a low mutation rate
per individual.

##  V Heterogeneous mutation rates

Since a finite mutation rate is needed to maintain a finite diversity, we have
an ecosystem collapse if the mutation rate is taken to zero. For instance, in
a system of purely hierarchically interacting species, the dominant species
will prefer a low mutation rate thus leading to a collapse of ecosystem
diversity. In the present system, however, there is no global optimum, and a
high mutations rate means quick adaptability and increases the chance of an
individual to out-compete others by an evolutionary advantage. A high mutation
rate is not strictly an advantage though, since it also means a high rate of
bad mutation events towards either locally or globally worse genomes.

In Fig. 4(inset), we show the competition between two populations with
$\mu_{A}=0.2$ and $\mu_{B}=0.001$, respectively. For early times, a low
mutation rate gives an advantage because there is a low rate of genomic decay
and thus we see population $B$ winning initially. This reflects the advantage
of localizing a population around a local fitness maximum over more diffuse
quasispecies at higher mutation rates eigen1989molecular . However, at some
point population $A$ finds a competitive advantage over the slowly adapting
population $B$ and annihilates the latter completely. Varying $\mu_{A}$ and
$\mu_{B}$, the average outcome of these scenarios is shown in the main part of
Fig. 4. The exact results depend on the initialization of the dice, but in
this case, we see that $\mu\approx 0.1$ is generally advantageous.

Figure 4: Annihilation statistics of two populations with different mutation
rates. Inset shows a simulation of $2\times 10,\\!000$ dice with one half
having $\mu_{A}=0.2$ and the other $\mu_{B}=0.001$. The time of annihilation
of population $B$ is $T=130$ time steps. Main plot shows the average of $1/T$
measured with a negative sign if $B$ wins. High values thus indicate that $B$
tends to be annihilated and negative values that $A$ tends to be annihilated.
Values near zero indicate a system dominated by stochastic extinction or one
where annihilation takes a very long time. All dice were initialized to
$(1,1,1,6,6,6)$ at the beginning of the simulations.  Figure 5: Mutating the
mutation rate. Simulations for which each individual mutates its mutation rate
with a 1 % chance of a 1 % change per time step (Gaussian multiplicative
noise). (abc) Simulations for $n=6,\,9,\,14$ showing the system average
mutation rate $\langle\mu\rangle$ as a function of time steps. Orange lines
show average over all simulations, whereas pink line shows an average over
only those simulations that at $t=10^{7}$ have $\langle\mu\rangle>10^{-3}$.
The full distribution of $\mu$ is narrowly peaked around $\langle\mu\rangle$
(see SI paperSI ). Inset in (c) shows, for each realization, the average
mutation rate versus the system diversity $D$ (dark curves) for $t>5\cdot
10^{5}$, with two trajectories emphasized, one of which collapses towards
$\langle\mu\rangle=0$. All simulations use $\delta=3$, but obtain similar
results for other $\delta$. The fraction of systems in the high mutation state
depends greatly on initial mutation rate, even for the $n=6$ die (see SI
paperSI ).

We complete the design of our model by, finally, also permitting mutations in
the mutation rate itself. In a hierarchical setting, this would lead to the
immediate collapse of both the mutation rates and diversity. Fig. 4 indicates,
however, that in the present system there is also an advantage to having a
finite mutation rate. Each panel in Fig. 5 show realizations of a system of
individuals, all initialized with the same genome and an initial mutation rate
of $\mu=0.01$. We observe two distinct outcomes: most trajectories reach a
metastable state with a high mutations rate (die size $n=6$ never reaches this
state, but would do so if we instead started with initial $\mu=0.1$, see SI
paperSI ), and some that decay towards zero mutation rates.

A characteristic of the metastable state is the rare but sudden decay events
of both mutation rate and diversity of the entire system [Fig. 5(b–c)]. As the
mutation rate decreases, it becomes less and less likely to escape the low
diversity situation thereby creating a positive feedback loop for decreasing
the mutation rate even further. Once collapsed, very large perturbations are
needed to bring the system out of this situation. Even changing the mutation
rate of half of the collapsed system to $\mu=0.1$ is typically not enough to
return to the high diversity state [see SI paperSI ]. In contrast, for the
surviving high diversity ecosystems the average number of mutations that
separate two random individuals is large. For instance,
$\langle\sum_{i}|X_{i}-Y_{i}|\rangle_{X,Y}\approx 12$ for $n=14$, which is
about $2/3$ of the average obtained between random dice.

##  VI Perspective

We have presented a theoretical model that, on one hand, is exceedingly simple
to define, and at the same time successfully describes a host of complex
phenomena related to co-evolution. At fixed, finite mutation rates, the model
permits a state of finite diversity in co-evolutionary balance. For
hierarchically interacting systems, allowing mutations in the mutation rates
themselves, will lead to an ecosystem collapse. In contrast, we find a
metastable state of finite diversity, whose stability increases quickly with
genomic complexity, measured by the number of sides of the dice $n$.

We only considered a well-mixed system, meaning that at all times each
individual could meet any other individual. Introducing space to the model,
e.g. putting the agents on a lattice, will most likely stabilize the observed
effects even further; with spatial dynamics, intransitive relations will decay
very slowly Mitarai2012 , and thus the rate of species extinctions decreases.
Furthermore, speciation events should increase in frequency as space allows
for transient allopatric speciation.

Precise quantification of intransitivity in the dynamical system is another
interesting avenue for further research: despite being dominated by few
intransitive cycles, a static view of the systems at any given time will not
reveal the dominance of these cycles as at least one species will have a low
population count due to the oscillatory Lotka–Volterra-like dynamics imposed
by the dynamics of the intransitive loops.

In conclusion, from the simple rules of competing dice emerge a natural
balance of mutation rates and diversity. Too high a mutation rate risks
genomic decay and the disintegration of quasispecies: “mutate and die”. Too
few mutations are disfavoured in analogy to the Red Queen hypothesis: “mutate
or die”.

###### Acknowledgements.

This project has received funding from the Novo Nordisk Foundation Grant
Agreement NNF20OC0062047 and from the European Research Council (ERC) under
the European Union’s Horizon 2020 Research and Innovation Programme, Grant
Agreement No. 740704.

## References

  * (1) Paul R Ehrlich and Peter H Raven.  Butterflies and plants: a study in coevolution.  Evolution, pages 586–608, 1964.
  * (2) Mark EJ Woolhouse, Joanne P Webster, Esteban Domingo, Brian Charlesworth, and Bruce R Levin.  Biological and biomedical implications of the co-evolution of pathogens and their hosts.  Nature genetics, 32(4):569–577, 2002.
  * (3) Stuart A Kauffmann.  The Origins of Order.  Oxford University Press, 1993.
  * (4) Per Bak, Henrik Flyvbjerg, and Benny Lautrup.  Coevolution in a rugged fitness landscape.  Physical Review A, 46(10):6724–6730, 1992.
  * (5) Per Bak and Kim Sneppen.  Punctuated Equilibribum and Criticality in a simple model of evolution.  Physical Review Letters, 71(24):4083–4086, 1993.
  * (6) J S Weitz, H Hartman, and S A Levin.  Coevolutionary arms races between bacteria and bacteriophage.  Proceedings of the National Academy of Sciences of the United States of America, 102(27):9535–9540, 2005.
  * (7) Chi Xue and Nigel Goldenfeld.  Coevolution Maintains Diversity in the Stochastic ”kill the Winner” Model.  Physical Review Letters, 119(26):1–5, 2017.
  * (8) Leigh Van Valen.  A new evolutionary law.  Evol theory, 1:1–30, 1973.
  * (9) Lee Hsiang Liow, Leigh Van Valen, and Nils Chr Stenseth.  Red queen: from populations to taxa and communities.  Trends in ecology & evolution, 26(7):349–358, 2011.
  * (10) JBC Jackson and LEO Buss.  Alleopathy and spatial competition among coral reef invertebrates.  Proceedings of the National Academy of Sciences, 72(12):5160–5163, 1975.
  * (11) Douglas R Taylor and Lonnie W Aarssen.  Complex competitive relationships among genotypes of three perennial grasses: implications for species coexistence.  The American Naturalist, 136(3):305–327, 1990.
  * (12) Duncan D Cameron, Andy White, and Janis Antonovics.  Parasite–grass–forb interactions and rock–paper–scissor dynamics: predicting the effects of the parasitic plant rhinanthus minor on host plant communities.  Journal of Ecology, 97(6):1311–1319, 2009.
  * (13) Benjamin C Kirkup and Margaret A Riley.  Antibiotic-mediated antagonism leads to a bacterial game of rock–paper–scissors in vivo.  Nature, 428(6981):412–414, 2004.
  * (14) Benjamin Kerr, Margaret A. Riley, Marcus W. Feldman, and Brendan J.M. Bohannan.  Local dispersal promotes biodiversity in a real-life game of rock-paper-scissors.  Nature, 418(6894):171–174, 2002.
  * (15) Manfred Eigen and Peter Schuster.  The hypercycle.  Naturwissenschaften, 65(1):7–41, 1978.
  * (16) Sanjay Jain and Sandeep Krishna.  Autocatalytic sets and the growth of complexity in an evolutionary model.  Physical Review Letters, 81(25):5684, 1998.
  * (17) Daniel Segré and Doron Lancet.  Composing life.  EMBO reports, 1(3):217–222, 2000.
  * (18) Robert A. Laird and Brandon S. Schamp.  Competitive intransitivity promotes species coexistence.  American Naturalist, 168(2):182–193, 2006.
  * (19) Tobias Reichenbach, Mauro Mobilia, and Erwin Frey.  Coexistence versus extinction in the stochastic cyclic Lotka-Volterra model.  Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 74(5):1–11, 2006.
  * (20) Tobias Reichenbach, Mauro Mobilia, and Erwin Frey.  Mobility promotes and jeopardizes biodiversity in rock-paper-scissors games.  Nature, 448(7157):1046–1049, 2007.
  * (21) Joachim Mathiesen, Namiko Mitarai, Kim Sneppen, and Ala Trusina.  Ecosystems with mutually exclusive interactions self-organize to a state of high diversity.  Physical Review Letters, 107(18):1–5, 2011.
  * (22) Namiko Mitarai, Joachim Mathiesen, and Kim Sneppen.  Emergence of diversity in a model ecosystem.  Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 86(1):1–6, 2012.
  * (23) Alexander Dobrinevski and Erwin Frey.  Extinction in neutrally stable stochastic Lotka-Volterra models.  Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 85(5):1–12, 2012.
  * (24) Werner Ulrich, Santiago Soliveres, Wojciech Kryszewski, Fernando T. Maestre, and Nicholas J. Gotelli.  Matrix models for quantifying competitive intransitivity from species abundance data.  Oikos, 123(9):1057–1070, 2014.
  * (25) Attila Szolnoki, Mauro Mobilia, Luo-Luo Jiang, Bartosz Szczesny, Alastair M Rucklidge, and Matjaž Perc.  Cyclic dominance in evolutionary games: a review.  Journal of the Royal Society Interface, 11(100):20140735, 2014.
  * (26) Jonathan M. Levine, Jordi Bascompte, Peter B. Adler, and Stefano Allesina.  Beyond pairwise mechanisms of species coexistence in complex communities.  Nature, 546(7656):56–64, 2017.
  * (27) Santiago Soliveres and Eric Allan.  Everything you always wanted to know about intransitive competition but were afraid to ask.  Journal of Ecology, 106(3):807–814, 2018.
  * (28) Mark Finkelstein and Edward O Thorp.  Nontransitive Dice With Equal Means *.  pages 1–24, 2006.
  * (29) Brian Conrey, James Gabbard, Katie Grant, Andrew Liu, and Kent E. Morrison.  Intransitive Dice.  Mathematics Magazine, 89(2):133–143, 2016.
  * (30) V. A. Traag, L. Waltman, and N. J. van Eck.  From Louvain to Leiden: guaranteeing well-connected communities.  Scientific Reports, 9(1):1–12, 2019.
  * (31) See supplemental material at [url] for more details.
  * (32) Robert M May.  Will a large complex system be stable?  Nature, 238(5364):413–414, 1972.
  * (33) Garrett Hardin.  The competitive exclusion principle: An idea that took a century to be born has implications in ecology, economics, and genetics.  science, 131(3409):1292–1297, 1960.
  * (34) Simon A Levin.  Community equilibria and stability, and an extension of the competitive exclusion principle.  The American Naturalist, 104(939):413–423, 1970.
  * (35) Jan O Haerter, Namiko Mitarai, and Kim Sneppen.  Food web assembly rules for generalized lotka-volterra equations.  PLoS computational biology, 12(2):e1004727, 2016.
  * (36) Caroline M Tucker, Marc W Cadotte, Silvia B Carvalho, T Jonathan Davies, Simon Ferrier, Susanne A Fritz, Rich Grenyer, Matthew R Helmus, Lanna S Jin, Arne O Mooers, et al.  A guide to phylogenetic metrics for conservation, community ecology and macroecology.  Biological Reviews, 92(2):698–715, 2017.
  * (37) Manfred Eigen, John McCaskill, and Peter Schuster.  The molecular quasi-species.  Advances in chemical physics, 75:149–263, 1989.