on $b$ and $\mathcal{L}(X_{\cdot})$ is less straightforward.

First observe that Young’s inequality still provides $b^{\mu}\in
C^{0}_{T}C^{\alpha}_{x}$, so that by Definition 5.3 the DDSDE is meaningful in
the classical integral sense. In order to check Assumption 3.12 for $b^{\mu}$
(which implies $X$ being strong), it remains to show that $b\ast\mu\in
C^{\tilde{\alpha}H}_{T}C^{0}_{x}$ for some $\tilde{\alpha}$ such that

$\displaystyle 1-\frac{1}{2H}<\tilde{\alpha}\leq\alpha.$

By addition and subtraction,
$b^{\mu}_{t}-b^{\mu}_{s}=(b_{t}-b_{s})\ast\mu_{t}+b_{s}\ast(\mu_{t}-\mu_{s})$;
by the hypothesis on $b,\,\mu,$ we can estimate the first term by

$\|(b_{t}-b_{s})\ast\mu_{s}\|_{C^{0}_{x}}\leq\|b_{t}-b_{s}\|_{L^{p}_{x}}\|\mu\|_{L^{\infty}_{T}L^{p^{\prime}}_{x}}\lesssim|t-s|^{\alpha
H}.$ (5.4)

Since $X$ is solution to the SDE
$X_{t}=\xi+\int_{0}^{t}b^{\mu}_{s}(X_{s})\mathrm{d}s+W_{t}$, for any
$r^{\prime}\in(1,\infty)$ it holds

$d_{r^{\prime}}(\mu_{t},\mu_{s})\leq\|X_{t}-X_{s}\|_{L^{r^{\prime}}_{\Omega}}\leq\|b\ast\mu^{i}\|_{L^{\infty}_{T}C^{0}_{x}}\,|t-s|+\|W_{t}-W_{s}\|_{L^{r^{\prime}}_{\Omega}}\lesssim_{r^{\prime},T}|t-s|^{H}.$

Now similarly to the case $H\leq 1/2$, choose $(s,r)\in(1,\infty)^{2}$ such
that

$\tilde{\alpha}:=\alpha-\frac{d}{s}>1-\frac{1}{2H},\quad
1+\frac{r}{s}\leq\frac{r}{p}+\frac{1}{p^{\prime}};$

Applying Corollary A.9 and the previous estimate for
$d_{r^{\prime}}(\mu_{t},\mu_{s})$, we then find

$\displaystyle\|b_{t}\ast(\mu^{i}_{t}-\mu^{i}_{s})\|_{B^{\tilde{\alpha}-1}_{\infty,\infty}}\lesssim
d_{r^{\prime}}(\mu_{t},\mu_{s})\lesssim|t-s|^{H}.$

On the other hand, since $b\in C^{0}_{T}B^{\alpha}_{p,p}$ and $\mu\in
L^{\infty}_{T}L^{p^{\prime}}_{x}$, by Young’s inequality, we have
$\|b_{t}\ast(\mu^{i}_{t}-\mu^{i}_{s})\|_{B^{\alpha}_{\infty,\infty}}\lesssim
1$. We can now interpolate between the two estimates: choose
$\theta=\tilde{\alpha}\in(0,1)$, so that
$\theta(\tilde{\alpha}-1)+(1-\theta)\alpha=(1-\tilde{\alpha})(\alpha-\tilde{\alpha})>0$,
then by the embedding $B^{\varepsilon}_{\infty,\infty}\hookrightarrow
C^{0}_{x}$ for $\varepsilon>0$, we obtain

$\displaystyle\|b_{t}\ast(\mu^{i}_{t}-\mu^{i}_{s})\|_{C^{0}_{x}}\lesssim\|b\ast(\mu^{i}_{t}-\mu^{i}_{s})\|_{B^{\alpha}_{\infty,\infty}}^{1-\tilde{\alpha}}\,\|b_{t}\ast(\mu^{i}_{t}-\mu^{i}_{s})\|_{B^{\tilde{\alpha}-1}_{\infty,\infty}}^{\tilde{\alpha}}\lesssim|t-s|^{\tilde{\alpha}H}.$

So we conclude that $b\ast\mu^{i}\in C^{0}_{T}C^{\tilde{\alpha}}_{x}\cap
C^{\tilde{\alpha}H}_{T}C^{0}_{x}$, where by construction
$\tilde{\alpha}>1-1/(2H)$.

The second part of the argument, concerning the comparison of two solutions
$X^{i}$ satisfying (5.2), now proceeds identically as in the case $H\leq 1/2$.
∎

###  5.1. Distributional kernels with bounded divergence

Proposition 5.4 reduces the problem of uniqueness of solutions (in a suitable
class) to that of establishing their regularity, in the sense of equation
(5.2).

One classical way to show that the condition $\mu_{0}\in L^{p^{\prime}}_{x}$
is propagated at positive times, which has been exploited systematically after
[12], is to impose boundedness of $\operatorname{div}b$; in the setting of
DDSDEs with general additive noise and regular drift $b$, an analogous
statement can be found in [15, Proposition 4.3].

######  Proposition 5.5.

Let $H\in(0,1)$, $q\in(2,\infty]$, $p\in[1,\infty)$, $p^{\prime}$ its
conjugate exponent. Let $b$ be a distributional drift such that
$\operatorname{div}b\in L^{1}_{T}L^{\infty}_{x}$ and either:

  1. i.

If $H>1/2$, then $b\in C^{\alpha H}_{T}L^{p}_{x}\cap
C^{0}_{T}B^{\alpha}_{p,p}$ for some $\alpha>1-\frac{1}{2H}$.

  2. ii.

If $H\leq 1/2$, then $b\in L^{q}_{T}B^{\alpha}_{p,p}$ for some
$\alpha>1-\frac{1}{2H}+\frac{1}{Hq}$.

Then for any $\mu_{0}\in L^{p^{\prime}}_{x}$ there exists a strong solution to
the DDSDE (5.1) associated to $(\mu_{0},b)$, which moreover satisfies
$\mathcal{L}(X_{\cdot})\in L^{\infty}_{T}L^{p^{\prime}}_{x}$.

###### Proof.

We start by dealing with the case $H\leq 1/2$; at the end of the proof we
explain how the reasoning needs to be modified for $H>1/2$.

The case $H\leq 1/2$. In this case we can assume w.l.o.g. $q<\infty$; recall
that if $f^{n}$ is a bounded sequence in
$L^{q}_{T}B^{\alpha}_{\infty,\infty}$, for $(\alpha,q)$ satisfying Assumption
3.12 such that $f^{n}\to f$ in $L^{q}_{T}B^{\alpha-1}_{\infty,\infty}$, then
by Corollary 3.17 the associated solutions

$X^{n}_{t}=\xi+\int_{0}^{t}f^{n}(X^{n}_{s})\,\mathrm{d}s+W_{t}$

converge to the unique strong solution $X$ of the SDE associated to
$(\xi,W,f)$ (we can assume $\\{X^{n}\\}_{n\geq 1}$ and $X$ to be defined on
the same probability space for the same $(\xi,W)$).

Given $b$ as in the hypothesis, consider a sequence of smooth, bounded
functions $b^{n}$ such that $b^{n}\to b$ in $L^{q}_{T}B^{\alpha}_{p,p}$ with
$\|\operatorname{div}b^{n}\|_{L^{1}_{T}L^{\infty}_{x}}\leq\|\operatorname{div}b\|_{L^{1}_{T}L^{\infty}_{x}}$;
let $X^{n}$ be the solutions to

$X^{n}_{t}=\xi+\int_{0}^{t}b^{n}\ast\mathcal{L}(X^{n}_{s})(X^{n}_{s})\mathrm{d}s+W_{t},$

whose existence is granted by classical results (see e.g. [10, Theorem 7]) and
set $\mu^{n}_{t}=\mathcal{L}(X^{n}_{t})$. By [15, Proposition 4.3], there
exists $C=C(\|\operatorname{div}b\|_{L^{1}_{T}L^{\infty}_{x}})>0$ such that

$\displaystyle\sup_{n\in\mathbb{N}}\sup_{t\in[0,T]}\|\mu^{n}_{t}\|_{L^{p^{\prime}}_{x}}\leq
C\|\mu_{0}\|_{L^{p^{\prime}}_{x}}<\infty.$

As a consequence, each $X^{n}$ solves an SDE with drift
$f^{n}=b^{n}\ast\mu^{n}$ satisfying

$\sup_{n\in\mathbb{N}}\|f^{n}\|_{L^{q}_{T}B^{\alpha}_{\infty,\infty}}\lesssim\sup_{n\in\mathbb{N}}\|b^{n}\|_{L^{q}_{T}B^{\alpha}_{p,p}}\,\sup_{n\in\mathbb{N}}\sup_{t\in[0,T]}\|\mu^{n}_{t}\|_{L^{p^{\prime}}_{x}}\lesssim
C\|b\|_{L^{q}_{T}B^{\alpha}_{p,p}}\|\mu_{0}\|_{L^{p^{\prime}}_{x}}<\infty.$

In turn this implies by Remark 3.19 that for any fixed $\varepsilon>0$ we have
the uniform estimate $\sup_{n}\mathbb{E}[\llbracket
X^{n}\rrbracket_{C^{H-\varepsilon}}]<\infty$; since moreover $X^{n}_{0}=\xi$
for all $n\in\mathbb{N}$, we can conclude by Ascoli–Arzelà that the sequence
$\\{X^{n}\\}_{n\geq 1}$ is tight in $C_{T}$. We can then extract a (not
relabelled) subsequence such that $\mathcal{L}(X^{n})$ converge weakly to some
$\mu\in\mathcal{P}(C_{T})$; consequently $\mu^{n}_{t}\rightharpoonup\mu_{t}$
in $\mathcal{P}(\mathbb{R}^{d})$ for any $t\in[0,T]$, where
$\mu_{t}=e_{t}\sharp\mu$ and $e_{t}:C_{T}\rightarrow\mathbb{R}^{d}$ is the
evaluation map. It follows from the uniform estimates that
$\|\mu_{t}\|_{L^{p^{\prime}}_{x}}\leq C\|\mu_{0}\|_{L^{p^{\prime}}}$ as well.

We claim that the drifts $f^{n}_{t}=b^{n}_{t}\ast\mu^{n}_{t}$ converge to
$f_{t}:=b_{t}\ast\mu_{t}$ in $L^{q}_{T}B^{\alpha-1}_{\infty,\infty}$. Once
this is shown, by the initial observation the solutions $X^{n}$ must converge
to the unique solution $X$ associated to $(\xi,W,f)$; then it must hold
$\mathcal{L}(X_{t})=\mu_{t}$, $f_{t}=b_{t}\ast\mathcal{L}(X_{t})$ and so we
can conclude that $X$ is a solution to (5.1) with the desired regularity.

It remains to show the claim; to this end, we set

$f^{n}_{t}-f_{t}=b_{t}^{n}\ast(\mu^{n}_{t}-\mu_{t})+(b_{t}^{n}-b_{t})\ast\mu_{t}=:g^{n}_{t}+h^{n}_{t}.$

By Corollary A.5 in Appendix A, $g^{n}_{t}\to 0$ in
$B^{\tilde{\alpha}}_{\infty,\infty}$ for all $\tilde{\alpha}<\alpha$ and a.e.
$t\in[0,T]$; the bound
$|g^{n}_{t}|\leq\|b_{t}\|_{B^{\alpha}_{p,p}}(\|\mu^{n}_{t}\|_{L^{p^{\prime}}_{x}}+\|\mu_{t}\|_{L^{p^{\prime}}_{x}})$
and dominated convergence imply that $g^{n}\rightarrow g$ in
$L^{q}_{T}B^{\tilde{\alpha}}_{\infty,\infty}$ for all $\tilde{\alpha}<\alpha$.
For $h^{n}$ we have the estimate

$\lim_{n\to\infty}\|h^{n}\|_{L^{q}_{T}B^{\alpha}_{\infty.\infty}}\leq\sup_{t\in[0,T]}\|\mu_{t}\|_{L^{p^{\prime}}_{x}}\,\lim_{n\to\infty}\|b^{n}-b\|_{L^{q}_{T}B^{\alpha}_{p,p}}=0.$

Hence we have shown the claim and thus the conclusion in this case.

The case $H>1/2$. As in the proof of Proposition 5.4, in this regime
$\mathcal{L}(X_{\cdot})\in L^{\infty}_{T}L^{p^{\prime}}_{x}$ is not enough to
deduce straightaway that $b\ast\mathcal{L}(X_{\cdot})$ satisfies Assumption
3.12; however up to technical details, the proof is almost the same as above.

Specifically, we can consider a sequence $\\{b^{n}\\}_{n}$ of smooth
functions, uniformly bounded in $C^{\alpha H}_{T}L^{p}_{x}\cap
C^{0}_{T}B^{\alpha}_{p,p}$, with $\operatorname{div}b^{n}$ uniformly bounded
in $L^{1}_{T}L^{\infty}_{x}$ and such that $b^{n}\to b$ in
$L^{q}_{T}B^{\alpha}_{p,p}$ for any $q<\infty$. Then exploiting the a priori
bound from [15, Proposition 4.3] and the argument from Proposition 5.4, one
can derive uniform estimates for the solutions $X^{n}$ associated to $X$ and
finally pass to the limit with the help of Corollary 3.17.

Alternatively, let us mention that the existence of a weak solution $X$
satisfying $\mathcal{L}(X_{\cdot})\in L^{\infty}_{T}L^{p^{\prime}}_{x}$ in
this setting can be obtained by an application of [15, Proposition 4.4]. ∎

###### Proof of Theorem 2.6.

It is now an immediate consequence of Propositions 5.4 and 5.5. ∎

###  5.2. Integrable kernels

We now restrict ourselves to the case $H\leq 1/2$ and drifts $b\in
L^{q}_{T}L^{p}_{x}$; in this setting we can present a second route to
establishing existence of a solution with sufficiently regular law, to which
we can apply Proposition 5.4.

Before proceeding further, let us explain why it is reasonable to expect so.
By the Besov embedding $L^{p}_{t}\hookrightarrow B^{-d/p}_{\infty,\infty}$,
drifts $b\in L^{q}_{T}L^{p}_{x}$ satisfy Assumption 3.12 if and only if

$\frac{1}{q}+\frac{Hd}{p}<\frac{1}{2}-H;$ (5.5)

however, differently from the class $L^{q}_{T}B^{\alpha}_{p,p}$, for $b\in
L^{q}_{T}L^{p}_{x}$ it is known after the works [36, 31] that Girsanov
transform (and thus weak existence and uniqueness in law for associated SDEs)
is available as soon as

$\frac{1}{q}+\frac{Hd}{p}<\frac{1}{2}.$ (5.6)

As already seen in Section 3.4, Girsanov transform allows to deduce
information on the regularity of $\mathcal{L}(X_{t})$, which in turn provides
higher regularity of the effective drift $b^{\mu}$ for the convolutional
DDSDE. In particular, we may hope that starting from $b\in L^{q}_{T}L^{p}_{x}$
for $(q,p)$ satisfying (5.6), we end up with $b^{\mu}\in
L^{\tilde{q}}_{T}L^{\tilde{p}}_{x}$ with $(\tilde{q},\tilde{p})$ satisfying
(5.5).

At a technical level, we will proceed similarly as in Section 5.1, first
establishing uniform a priori estimates for regular $b$ and then running an
approximation procedure. We start by establishing the recalling and improving
the available results on Girsanov transform; as we are only interested in
smooth approximations, for simplicity we restrict to regular drifts.

######  Lemma 5.6.

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, $(\xi,W)$ a
$\mathbb{R}^{d}\times C_{T}$-valued r.v. on it with
$\mathcal{L}_{\mathbb{P}}(\xi,W)=\mu_{0}\otimes\mu^{H}$ for some $H\leq 1/2$
and let $f:[0,T]\times\mathbb{R}^{d}\to\mathbb{R}^{d}$ be a globally Lipschitz
drift, $f\in L^{q}_{T}L^{p}_{x}$ for parameters $(p,q)\in[1,\infty]^{2}$
satisfying (5.6); let $X$ be the unique strong solution to

$X_{t}=\xi+\int_{0}^{t}f_{s}(X_{s})\mathop{}\\!\mathrm{d}s+W_{t}\quad\forall\,t\in[0,T].$

Then there exists a measure $\mathbb{Q}$ equivalent to $\mathbb{P}$ such that
$\mathcal{L}_{\mathbb{Q}}(X)=\mathcal{L}_{\mathbb{P}}(\xi+W)$ and there exists
an increasing function $F$, depending on $H,T,p,q$, such that

$\mathbb{E}_{\mathbb{Q}}\bigg{[}\Big{(}\frac{\mathop{}\\!\mathrm{d}\mathbb{P}}{\mathop{}\\!\mathrm{d}\mathbb{Q}}\Big{)}^{n}\bigg{]}+\mathbb{E}_{\mathbb{Q}}\bigg{[}\Big{(}\frac{\mathop{}\\!\mathrm{d}\mathbb{Q}}{\mathop{}\\!\mathrm{d}\mathbb{P}}\Big{)}^{n}\bigg{]}\leq
F(n,\|f\|_{L^{q}_{T}L^{p}_{x}})<\infty\quad\forall\,n\in\mathbb{N}$

where the estimate does not depend on $\mu_{0}$ nor the specific function $f$.

###### Proof.

For deterministic initial data $\xi=x_{0}\in\mathbb{R}^{d}$ (equiv.
$\mu_{0}=\delta_{x_{0}}$), the statement is a direct consequence of [31, Lemma
6.7], where it is already stressed that the estimates only depend on
$\|f\|_{L^{q}_{T}L^{p}_{x}}$ but not on $x_{0}$ nor the specific $f$. The
proof for random initial data $\xi$ independent of $W$ is now identical to
that of Corollary 3.18; the estimate not depending on $\xi$ follows from the
property that
$\|f(x_{0}+\cdot)\|_{L^{q}_{T}L^{p}_{x}}=\|f\|_{L^{q}_{T}L^{p}_{x}}$ for all
$x_{0}\in\mathbb{R}^{d}$. ∎

The next lemma shows that the initial regularity of $\mu_{0}$ is propagated at
positive times, establishing useful a priori estimates; the proof is similar
to that of Proposition 3.21.

######  Lemma 5.7.

Let $\xi,\,W,\,X,\,f,\,(p,q)$ be as in Lemma 5.6 and assume $\mu_{0}\in
L^{r}_{x}$ for some $r\in(1,\infty)$; then

$\displaystyle\sup_{t\in[0,T]}\|\mathcal{L}_{\mathbb{P}}(X_{t})\|_{L^{\tilde{r}}_{x}}<\infty\quad\forall\,\tilde{r}\in(1,r).$

###### Proof.

Fix $\tilde{r}<r$ and denote by $\tilde{r}^{\prime}$ the conjugate exponent of
$\tilde{r}$; take $\varepsilon>0$ such that
$r^{\prime}(1+\varepsilon)=\tilde{r}^{\prime}$. Let $\mathbb{Q}$ be the
measure given by Lemma 5.6 such that
$\mathcal{L}_{\mathbb{Q}}(X)=\mathcal{L}_{\mathbb{P}}(\xi+W)$; since
$\mathrm{d}\mathbb{P}/\mathrm{d}\mathbb{Q}$ admits moments of any order, for
any $g\in C^{\infty}_{c}(\mathbb{R}^{d})$, by Hölder

$\displaystyle|\langle g,\mathcal{L}(X_{t})\rangle|$
$\displaystyle\leq\mathbb{E}_{\mathbb{P}}[|g|(X_{t})]=\mathbb{E}_{\mathbb{Q}}\bigg{[}|g|(\xi+W_{t})\,\frac{\mathrm{d}\mathbb{P}}{\mathrm{d}\mathbb{Q}}\bigg{]}$
$\displaystyle\leq\mathbb{E}_{\mathbb{Q}}\big{[}|g|^{1+\varepsilon}(\xi+W_{t})\big{]}^{\frac{1}{1+\varepsilon}}\,\mathbb{E}_{\mathbb{Q}}\bigg{[}\Big{(}\frac{\mathrm{d}\mathbb{P}}{\mathrm{d}\mathbb{Q}}\Big{)}^{1+\frac{1}{\varepsilon}}\bigg{]}^{\frac{\varepsilon}{1+\varepsilon}}$
$\displaystyle\lesssim_{\varepsilon}\langle|g|^{1+\varepsilon},\mu_{0}\ast\mathcal{L}(W_{t})\rangle^{\frac{1}{1+\varepsilon}}$

where in the last passage we used the fact that $\xi$ and $W_{t}$ are
independent. Recalling that $\mathcal{L}(W_{t})$ is a probability measure, by
Hölder’s and then Young’s inequality we arrive at

$|\langle
g,\mathcal{L}(X_{t})\rangle|\lesssim_{\varepsilon}\||g|^{1+\varepsilon}\|_{L^{r^{\prime}}_{x}}^{\frac{1}{1+\varepsilon}}\,\|\rho\ast\mathcal{L}(W_{t})\|_{L^{r}_{x}}^{\frac{1}{1+\varepsilon}}\lesssim\|g\|_{L^{r^{\prime}(1+\varepsilon)}_{x}}\,\|\mu_{0}\|_{L^{r}_{x}}^{\frac{1}{1+\varepsilon}}=\|g\|_{L^{\tilde{r}^{\prime}}_{x}}\,\|\mu_{0}\|_{L^{r}_{x}}^{\frac{1}{1+\varepsilon}}$

As the estimate is uniform over all $g\in C^{\infty}_{c}(\mathbb{R}^{d})$ and
$t\in[0,T]$, by duality we deduce that

$\sup_{t\in[0,T]}\|\mathcal{L}(X_{t})\|_{L^{\tilde{r}}_{x}}\lesssim_{\varepsilon}\|\mu_{0}\|_{L^{r}_{x}}^{\frac{1}{1+\varepsilon}};$

as the reasoning holds for all $\tilde{r}<r$, the conclusion follows. ∎

We are now ready to prove the existence of solutions to the DDSDE (5.1) for
$b\in L^{q}_{T}L^{p}_{x}$ and sufficiently integrable $\mu_{0}$.

######  Proposition 5.8.

Let $H\leq 1/2$, $(p,r,q)\in[1,\infty)\times(1,\infty)\times(2,\infty]$ such
that

$\frac{1}{q}+Hd\bigg{(}\frac{1}{p}+\frac{1}{r}-1\bigg{)}<\frac{1}{2}-H,\quad\frac{1}{q}+\frac{Hd}{p}<\frac{1}{2}.$
(5.7)

Then for any $b\in L^{q}_{T}L^{p}_{x}$ and any $\mu_{0}\in L^{r}_{x}$ there
exists a strong solution $X$ to the associated DDSDE (5.1), which moreover
satisfies $\mathcal{L}(X_{\cdot})\in L^{\infty}_{T}L^{\tilde{r}}_{x}$ for any
$\tilde{r}\in[1,r)$.

###### Proof.

We pursue the same general strategy as in the proof of Proposition 5.5.

As condition (5.7) only contains strict inequalities, w.l.o.g. we can assume
$q<\infty$; consider a sequence $\\{b^{n}\\}_{n}$ of Lipschitz, compactly
supported functions such that $b^{n}\to b$ in $L^{q}_{T}L^{p}_{x}$ and
$\|b^{n}\|_{L^{q}_{T}L^{p}_{x}}\leq\|b\|_{L^{q}_{T}L^{p}_{x}}$. It follows
from [10, Theorem 7] that for every $n$ there exists a unique solution $X^{n}$
to the approximating DDSDE

$X^{n}_{t}=\xi+\int_{0}^{t}(b^{n}_{s}\ast\mu^{n}_{s})(X^{n}_{s})\,\mathop{}\\!\mathrm{d}s+W_{t},\quad\mu^{n}_{t}=\mathcal{L}(X^{n}_{t}).$

In particular, each $X^{n}$ is also a solution to an SDE with drift
$f^{n}_{t}:=b^{n}_{t}\ast\mathcal{L}(X^{n}_{t})$ and by Young’s inequality

$\displaystyle\sup_{n}\|f^{n}\|_{L^{q}_{T}L^{p}_{x}}\leq\sup_{n}\|b^{n}\|_{L^{q}_{T}L^{p}_{x}}\leq\|b\|_{L^{q}_{T}L^{p}_{x}}.$

Therefore we may apply Lemmas 5.6 and 5.7 to obtain the uniform bound

$\sup_{n}\|\mu^{n}_{\cdot}\|_{L^{\infty}_{T}L^{\tilde{r}}_{x}}<\infty$

for all $\tilde{r}\in[1,r)$. Applying Hölder’s inequality to the integral in
time and and Young’s inequalities to the convolution in space, we find

$\sup_{n}\|b^{n}\ast\mu^{n}\|_{L^{q}_{T}L^{\tilde{p}}_{x}}<\infty$

for any $\tilde{p}<\bar{p}$, where

$1+\frac{1}{\bar{p}}=\frac{1}{r}+\frac{1}{p}.$

Using the fact that $\tilde{p}$ can be chosen arbitratrily close to $\bar{p}$
and that the first inequality in (5.7) is strict, we see that the family
$\\{b^{n}\ast\mu^{n}\\}$ is bounded in $L^{q}_{T}L^{\tilde{p}}_{x}$ for
parameters $(q,\tilde{p})$ satisfying

$\frac{1}{q}+\frac{Hd}{\tilde{p}}<\frac{1}{2}-H;$

but this is exactly condition (5.5), i.e. the regularity regime in which we
know how to solve the SDE in a strong sense. On the other hand, the uniform
bound for $\|b^{n}\|_{L^{q}_{T}L^{p}_{x}}$ and the use of Girsanov transform
allows to derive a uniform bound for $\mathbb{E}[\llbracket
X^{n}\rrbracket_{H-\varepsilon}]$ for any $\varepsilon>0$; together with
$X^{n}_{0}=\xi$ for all $n$ this implies tightness of $\\{X^{n}\\}_{n}$, so
that we can extract a (not relabelled) subsequence such that
$\mathcal{L}(X^{n})\rightharpoonup\mu$ in $\mathcal{P}(C_{T})$,
$\mu^{n}_{t}\rightharpoonup\mu_{t}=e_{t}\sharp\mu$ for all $t\in[0,T]$.

From here, the argument is almost identical to that of Proposition 5.5: once
we show that $b^{n}\ast\mu^{n}\to b\ast\mu$ in a sufficiently strong topology,
then by Corollary 3.17 the solutions $X^{n}$ will converge to the unique
strong solution $X$ associated to $(b\ast\mu,\xi)$, which must therefore be a
solution to the DDSDE associated to $\mu_{0}=\mathcal{L}(\xi)$ and $b$. By the
uniform bounds on $\\{\mu^{n}\\}_{n}$ and weak convergence
$\mu^{n}_{t}\rightharpoonup\mu_{t}=\mathcal{L}(X_{t})$, we also deduce that
$\mathcal{L}(X_{\cdot})\in L^{\infty}_{T}L^{\tilde{r}}_{x}$ for all
$\tilde{r}<r$.

Since by construction $b^{n}\to b$ in $L^{q}_{t}L^{p}_{x}$,
$\mu^{n}_{t}\rightharpoonup\mu_{t}$ and $\\{b^{n}\ast\mu^{n}\\}_{n}$ is
bounded in $L^{q}_{T}L^{\tilde{p}}_{x}$ for some $(q,\tilde{p})$ satisfying
(5.5), we can apply Corollary A.6 from Appendix A to deduce that (up to
further relabelling $\tilde{p}-\varepsilon$ into $\tilde{p}$)
$b^{n}\ast\mu^{n}\to b\ast\mu$ in $L^{q}_{T}L^{\tilde{p}}_{x}$. In light of
the embedding $L^{q}_{T}L^{\tilde{p}}_{x}\hookrightarrow
L^{q}_{T}B^{-d/\tilde{p}}_{\infty,\infty}$ and Corollary 3.17, this implies
the conclusion. ∎

We are now ready to prove the main result of this subsection.

######  Theorem 5.9.

Let $H\leq 1/2$, $(p,r,q)\in[1,\infty)\times(1,\infty)\times(2,\infty]$
satisfy (5.7). Then for any $b\in L^{q}_{T}L^{p}_{x}$ and $\mu_{0}\in
L^{r}_{x}$, then there exists a strong solution $X$ to (5.1), which satisfies
$\mathcal{L}(X_{\cdot})\in L^{\infty}_{T}L^{\tilde{r}}_{x}$ for any
$\tilde{r}<r$; pathwise uniqueness and uniqueness in law hold in the class of
solutions satisfying this condition.

###### Proof.

The proof is based on a (non-trivial) combination of Propositions 5.4 and 5.8.

Under our assumptions, the existence of a strong solution such that
$\mathcal{L}(X_{\cdot})\in L^{\infty}_{T}L^{\tilde{r}}_{x}$ for any
$\tilde{r}<r$ is granted; in particular if $r>p^{\prime}$, then we can choose
$\tilde{r}=p^{\prime}$ and then assumptions of Proposition 5.5 in this case
are satisfied thanks to the embedding $L^{q}_{T}L^{p}_{x}\hookrightarrow
L^{q}_{T}B^{-\varepsilon}_{p,p}$ for any $\varepsilon>0$, giving the
uniqueness part of the statement. Up to technicalities the borderline case
$r=p^{\prime}$ can be treated similarly, exploiting the embedding
$L^{q}_{T}L^{p}_{x}\hookrightarrow
L^{q}_{T}B^{-\varepsilon}_{\tilde{p},\tilde{p}}$ for some
$\tilde{p}=\tilde{p}(\varepsilon)>p$ chosen so that
$1/\tilde{r}+1/\tilde{p}=1$.

Thus it remains to study the regime $r<p^{\prime}$, equivalently
$r^{\prime}>p$; in this case we can choose $\tilde{r}<r$ such that
$\tilde{r}^{\prime}>p$ as well. By Besov embedding it then holds

$\displaystyle L^{q}_{T}L^{p}_{x}\hookrightarrow
B^{\alpha}_{\tilde{r}^{\prime},\tilde{r}^{\prime}}\quad\text{for}\quad\alpha:=-d\bigg{(}\frac{1}{p}-\frac{1}{\tilde{r}^{\prime}}\bigg{)}=-d\bigg{(}\frac{1}{p}+\frac{1}{\tilde{r}}-1\bigg{)};$

to verify that $b$ satisfies the assumptions of Proposition 5.4, it then
suffices to check that

$\displaystyle\frac{1}{q}+Hd\bigg{(}\frac{1}{p}+\frac{1}{\tilde{r}}-1\bigg{)}<\frac{1}{2}-H;$

since $\tilde{r}$ can be taken arbitrarily close to $r$, this follows from the
first strict inequality in (5.7). ∎

######  Remark 5.10.

For $r>d/(d-1)$, condition (5.6) implies condition (5.7). Therefore Theorem
2.7 is a particular subcase of Theorem 5.9.

##  Appendix A Some useful lemmas

We collect in this appendix several technical lemmas of analytic nature that
have been used throughout the paper. We start with useful facts on the
identification of elements of dual spaces.

######  Lemma A.1.

Suppose $f\in L^{1}_{loc}([0,T]\times\mathbb{R}^{d})$, $q\in[1,\infty)$ and
there exists a constant $C$ such that

$\int_{[0,T]\times\mathbb{R}^{d}}f(t,x)\varphi(t,x)\,\mathop{}\\!\mathrm{d}t\mathop{}\\!\mathrm{d}x\leq
C\|\varphi\|_{L^{q^{\prime}}_{T}L^{\infty}_{x}}$

for all compactly supported $\varphi\in
L^{\infty}([0,T]\times\mathbb{R}^{d})$. Then $f\in L^{q}_{T}L^{1}_{x}$ and
$\|f\|_{L^{q}_{T}L^{1}_{x}}\leq C$.

###### Proof.

Fix $M>0$ and set

$g_{t}:=\int_{\mathbb{R}^{d}}|f(t,x)|\mathbbm{1}_{|f(t,\cdot)|\leq
M}(x)\,\mathbbm{1}_{|x|\leq M}\mathop{}\\!\mathrm{d}x,\quad
h(t,x):=g_{t}^{q-1}\text{sgn}(f(t,x))\mathbbm{1}_{|f(t,\cdot)|\leq
M}(x)\,\mathbbm{1}_{|x|\leq M};$

then by assumption

$\int_{[0,T]}|g_{t}|^{q}\mathop{}\\!\mathrm{d}t=\int_{[0,T]\times\mathbb{R}^{d}}f(t,x)h(t,x)\,\mathop{}\\!\mathrm{d}t\mathop{}\\!\mathrm{d}x\leq
C\|h\|_{L^{q^{\prime}}_{t}L^{\infty}_{x}}=C\bigg{(}\int_{[0,T]}|g_{t}|^{q}\mathop{}\\!\mathrm{d}t\bigg{)}^{1-1/q},$

namely

$\Bigg{(}\int_{[0,T]}\bigg{(}\int_{\mathbb{R}^{d}}|f(t,x)|\mathbbm{1}_{|f(t,\cdot)|\leq
M}(x)\,\mathbbm{1}_{|x|\leq
M}\mathop{}\\!\mathrm{d}x\bigg{)}^{q}\mathop{}\\!\mathrm{d}t\Bigg{)}^{1/q}\leq
C.$

Taking the limit $M\to\infty$ gives the conclusion. ∎

######  Lemma A.2.

Suppose $f\in L^{1}_{loc}([0,T]\times\mathbb{R}^{d})$, $q\in(1,\infty)$ and
there exists a constant $C$ such that

$\int_{[0,T]\times\mathbb{R}^{d}}f(t,x)\varphi(t,x)\,\mathop{}\\!\mathrm{d}t\mathop{}\\!\mathrm{d}x\leq
C\|\varphi\|_{L^{q^{\prime}}_{T}L^{1}_{x}}$

for all compactly supported $\varphi\in
L^{\infty}([0,T]\times\mathbb{R}^{d})$. Then $f\in L^{q}_{T}L^{\infty}_{x}$
and $\|f\|_{L^{q}_{T}L^{\infty}_{x}}\leq C$.

###### Proof.

Fix $R>0$ and set $D=\\{x\in\mathbb{R}^{d}:|x|\leq R\\}$; we can identify
$L^{p}_{D}:=L^{p}(D)$ as the subset of $L^{p}(\mathbb{R}^{d})$ made of
functions supported on $D$, similarly $L^{q}_{T}L^{p}_{D}$ as a subset of
$L^{q}_{T}L^{p}_{x}$. Observe that for $p>1$,
$L^{q^{\prime}}_{T}L^{p}_{D}\hookrightarrow L^{q^{\prime}}_{T}L^{1}_{D}$ with
$\|f\|_{L^{q^{\prime}}_{T}L^{1}_{D}}\leq(c_{d}R^{d})^{1/p^{\prime}}\|f\|_{L^{q^{\prime}}_{T}L^{p}_{D}}$.
By hypothesis $f$ defines an element of the dual of
$L^{q^{\prime}}_{T}L^{p}_{D}$, thus by duality

$\|f\|_{L^{q}_{T}L^{p^{\prime}}_{D}}\leq
C(c_{d}R^{d})^{1/p^{\prime}}\quad\forall p^{\prime}\in(1,\infty).$

Since $D$ has finite measure,
$\|g\|_{L^{\infty}_{D}}=\lim_{p\to\infty}\|g\|_{L^{p}_{D}}$ for all measurable
$g$; taking $p^{\prime}\to\infty$ in the above by Fatou’s lemma we obtain

$\|f\|_{L^{q}_{T}L^{\infty}_{D}}\leq C;$

as the reasoning holds for any $R>0$, the conclusion follows talking
$R\to\infty$. ∎

The next statements concern the compactness properties of convolutions,
specifically how weak convergence of measures is enhanced to strong
convergence of associated functionals $b\ast\mu$.

######  Lemma A.3.

Let $b\in L^{p}_{x}$ with $p\in[1,\infty)$ and
$\\{\mu^{n}\\}_{n}\subset\mathcal{P}(\mathbb{R}^{d})$ such that
$\mu^{n}\rightharpoonup\mu$ weakly, then

$\displaystyle\lim_{n\to\infty}\|b\ast\mu^{n}-b\ast\mu\|_{L^{p}_{x}}=0.$

If moreover $\\{\mu^{n}\\}_{n}$ is bounded in $L^{r}_{x}$ with $r>1$, then
$b\ast\mu^{i}\to b\ast\mu$ in $L^{\tilde{p}}_{x}$ for any
$\tilde{p}\in[p,\infty)$ s.t.

$1+\frac{1}{\tilde{p}}>\frac{1}{p}+\frac{1}{r}.$ (A.1)

###### Proof.

Given $h\in\mathbb{R}^{d}$, define the translation operator $\tau_{h}:f\mapsto
f(\cdot+h)$ acting on $L^{p}_{x}$. Recall that any given $b\in L^{p}_{x}$ is
equicontinuous, in the sense that $\tau_{h^{n}}b\to\tau_{h}b$ in $L^{p}_{x}$
for $h^{n}\to h$ in $\mathbb{R}^{d}$.

Since $\mu^{n}\rightharpoonup\mu$, by Skorokhod’s representation theorem we
can construct a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a
family of r.v.s $\\{X^{n}\\}_{n},\,X$ on it such that
$\mathcal{L}_{\mathbb{P}}(X^{n})=\mu^{n},\,\mathcal{L}_{\mathbb{P}}(X)=\mu$
and $X^{n}\to X$ $\mathbb{P}$-a.s.; it then holds

$\displaystyle\|b\ast\mu^{n}-b\ast\mu\|_{L^{p}_{x}}^{p}$
$\displaystyle=\int_{\mathbb{R}^{d}}|\mathbb{E}[b(x-X^{n})-b(x-X)]|^{p}\,\mathop{}\\!\mathrm{d}x$
$\displaystyle\leq\mathbb{E}\bigg{[}\int_{\mathbb{R}^{d}}|b(x-X^{n})-b(x-X)|^{p}\,\mathop{}\\!\mathrm{d}x\bigg{]}=\mathbb{E}\Big{[}\|\tau_{X^{n}}b-\tau_{X}b\|_{L^{p}_{x}}^{p}\Big{]}$

where in the second passage we used Jensen’s inequality. By the aforementioned
equicontinuity it holds $\|\tau_{X^{n}}b-\tau_{X}b\|_{L^{p}_{x}}\to 0$
$\mathbb{P}$-a.s. and we have the uniform bound
$\|\tau_{X^{n}}b-\tau_{X}b\|_{L^{p}_{x}}\leq 2\|b\|_{L^{p}_{x}}$, thus the
first claim follows from dominated convergence.

Regarding the second claim, Young’s inequality gives a uniform bound for
$\\{b\ast\mu^{n}\\}$ in $L^{\bar{p}}_{x}$ for $1+1/\bar{p}=1/p+1/r$; combined
with convergence in $L^{p}_{x}$ and interpolation estimates, we deduce
convergence in $L^{\tilde{p}}_{x}$ for any $\tilde{p}\in[p,\bar{p})$. ∎

######  Remark A.4.

In the borderline case of $b\in L^{p}_{x}$ and $\\{\mu^{n}\\}_{n}$ bounded in
$L^{p^{\prime}}_{x}$, then $\\{b\ast\mu^{n}\\}_{n}$ is a bounded sequence in
$C_{0}(\mathbb{R}^{d})$, which denotes the Banach space of continuous
functions vanishing at infinity, endowed with the supremum norm. In this case
it can be shown that $\\{b\ast\mu^{n}\\}_{n}$ is also equicontinuous, so by
Ascoli–Arzelà it converges to $b\ast\mu$ uniformly on compact sets.

######  Corollary A.5.

Let $b\in B^{\alpha}_{p,p}$ with $p\in[1,\infty)$, $\alpha\in\mathbb{R}$, and
$\\{\mu^{n}\\}_{n}\subset\mathcal{P}(\mathbb{R}^{d})$ such that
$\mu^{n}\rightharpoonup\mu$ weakly, then $b\ast\mu^{n}\to b\ast\mu$ in
$B^{\alpha}_{p,p}$; if moreover $\\{\mu^{n}\\}_{n}$ is bounded in $L^{r}_{x}$
with $r>1$, then $b\ast\mu^{n}\to b\ast\mu$ in
$B^{\alpha}_{\tilde{p},\tilde{p}}$ for any $\tilde{p}\in[p,\infty)$ satisfying
(A.1). Finally, if

$\displaystyle\frac{1}{p}+\frac{1}{r}<1,$

then $b\ast\mu^{n}\to b\ast\mu$ in $B^{\tilde{\alpha}}_{\infty,\infty}$ for
any $\tilde{\alpha}<\alpha$.

###### Proof.

Observe that
$\Delta_{j}(b\ast\mu^{n}-b\ast\mu)=(\Delta_{j}b)\ast(\mu^{n}-\mu)$, so by
Lemma A.3 $\|\Delta_{j}(b\ast\mu^{n}-b\ast\mu)\|_{L^{p}_{x}}\to 0$ for any
fixed $j\in\mathbb{N}$. Moreover

$\|(\Delta_{j}b)\ast(\mu^{n}-\mu)\|_{L^{p}_{x}}^{p}\lesssim_{p}\|\Delta_{j}b\|_{L^{p}_{x}}^{p}\quad\text{with}\quad\sum_{j}2^{\alpha
jp}\|\Delta_{j}b\|_{L^{p}_{x}}^{p}<\infty$

therefore by dominated convergence

$\lim_{i}\|b\ast\mu^{n}-b\ast\mu\|_{B^{\alpha}_{p,p}}^{p}=\lim_{i}\sum_{j}2^{\alpha
jp}\|(\Delta_{j}b)\ast(\mu^{n}-\mu)\|_{L^{p}_{x}}^{p}=0.$

The second statement follows as before by interpolation between convergence in
$B^{\alpha}_{p,p}$ and boundedness in $B^{\alpha}_{\bar{p},\bar{p}}$ with
$\bar{p}$ satisfying the equality in (A.1).

Finally, if $1>1/p+1/r$, then condition (A.1) is satisfied for all $\tilde{p}$
big enough; the conclusion then follows from convergence in
$B^{\alpha}_{\tilde{p},\tilde{p}}$ and the Besov embedding
$B^{\alpha}_{\tilde{p},\tilde{p}}\hookrightarrow
B^{\alpha-d/\tilde{p}}_{\infty,\infty}$. ∎

######  Corollary A.6.

Let $p,q\in[1,\infty)$, $\\{b^{n}\\}_{n}\subset L^{q}_{T}L^{p}_{x}$ be a
sequence such that $b^{n}\to b$ in $L^{q}_{T}L^{p}_{x}$; moreover let
$\\{\mu^{n},\mu\\}\subset C_{T}\mathcal{P}(\mathbb{R}^{d})$ be such that
$\mu^{n}_{t}\rightharpoonup\mu_{t}$ weakly for every $t\in[0,T]$ and such that

$\displaystyle\sup_{n\in\mathbb{N}}\sup_{t\in[0,T]}\|\mu^{n}\|_{L^{r}_{x}}+\sup_{t\in[0,T]}\|\mu\|_{L^{r}_{x}}<\infty$

for some $r\in[1,\infty]$. Then $b^{n}\ast\mu^{n}\to b\ast\mu$ in
$L^{q}_{T}L^{\tilde{p}}_{x}$ for all $\tilde{p}\in[p,\infty)$ satisfying
(A.1).

###### Proof.

It suffices to show that $b^{n}\ast\mu^{n}\to b\ast\mu$ in
$L^{q}_{T}L^{p}_{x}$; once this is done, convergence in
$L^{q}_{T}L^{\tilde{p}}_{x}$ follows as usual by interpolation and boundedness
in $L^{q}_{T}L^{\bar{p}}_{x}$, which comes from the assumptions and Young’s
inequality. It holds

$\displaystyle\|b^{n}\ast\mu^{n}\|_{L^{q}_{T}L^{p}_{x}}\leq\|(b^{n}-b)\ast\mu^{n}\|_{L^{q}_{T}L^{p}_{x}}+\|b\ast\mu^{n}-b\ast\mu\|_{L^{q}_{T}L^{p}_{x}}$

where we can estimate the first term by
$\|(b^{n}-b)\ast\mu^{n}\|_{L^{q}_{T}L^{p}_{x}}\leq\|b^{n}-b\|_{L^{q}_{T}L^{p}_{x}}\to
0$. For the second term, by Lemma A.3 and the assumptions it holds
$\|b_{t}\ast(\mu^{n}_{t}-\mu_{t})\|_{L^{p}_{x}}\to 0$ for Lebesgue a.e.
$t\in[0,T]$, as well as $\|b_{t}\ast(\mu^{n}_{t}-\mu_{t})\|_{L^{p}_{x}}\leq
2\|b_{t}\|_{L^{p}_{x}}$; thus by dominated convergence we infer
$\|b\ast\mu^{n}-b\ast\mu\|_{L^{q}_{T}L^{p}_{x}}\to 0$ as well. ∎

######  Lemma A.7.

For any $p\in[1,\infty)$ and $\alpha\in\mathbb{R}$ there exists a constant
$C=C(\alpha)$ such that

$\displaystyle\|b\ast(\mu-\nu)\|_{B^{\alpha-1}_{\infty,\infty}}\leq
C\,\|b\|_{B^{\alpha}_{\infty,\infty}}\,d_{p}(\mu,\nu)$

for all $b\in B^{\alpha}_{\infty,\infty}$ and
$\mu,\nu\in\mathcal{P}(\mathbb{R}^{d})$.

###### Proof.

It’s enough give the proof for $p=1$, as the general case follows from
$d_{1}(\mu,\nu)\leq d_{p}(\mu,\nu)$; we can assume $d_{1}(\mu,\nu)<\infty$,
otherwise the inequality is trivial. By Bernstein estimates, reasoning on
Littlewood-Paley blocks, we have

$\displaystyle\|b\ast(\mu-\nu)\|_{B^{\alpha-1}_{\infty,\infty}}$
$\displaystyle=\sup_{n}\big{\\{}2^{n(\alpha-1)}\|(\Delta_{n}b)\ast(\mu-\nu)\|_{L^{\infty}_{x}}\big{\\}}$
$\displaystyle\leq\sup_{n}\big{\\{}2^{n(\alpha-1)}\|\Delta_{n}b\|_{W^{1,\infty}_{x}}\big{\\}}\,d_{1}(\mu,\nu)$
$\displaystyle\lesssim\sup_{n}\big{\\{}2^{n\alpha}\|\Delta_{n}b\|_{L^{\infty}_{x}}\big{\\}}\,d_{1}(\mu,\nu)=\|b\|_{B^{\alpha}_{\infty,\infty}}\,d_{1}(\mu,\nu)$

which gives the claim. ∎

The last statements we are going to provide concern the continuity of the map
$\mu\mapsto b\ast\mu$ in suitable topologies. Their proof require the use of
maximal functions and their basic properties, which we recall first; we refer
the interested reader to [41] for their proofs.

Given $b\in L^{p}(\mathbb{R}^{d})$, $p\in[1,\infty]$, its maximal function
$Mb$ is defined by

$Mb(x):=\sup_{r>0}\frac{1}{\lambda_{d}\,r^{d}}\int_{B(x,r)}|b(y)|\,\mathop{}\\!\mathrm{d}y$

where $\lambda_{d}$ stands for the Lebesgue measure of $B(0,1)$ in
$\mathbb{R}^{d}$. It is well known that if $p\in(1,\infty]$, then $Mf\in
L^{p}(\mathbb{R}^{d})$ and

$\|Mb\|_{p}\leq c_{d,p}\|b\|_{p}$

for some constant $c_{d,p}>0$; similar definitions and properties hold in the
case of vector-valued drifts $b\in L^{p}(\mathbb{R}^{d};\mathbb{R}^{m})$ (in
which case $c=c_{d,p,m}$).

If $b\in W^{1,p}(\mathbb{R}^{d};\mathbb{R}^{d})$, then there exists a
Lebesgue-negligible set $N\subset\mathbb{R}^{d}$ and a constant $c_{d}>0$ such
that the Hajlasz inequality holds:

$|b(x)-b(y)|\leq
c_{d}\,|x-y|\,(MDb(x)+MDb(y))\quad\forall\,x,y\in\mathbb{R}^{d}\setminus N.$
(A.2)

The above and similar inequalities (see [8, 7] for recent asymmetric
extensions) allow to control the map $\mu\mapsto b\ast\mu$ in Wasserstein
spaces.

######  Lemma A.8.

Let $(p,q,r,s)\in(1,\infty)^{4}$ be such that $r\leq p\wedge s$ and

$1+\frac{r}{s}\leq\frac{r}{p}+\frac{1}{q}.$ (A.3)

Then there exists a constant $C$, depending on $d$ and the above parameters,
such that for any $b\in W^{1,p}_{x}$ and $\mu,\nu\in L^{q}_{x}$ it holds

$\|b\ast(\mu-\nu)\|_{L^{s}_{x}}\leq
C\|b\|_{W^{1,p}_{x}}\,\big{(}\|\mu\|_{L^{q}_{x}}^{1/r}+\|\nu\|_{L^{q}_{x}}^{1/r}\big{)}\,d_{r^{\prime}}(\mu,\nu)$

###### Proof.

If $d_{r^{\prime}}(\mu,\nu)=+\infty$ the inequality is trivially true, so we
can assume $d_{r^{\prime}}(\mu,\nu)<\infty$; moreover since $\mu,\nu\in
L^{\tilde{q}}_{x}$ for any $\tilde{q}\in[1,q]$, w.l.o.g. we can assume
equality holds in (A.3).

Let $m\in\Pi(\mu,\nu)$ be an optimal coupling of $(\mu,\nu)$ for
$d_{r^{\prime}}(\mu,\nu)$ and let $N\subset\mathbb{R}^{d}$ be as in (A.2);
since $\mu,\nu$ are absolutely continuous w.r.t. Lebesgue, it holds that
$m((N\times\mathbb{R}^{d})\cup(\mathbb{R}^{d}\times N))=0$. Therefore we can
apply (A.2) for any fixed $x\in\mathbb{R}^{d}$ to find

$\displaystyle|b\ast(\mu-\nu)(x)|$
$\displaystyle=\bigg{|}\int_{\mathbb{R}^{2d}}[b(x-y)-b(x-z)]\,m(\mathop{}\\!\mathrm{d}y,\mathop{}\\!\mathrm{d}z)]\bigg{|}$
$\displaystyle\lesssim\int_{\mathbb{R}^{2d}}|y-z|\,(MDb(x-y)+MDb(x-z))\,m(\mathop{}\\!\mathrm{d}y,\mathop{}\\!\mathrm{d}z)$
$\displaystyle\leq\bigg{(}\int_{\mathbb{R}^{2d}}|y-z|^{r^{\prime}}\,m(\mathop{}\\!\mathrm{d}y,\mathop{}\\!\mathrm{d}z)\bigg{)}^{1/r^{\prime}}\,\bigg{(}\int_{\mathbb{R}^{2d}}|MDb(x-y)+MDb(x-z)|^{r}\,m(\mathop{}\\!\mathrm{d}y,\mathop{}\\!\mathrm{d}z)\bigg{)}^{1/r}$
$\displaystyle\leq
d_{r^{\prime}}(\mu,\nu)\,\bigg{[}\Big{(}\int_{\mathbb{R}^{d}}|MDb(x-y)|^{r}\mu(\mathop{}\\!\mathrm{d}y)\Big{)}^{1/r}+\Big{(}\int_{\mathbb{R}^{d}}|MDb(x-z)|^{r}\nu(\mathop{}\\!\mathrm{d}z)\Big{)}^{1/r}\bigg{]}.$

Observe that

$\displaystyle\Big{(}\int_{\mathbb{R}^{d}}|MDb(\cdot-y)|^{r}\mu(\mathop{}\\!\mathrm{d}y)\Big{)}^{1/r}=(|MDb|^{r}\ast\mu)^{1/r}(\cdot)$

and by assumption (A.3) and Young’s inequality it holds

$\displaystyle\|(|MDb|^{r}\ast\mu)^{1/r}\|_{L^{s}_{x}}=\||MDb|^{r}\ast\mu\|_{L^{s/r}_{x}}^{1/r}\lesssim\||MDb|^{r}\|_{L^{p/r}_{x}}^{1/r}\,\|\mu\|_{L^{q}_{x}}^{1/r}=\|MDb\|_{L^{p}_{x}}\,\|\mu\|_{L^{q}_{x}}^{1/r}.$

Together with a similar estimate for $|MDb|\ast\nu$ and the property
$\|MDb\|_{L^{p}_{x}}\lesssim\|b\|_{W^{1,p}_{x}}$, combining everything we
arrive at

$\|b\ast(\mu-\nu)\|_{L^{s}_{x}}\lesssim\|b\|_{W^{1,p}_{x}}(\|\mu\|_{L^{q}_{x}}^{1/r}+\|\nu\|_{L^{q}_{x}}^{1/r})\,d_{r^{\prime}}(\mu,\nu)$

which is the claim. ∎

######  Corollary A.9.

Let $\alpha\in\mathbb{R}$, $(p,q,r,s)$ as in Lemma A.8. Then there exists a
constant $C$, depending on $d$ and the above parameters, such that for any
$b\in B^{\alpha}_{p,p}$ and $\mu,\nu\in L^{q}_{x}$

$\|b\ast(\mu-\nu)\|_{B^{\alpha-1}_{s,s}}\leq
C\|b\|_{B^{\alpha}_{p,p}}\,\big{(}\|\mu\|_{L^{q}_{x}}^{1/r}+\|\nu\|_{L^{q}_{x}}^{1/r}\big{)}\,d_{r^{\prime}}(\mu,\nu)$

###### Proof.

Let $\Delta_{j}b$ denote the Littlewood–Paley blocks of $b$, then
$\Delta_{j}(b\ast(\mu-\nu))=(\Delta_{j}b)\ast(\mu-\nu)$. By the previous lemma
and Bernstein estimates, one has that

$\displaystyle\|(\Delta_{j}b)\ast(\mu-\nu)\|_{L^{s}_{x}}$
$\displaystyle\lesssim\|\Delta_{j}b\|_{W^{1,p}_{x}}\big{(}\|\mu\|_{L^{q}_{x}}^{1/r}+\|\nu\|_{L^{q}_{x}}^{1/r}\big{)}\,d_{r^{\prime}}(\mu,\nu)$
$\displaystyle\lesssim
2^{j}\,\|\Delta_{j}b\|_{p}\big{(}\|\mu\|_{L^{q}_{x}}^{1/r}+\|\nu\|_{L^{q}_{x}}^{1/r}\big{)}\,d_{r^{\prime}}(\mu,\nu);$

moreover assumption (A.3) implies $s\geq p$. Therefore we have

$\displaystyle\|b\ast(\mu-\nu)\|_{B^{\alpha}_{s,s}}$
$\displaystyle=\bigg{(}\sum_{j}\Big{[}2^{(\alpha-1)j}\|(\Delta_{j}b)\ast(\mu-\nu)\|_{L^{s}_{x}}\Big{]}^{s}\bigg{)}^{1/s}$
$\displaystyle\lesssim\bigg{(}\sum_{j}\Big{[}2^{\alpha
j}\|\Delta_{j}b\|_{L^{s}_{x}}\Big{]}^{s}\bigg{)}^{1/s}\big{(}\|\mu\|_{L^{q}_{x}}^{1/r}+\|\nu\|_{L^{q}_{x}}^{1/r}\big{)}\,d_{r^{\prime}}(\mu,\nu)$
$\displaystyle\lesssim\bigg{(}\sum_{j}\Big{[}2^{\alpha
j}\|\Delta_{j}b\|_{L^{s}_{x}}\Big{]}^{p}\bigg{)}^{1/p}\big{(}\|\mu\|_{L^{q}_{x}}^{1/r}+\|\nu\|_{L^{q}_{x}}^{1/r}\big{)}\,d_{r^{\prime}}(\mu,\nu)$

which gives the conclusion. ∎

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