# Hyperbolic Anderson model 2:
Strichartz estimates and Stratonovich setting

Xia Chen  X. Chen: Department of Mathematics, University of Tennessee,
Knoxville<EMAIL_ADDRESS>,  Aurélien Deya  A. Deya: Institut Elie Cartan,
University of Lorraine<EMAIL_ADDRESS>,  Jian Song  J. Song:
Research Center for Mathematics and Interdisciplinary Sciences, Shandong
University; School of Mathematics, Shandong University<EMAIL_ADDRESS>and
Samy Tindel  S. Tindel: Department of Mathematics, Purdue University
<EMAIL_ADDRESS>
###### Abstract.

We study a wave equation in dimension $d\in\\{1,2\\}$ with a multiplicative
space-time Gaussian noise. The existence and uniqueness of the Stratonovich
solution is obtained under some conditions imposed on the Gaussian noise. The
strategy is to develop some Strichartz type estimates for the wave kernel in
weighted Besov spaces, by which we can prove the wellposedness of an
associated Young-type equation. Those Strichartz bounds are of independent
interest.

###### Key words and phrases:

Stochastic wave equation, Stratonovich equation, weighted Besov space,
Strichartz estimate

###### 2010 Mathematics Subject Classification:

60H15, 60G15, 35L05

###### Contents

  1. 1 Introduction
  2. 2 Weighted Besov spaces
    1. 2.1 Weighted Besov spaces
    2. 2.2 A few properties of the weighted Besov spaces
  3. 3 Smoothing effect of the wave kernel
    1. 3.1 Definition of the wave kernel
    2. 3.2 Strichartz type estimates
  4. 4 Young wave equation
    1. 4.1 Young wave integral
    2. 4.2 Wellposedness of the Young wave equation for $\mathbf{d=1,2}$
  5. 5 Application to Gaussian noises
  6. A
    1. A.1 A local reproducing formula
    2. A.2 Proof of Lemma 2.13
    3. A.3 Proof of Lemma 3.5

##  1\. Introduction

In [4] we have started a long term project aiming at defining wave equations
driven by rough noises. More specifically, [4] focused on the following
Skorohod type equation on $\mathbb{R}_{+}\times\mathbb{R}^{d}$ for
$d\in\\{1,2,3\\}$:

$\frac{\partial^{2}u}{\partial t^{2}}(t,x)=\Delta
u(t,x)+u\diamond\dot{W}(t,x),$ (1.1)

where $\diamond$ stands for the Wick product and where $\dot{W}$ is a centered
Gaussian noise. In [4], the covariance function for $\dot{W}$ was fractional
in time with proper decay in space. That is for $s,t\in\mathbb{R}_{+}$ and
$x,y\in\mathbb{R}^{d}$ we had

$\mathbb{E}[\dot{W}(s,x)\dot{W}(t,y)]=|s-t|^{-a_{0}}\gamma(x-y),$ (1.2)

where $a_{0}\in[0,1)$ and $\gamma$ is a nonnegative and nonnegative definite
(generalized) function admitting a spectral measure $\mu$. Then we obtained
that under appropriate conditions on the initial conditions, the following
relation is necessary and sufficient in order to solve (1.1):

$\int_{\mathbb{R}^{d}}\left(\frac{1}{1+|\xi|}\right)^{3-a_{0}}\mu(d\xi)<\infty.$
(1.3)

Notice that (1.3) quantifies how irregular in space our noise can be according
to its regularity in time. In particular, if the function $\gamma$ in (1.2)
satisfies the scaling property

$\gamma(cx)=c^{-a}\gamma(x)$ (1.4)

for some $a\in(0,d]$, then condition (1.3) can be recast as

$a_{0}+a<3\,.$ (1.5)

In the current paper, we will contrast the neat condition obtained in (1.3)
with the situation for a pathwise version of the wave equation (interpreted in
the Young or Stratonovich sense). Namely, we consider the following
Stratonovich type wave equation on $\mathbb{R}_{+}\times\mathbb{R}^{d}$ for
$d\in\\{1,2\\}$,

$\frac{\partial^{2}u}{\partial t^{2}}(t,x)=\Delta u(t,x)+u\dot{W}(t,x),$ (1.6)

with initial conditions $u(0,x)=u_{0}(x)$ and $\frac{\partial}{\partial
t}u(0,x)=u_{1}(x)$. The Gaussian noise we consider in (1.6) also has the
covariance (1.2) with $\gamma$ being a nonnegative definite (generalized)
function. We prove the following result on the existence and uniqueness of the
Stratonovich solution to (1.6) (see Proposition 5.6 and Corollary 5.7 for a
precise account).

######  Theorem 1.1.

Assume $d\in\\{1,2\\}$ and the spectral measure $\mu$ of $W$ verifies

$\int_{\mathbb{R}^{d}}\left(\frac{1}{1+|\xi|}\right)^{\rho_{d}-a_{0}-\eta}\mu(d\xi)<\infty\,\quad\text{
for some }\eta>0,$ (1.7)

where the parameter $\rho_{d}$ is such that $\rho_{1}=1$ and
$\rho_{2}=\frac{1}{2}$. Then, under some regularity conditions on $u_{0}$ and
$u_{1}$, there exists a unique Stratonovich solution to (1.6) in a proper path
space. If $\gamma$ has the scaling property (1.4), condition (1.7) is
equivalent to

$\begin{cases}a_{0}+a<1,&\text{ if }d=1,\vspace{0.2cm}\\\
a_{0}+a<\frac{1}{2},&\text{ if }d=2.\end{cases}$ (1.8)

It is well known that Stratonovich solutions to stochastic PDEs demand a more
regular noisy input $\dot{W}$ than in the Skorohod setting. And indeed in our
case, it is readily checked that (1.5) is less restrictive than (1.8). In
fact, for $d\in\\{1,2\\}$, (1.5) is automatically fulfilled as soon as
$a_{0}\in[0,1)$ and the covariance function $\gamma$ is nonnegative (which
implies $a\in(0,d]$, see [4, Remark 1.4]). Nevertheless, Theorem 1.1 gives the
first condition on $a_{0}$ and $\mu(d\xi)$ in order to get a unique pathwise
solution to (1.6). In addition, as the reader will see, our main result also
unifies the treatment for $d=1$ and $d=2$.

Let us mention some recent progress, obtained by methods which are completely
different from ours, towards the definition of the stochastic wave equation in
a pathwise sense.

(i) As far as we know, the first pathwise developments for a stochastic wave
equation can be found in [10], dealing with the one-dimensional case. Based on
the specific expression of the wave kernel for $d=1$, the strategy therein
relies on a natural preliminary rotation of the model, that turns (1.6) into a
more tractable equation in the plane $\mathbb{R}^{2}$. It is then established
that when one injects a noise of the form (1.2)-(1.4) within the new equation
(thus corresponding to a “rotated” noise for the original equation), the
interpretation and wellposedness can be obtained for all $a_{0},a\in(0,1)$.
Unfortunately, this preliminary transform of the model - which will not occur
in our direct approach - essentially rules out the possibility to compare the
interpretation in [10] with ours, and accordingly to compare the conditions in
[10] with those in (1.8).

(ii) The recent publication [1] focuses on the wave equation with a noise
$\dot{W}$ independent of time. The condition obtained in [1] is

$\int_{\mathbb{R}^{d}}\frac{1}{1+|\xi|}\,\mu(d\xi)<\infty,$ (1.9)

which is equivalent to $a<1$ when $\gamma$ has the scaling property (1.4). A
time independent noise being morally equivalent to $a_{0}=0$, in this
situation our condition (1.8) can be read

$a<1\text{ for }d=1,\quad\text{ and }\quad a<\frac{1}{2}\text{ for }d=2.$

Our condition on $a$ is thus slightly suboptimal for $d=2$ if one compares it
to (1.9), although our method is currently the only one accommodating for a
time-dependent noise. Let us also mention that [1] relies on chaos expansions
for Stratonovich integrals. This strategy is interesting in its own right and
very different from the pathwise considerations in the current paper.

(iii) Some attention has been paid recently to models of wave equation with
additive noise and polynomial nonlinearities. A prototype for such an equation
can be written as

$\frac{\partial^{2}u}{\partial t^{2}}(t,x)=\Delta u-\rho u^{2}+\dot{W},$
(1.10)

with an additive fractional noise $\dot{W}$ and $\rho>0$. In this context,
renormalization procedures are implemented in [6, 8]. This yields existence
results for (1.10) in case of a rough noise $\dot{W}$.

As one can see, the study of wave equations in a rough setting is still wide
open. Our contribution aims at a better understanding of the Young regime
within this landmark.

As mentioned above, instead of the Skorohod setting advocated in [4], the
stochastic differential in (1.6) is interpreted in the Stratonovich sense.
This forces to a totally different approach which is based on pathwise type
considerations. We briefly elaborate our strategy below.

We will solve the wave equation under its so-called mild form, which will be
properly introduced in (4.29). The main technical issue will thus be to
understand the meaning of integrals like

$\int_{0}^{t}\mathcal{G}_{t-r}(u_{r}d\dot{W}_{r}),$ (1.11)

where $u$ is a candidate solution, $\mathcal{G}_{t}$ stands for the wave
operator at time $t$, and where $\dot{W}$ is our noisy input. Since we wish to
interpret (1.11) as a Young integral, we will see it as the limit of a
sequence of paths $\\{\mathcal{J}_{t}^{(n)};t\in[0,T]\\}$ defined by

$\mathcal{J}_{t}^{(n)}:=\sum_{k=0}^{m-1}\mathcal{G}_{t-t^{n}_{k}}(u_{t^{n}_{k}}\,\delta\dot{W}_{t^{n}_{k}t^{n}_{k+1}}),\quad\mbox{
for }t\in(t^{n}_{m-1},t^{n}_{m}]\,,$ (1.12)

where $t_{m}^{n}=mT/2^{n}$ for $0\leq m\leq 2^{n}$ and
$\delta\dot{W}_{t_{k}^{n}t_{k+1}^{n}}=\dot{W}_{t_{k+1}^{n}}-\dot{W}_{t_{k}^{n}}$.
Our most important endeavour is thus to quantify the smoothing effect of the
operator $\mathcal{G}_{t-t_{k}^{n}}$ in (1.12). This is achieved thanks to
some Strichartz type estimates for the operator $\mathcal{G}_{s}$ in some
weighted Besov spaces, which are new and should be considered as one of the
main contributions of the current paper.

Our findings concerning Strichartz type estimates are summarized in
Proposition 3.4 below. Roughly speaking, if $\mathcal{B}^{\alpha}$ designates
a weighted Besov space with regularity $\alpha\leq 0$, we claim that

$\displaystyle\|\\{\mathcal{G}_{t}-\mathcal{G}_{s}\\}f\|_{\mathcal{B}^{\alpha+\kappa}}$
$\displaystyle\lesssim|t-s|^{1-\kappa}\|f\|_{\mathcal{B}^{\alpha}},\quad\text{for
all}\quad\kappa\in[0,1],\ \text{ if }d=1,$ (1.13) $\displaystyle\text{ and
}\|\\{\mathcal{G}_{t}-\mathcal{G}_{s}\\}f\|_{\mathcal{B}^{\alpha+\kappa}}$
$\displaystyle\lesssim|t-s|^{1/2-\kappa}\|f\|_{\mathcal{B}^{\alpha}},\quad\text{for
all}\quad\kappa\in[0,1/2],\ \text{ if }d=2.$

These bounds thus differ from the classical Strichartz estimates (see e.g.
[7]) in two aspects. First, they offer a Hölder-type control (in time) on the
action of $\mathcal{G}$, a key ingredient toward a successful Young
integration procedure. Secondly, they involve Besov spaces with weights, which
is crucial in order to deal with a noise like $\dot{W}$ defined on the whole
space.

Our considerations are based on a weighted version of Littlewood-Paley
analysis introduced by Rychkov in [9]. This approach is convenient when one
wishes to handle kernels with polynomial decay in Fourier modes such as
$\mathcal{G}_{t}$. This explains in particular why we have decided to stick to
[9] instead of using the more recent method advocated in [5].

We close this section by highlighting some possible generalization of our
work:

(i) In the current article, we have focused on a noisy term of the form
$u\dot{W}$ in (1.6) for sake of conciseness. We firmly believe that a noise
term $\sigma(u)\dot{W}$ with a smooth enough $\sigma$ could also be covered by
our approach, at the price of longer technical considerations.

(ii) The treatment of rougher noises is expected to rely on higher-order
expansions of the model, together with renormalization procedures. These
challenging developments may require the adaptation of some ideas from
regularity structures theory or paracontrolled calculus in the wave setting.

(iii) The Strichartz type estimate alluded to above are based on convolution
estimates for functions. In order to reach $d=3$ or above, those estimates
should be extended to measures or distributions. At the moment, this is still
an open problem for us.

The rest of this paper is organized as follows. In Section 2, we recall some
preliminaries on weighted Besov spaces. The smoothing effect of the wave
kernel in weighted Besov spaces is obtained in Section 3. It is then applied
to study a Young-type equation associated with (1.6) in Section 4, where we
define Young wave integration and prove the wellposedness of the Young-type
equations. In Section 5, we apply the result on the Young-type equation
obtained in Section 4 to our wave equation (1.6) and obtain the existence and
uniqueness of the solution under condition (1.8). Finally, in Appendix A, we
provide details for some results used in the preceding sections.

######  Notation 1.2.

For $r=(r_{1},\dots,r_{d})$ with $r_{i}\in\mathbb{N}$ and
$x=(x_{1},\dots,x_{d})$ with $x_{i}\in\mathbb{R}^{d}$, we set

$|r|=\sum_{i=1}^{d}r_{i},\quad r!=r_{1}!\dots r_{d}!,\quad
x^{r}=x_{1}^{r_{1}}\dots x_{d}^{r_{d}},\quad
D^{r}f(x)=\left(\frac{\partial}{\partial
x_{1}}\right)^{r_{1}}\dots\left(\frac{\partial}{\partial
x_{d}}\right)^{r_{d}}.$

For a generic continuous function $\varphi$ defined on $\mathbb{R}^{d}$, we
set $\varphi_{j}(x)=2^{dj}\varphi(2^{j}x).$ We use $\mathbf{B}_{R}(x)$ to
denote the open ball in $\mathbb{R}^{d}$ centered at $x$ with radius $R>0$,
i.e., $\mathbf{B}_{R}(x)=\\{y\in\mathbb{R}^{d};|y-x|<R\\}$, and in particular
we set $\mathbf{B}_{R}:=\mathbf{B}_{R}(0).$ Finally, we denote by
$\mathcal{C}_{c}^{\infty}$ the set of smooth and compactly-supported functions
on $\mathbb{R}^{d}$.

##  2\. Weighted Besov spaces

Equation (1.6) will be solved in some properly weighted Besov spaces. In fact
for a sharp analysis of the wave kernel properties (in Section 3), we have
found convenient to use the general setting introduced by Rychkov in [9]. This
setting has the advantage to cover a large class of weights. The current
section is mostly dedicated to recall the main elements of the latter article.

###  2.1. Weighted Besov spaces

In this section we construct and give some basic properties of the weighted
Besov spaces used for the wave equation.

Let us first recall the definition of the class $A_{p}^{\textnormal{loc}}$ of
weights which was introduced in [9].

######  Definition 2.1.

We call weight any locally integrable and strictly positive function
$w:\mathbb{R}^{d}\to\mathbb{R}_{+}$. Next for every $1<p<\infty$, we define
$A^{\textnormal{loc}}_{p}$ as the set of weights $w$ for which

$\|w\|_{p,\textnormal{loc}}:=\sup_{|I|\leq
1}\frac{1}{|I|^{p}}\bigg{(}\int_{I}w(x)\,dx\bigg{)}\bigg{(}\int_{I}w(x)^{-\frac{p^{\prime}}{p}}\,dx\bigg{)}^{\frac{p}{p^{\prime}}}\
<\ \infty\,,$ (2.1)

where $p^{\prime}$ refers to the Hölder conjugate of $p$,
$I=[a_{1},b_{1}]\times\dots\times[a_{d},b_{d}]$ with $a_{i}<b_{i}$ for
$i=1,\dots,d$ and $|I|=\prod_{i=1}^{d}|a_{i}-b_{i}|$.

As pointed out in [9], this general class $A_{p}^{\textnormal{loc}}$ of
weights consists of the classical $A_{p}$ weights (see e.g. [12]) and the
locally regular weights that grow/decay at most exponentially at infinity. In
particular, for $1<p<\infty$, $A_{p}^{\textnormal{loc}}$ includes the
following exponential and polynomial weights that we will use in this paper:

$w_{\mu}(x):=e^{-\mu|x|}\quad\text{ for }\mu\in\mathbb{R},\quad\text{ and
}\quad P(x):=\left(1+|x|^{d+1}\right)^{-1}\,.$ (2.2)

For a generic weight $w$, we introduce the notion of weighted $L^{p}$ space
which will be at the heart of our analysis.

######  Definition 2.2.

Let $w$ be a weight function in the space $A_{p}^{\textnormal{loc}}$ for
$p>1$. For a function $f:\mathbb{R}^{d}\to\mathbb{R}$, we set

$\|f\|_{L^{p}_{w}}:=\left(\int_{\mathbb{R}^{d}}|f(x)|^{p}\,w(x)\,dx\right)^{1/p}\,.$

Whenever $w_{\mu}(x)=e^{-\mu|x|}$ is an exponential weight, we write
$L_{w_{\mu}}^{p}:=~{}L_{\mu}^{p}.$

We also label a notation for functions with centered moments in the definition
below.

######  Definition 2.3.

For every integer $L\geq 0$, we denote by $\mathcal{D}_{L}$ the set of
functions $\varphi\in\mathcal{C}_{c}^{\infty}$ such that for each multi-index
$r$ with $0\leq|r|\leq L$, one has

$\int_{\mathbb{R}^{d}}\,x^{r}\varphi(x)\,dx=0\,.$ (2.3)

The compatibility of weights with convolution products is a crucial and basic
feature for a proper Besov analysis. We state and prove a lemma in this sense.

######  Lemma 2.4.

Fix two constants $\mu_{\ast},K>0$. Then for all $0\leq\mu\leq\mu_{\ast}$ and
$\varphi\in\mathcal{C}_{c}^{\infty}$ such that
$\text{Supp}(\varphi)\subset\mathbf{B}_{K}$, one has

$\big{\|}\varphi\ast
f\big{\|}_{L^{p}_{\mu}}\lesssim\big{\|}|\varphi|\ast\big{[}w_{\mu}^{1/p}|f|\big{]}\big{\|}_{L^{p}}\,,$
(2.4)

for some proportional constant that only depends on $p$, $K$ and $\mu_{\ast}$.
In particular, the following compatibility relation holds true:

$\big{\|}\varphi\ast
f\big{\|}_{L^{p}_{\mu}}\lesssim\|\varphi\|_{L^{1}}\|f\|_{L^{p}_{\mu}}.$ (2.5)

###### Proof.

We start by writing the $L_{\mu}^{p}$-norm of $\varphi\ast f$ according to
Definition 2.2,

$\displaystyle\big{\|}\varphi\ast f\big{\|}_{L^{p}_{\mu}}^{p}$
$\displaystyle=\int_{\mathbb{R}^{d}}dx\,w_{\mu}(x)\bigg{|}\int_{\mathbb{R}^{d}}dy\,\varphi(x-y)f(y)\bigg{|}^{p}.$

We now insert the weight $w_{\mu}(y)$ in order to get

$\displaystyle\big{\|}\varphi\ast f\big{\|}_{L^{p}_{\mu}}^{p}$
$\displaystyle=\int_{\mathbb{R}^{d}}dx\,\bigg{|}\int_{\mathbb{R}^{d}}dy\,\bigg{[}\bigg{(}\frac{w_{\mu}(x)}{w_{\mu}(y)}\bigg{)}^{\frac{1}{p}}\varphi(x-y)\bigg{]}\Big{[}w_{\mu}(y)^{\frac{1}{p}}f(y)\Big{]}\bigg{|}^{p}$
$\displaystyle\leq\int_{\mathbb{R}^{d}}dx\,\bigg{|}\int_{\mathbb{R}^{d}}dy\,\bigg{|}\bigg{(}\frac{w_{\mu}(x)}{w_{\mu}(y)}\bigg{)}^{\frac{1}{p}}\varphi(x-y)\bigg{|}\Big{|}w_{\mu}(y)^{\frac{1}{p}}f(y)\Big{|}\bigg{|}^{p}\,.$
(2.6)

Recall that $\text{Supp}(\varphi)$ is a subset of $\mathbf{B}_{K}$. Hence for
all $x,y\in\mathbb{R}^{d}$, we have

$\bigg{|}\bigg{(}\frac{w_{\mu}(x)}{w_{\mu}(y)}\bigg{)}^{\frac{1}{p}}\varphi(x-y)\bigg{|}\leq
e^{\mu K/p}|\varphi(x-y)|.$

Plugging this inequality into (2.6) we immediately get (2.4), which ends our
proof. ∎

In order to implement the partition of unity necessary for a proper definition
of Besov spaces, we state a technical lemma about convolutions of rescaled
functions.

######  Lemma 2.5.

Fix an integer $L\geq 0$. Let $\varphi,\psi\in\mathcal{C}_{c}^{\infty}$ and
suppose $\psi\in\mathcal{D}_{L}$, where we recall that $\mathcal{D}_{L}$ is
introduced in Definition 2.3. Next for every $j\geq 1$, recall from Notation
1.2 that we have set $\varphi_{j}(x):=2^{dj}\varphi(2^{j}x)$,
$\psi_{j}(x):=2^{dj}\psi(2^{j}x)$. Then for all $0\leq j\leq\ell$ and $1\leq
r\leq\infty$, it holds that

$\big{\|}\varphi_{j}\ast\psi_{\ell}\big{\|}_{L^{r}}\lesssim
2^{dj\left(1-\frac{1}{r}\right)}2^{-L(\ell-j)}\,.$ (2.7)

###### Proof.

In order to ease our notation, let us agree on the following convention:

For the sake of clarity, we assume from now on that the supports of functions
in $\mathcal{C}_{c}^{\infty}$ are all included in the unit ball
$\mathbf{B}_{1}$.

We first prove inequality (2.7) for the case $1\leq r<\infty$. Writing the
definition of the convolution product and performing the change of variables
$2^{\ell}y\to y$ and $2^{j}x\to x$, one gets

$\displaystyle\big{\|}\varphi_{j}\ast\psi_{\ell}\big{\|}_{L^{r}}^{r}$
$\displaystyle=\int dx\,\bigg{|}\int
dy\,2^{dj}\varphi(2^{j}(x-y))2^{d\ell}\psi(2^{\ell}y)\bigg{|}^{r}$
$\displaystyle=2^{dj(r-1)}\int_{\mathbf{B}_{2}}dx\,\bigg{|}\int_{\mathbf{B}_{1}}dy\,\psi(y)\varphi(x-2^{-(\ell-j)}y)\bigg{|}^{r}.$

Next invoke the fact that $\psi\in\mathcal{D}_{L}$ in order to write

$\displaystyle\big{\|}\varphi_{j}\ast\psi_{\ell}\big{\|}_{L^{r}}^{r}$
$\displaystyle=$ $\displaystyle
2^{dj(r-1)}\int_{\mathbf{B}_{2}}dx\,\bigg{|}\int_{\mathbf{B}_{1}}dy\,\psi(y)\bigg{[}\varphi(x-2^{-(\ell-j)}y)-\sum_{0\leq|k|\leq
L-1}\frac{D^{k}\varphi(x)}{k!}(-2^{-(\ell-j)}y)^{k}\bigg{]}\bigg{|}^{r}\,.$
(2.8)

Furthermore, thanks to classical considerations on remainders for Taylor
expansions, for $x,y\in\mathbf{B}_{2}$, we have

$\bigg{|}\varphi(x-2^{-(\ell-j)}y)-\sum_{0\leq|k|\leq
L-1}\frac{D^{k}\varphi(x)}{k!}(-2^{-(\ell-j)}y)^{k}\bigg{|}\lesssim\,2^{-L(\ell-j)}.$
(2.9)

Plugging (2.9) into (2.1), this proves (2.7) for $1\leq r<\infty$.

For the case $r=\infty$ in (2.7), we have similarly

$\displaystyle\varphi_{j}\ast\psi_{\ell}(2^{-j}x)$ $\displaystyle=$
$\displaystyle\int
dy\,2^{dj}\varphi(2^{j}(2^{-j}x-y))2^{d\ell}\psi(2^{\ell}y)=2^{dj}\int_{\mathbf{B}_{1}}dy\,\psi(y)\varphi(x-2^{-(\ell-j)}y)$
$\displaystyle=$ $\displaystyle
2^{dj}\int_{\mathbf{B}_{1}}dy\,\psi(y)\bigg{[}\varphi(x-2^{-(\ell-j)}y)-\sum_{0\leq|k|\leq
L-1}\frac{D^{k}\varphi(x)}{k!}(-2^{-(\ell-j)}y)^{k}\bigg{]}\,,$

and (2.7) follows from (2.9). This finishes the proof. ∎

Eventually we will define a bump type function adapted to the construction of
our Besov type spaces.

######  Notation 2.6.

Pick an arbitrary function $\varphi_{0}\in\mathcal{C}_{c}^{\infty}$. Then we
denote by $\varphi$ the function defined for $x\in\mathbb{R}^{d}$ by

$\varphi(x):=\varphi_{0}(x)-2^{-d}\varphi_{0}\Big{(}\frac{x}{2}\Big{)}\,.$

Also recall that, for every continuous function
$g:\mathbb{R}^{d}\to\mathbb{R}$ and $j\geq 1$, we set
$g_{j}(x):=2^{dj}g(2^{j}x)$ for all $x\in\mathbb{R}$.

We are now ready to give a precise definition of the Besov spaces considered
in this article, which is borrowed from [9, Definition 2.4].

######  Definition 2.7.

Consider a function $\varphi_{0}\in\mathcal{C}_{c}^{\infty}$ and the
corresponding $\varphi$ as in Notation 2.6. For any fixed length $L$, we
assume that $\int_{\mathbb{R}^{d}}\varphi_{0}(x)\,dx\neq 0$ and
$\varphi\in\mathcal{D}_{L}$. Otherwise stated we have,

$\int_{\mathbb{R}^{d}}\varphi_{0}(x)\,dx\neq
0,\quad\text{and}\quad\int_{\mathbb{R}^{d}}x^{r}\varphi(x)\,dx=0\,,\text{ for
}|r|\leq L\,.$ ($\mathcal{A}_{\varphi}$)

Then for all $-\infty<s\leq 1$, all $1<p<\infty$ and $1<q\leq\infty$, we
define the space $\mathcal{B}^{s,\mu}_{p,q}(\varphi_{0})$ as the completion of
$\mathcal{C}_{c}^{\infty}$ with respect to the norm

$\|f\|_{\mathcal{B}^{s,\mu}_{p,q}(\varphi_{0})}:=\bigg{(}\sum_{j\geq
0}2^{jsq}\big{\|}\varphi_{j}\ast
f\big{\|}_{L^{p}_{\mu}}^{q}\bigg{)}^{\frac{1}{q}}\,.$ (2.10)

At first sight it seems that the spaces
$\mathcal{B}^{s,\mu}_{p,q}(\varphi_{0})$ depend on the specific function
$\varphi_{0}$ we have chosen. The following Proposition, borrowed from [9,
Corollary 2.7], shows that the Besov spaces do not exhibit this kind of
dependence.

######  Proposition 2.8.

Let $\varphi_{0},\tilde{\varphi}_{0}\in\mathcal{C}_{c}^{\infty}$ be two
functions satisfying Assumption ($\mathcal{A}_{\varphi}$), and fix
$\mu_{\ast}>0$. Then there exist two constants
$c_{\varphi_{0},\tilde{\varphi}_{0},\mu_{\ast}},C_{\varphi_{0},\tilde{\varphi}_{0},\mu_{\ast}}>0$
such that for all $-\infty<s\leq 1$, all $0\leq\mu\leq\mu_{\ast}$, and
$1<p,q<\infty$ we have

$c_{\varphi_{0},\tilde{\varphi}_{0},\mu_{\ast}}\|f\|_{\mathcal{B}^{s,\mu}_{p,q}(\tilde{\varphi}_{0})}\leq\|f\|_{\mathcal{B}^{s,\mu}_{p,q}(\varphi_{0})}\leq
C_{\varphi_{0},\tilde{\varphi}_{0},\mu_{\ast}}\|f\|_{\mathcal{B}^{s,\mu}_{p,q}(\tilde{\varphi}_{0})}\,.$
(2.11)

Since Proposition 2.8 states that the Besov spaces do not depend on the
particular choice of $\varphi_{0}$, one might just fix a particular
$\varphi_{0}$ and consider the corresponding spaces $\mathcal{B}$. This
convention is labeled below.

######  Notation 2.9.

From now on, we fix a function
$\boldsymbol{\varphi_{0}}\in\mathcal{C}_{c}^{\infty}$ satisfying
($\mathcal{A}_{\varphi}$) and consider the scale of spaces
$\mathcal{B}^{s,\mu}_{p,q}:=\mathcal{B}^{s,\mu}_{p,q}(\boldsymbol{\varphi_{0}})$,
for $1<p<\infty$, $1<q\leq\infty$, $-\infty<s\leq 1$ and $\mu\in\mathbb{R}$.

###  2.2. A few properties of the weighted Besov spaces

We now collect a few basic facts about Besov spaces, whose role for our
existence and uniqueness result will be crucial.

First most of our weights $w$ in the sequel are exponential. However, we also
deal occasionally with polynomial weights. The lemma below gives a link
between those classes.

######  Lemma 2.10.

Let $\mu_{\ast}>0$ be a fixed constant. Consider the polynomial weight $P(x)$
and the family $\\{w_{\mu},0<\mu\leq\mu_{\ast}\\}$ of exponential weights
given in (2.2). Then, for $1<p<\infty,1<q\leq\infty$ and $s\in\mathbb{R}$, we
have

$\big{\|}f\big{\|}_{\mathcal{B}^{s,w_{\mu}}_{p,q}}\lesssim\mu^{-\frac{d+1}{p}}\big{\|}f\big{\|}_{\mathcal{B}^{s,P}_{p,q}}\,,$

where the multiplicative constant on the right-hand side depends on
$\mu_{\ast}$ only.

###### Proof.

Invoking the fact that $0<\mu\leq\mu_{\ast}$, it is readily checked that

$e^{-\mu|x|}\lesssim P(x)\bigg{\\{}{\bf
1}_{\\{|x|<1\\}}+\frac{1}{\mu^{d+1}}{\bf 1}_{\\{|x|\geq
1\\}}\bigg{\\}}\lesssim\frac{P(x)}{\mu^{d+1}}\big{\\{}\mu_{\ast}^{d+1}+1\big{\\}}\,.$

Inserting this inequality into the definition (2.10) of Besov norms, our claim
is achieved in a straightforward way. ∎

Interpolation inequalities are part of the basic toolkit of Besov spaces. In
our weighted context, we can state the following result.

######  Lemma 2.11.

Consider nine parameters $s,s_{0},s_{1}\in(-\infty,1]$,
$p,p_{0},p_{1}\in(1,\infty)$ and $q,q_{0},q_{1}\in(1,\infty]$. We suppose that
there exists $\theta\in[0,1]$ such that

$s=(1-\theta)s_{0}+\theta
s_{1}\,,\qquad\frac{1}{p}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}}\,,\qquad\frac{1}{q}=\frac{1-\theta}{q_{0}}+\frac{\theta}{q_{1}}\,.$

Then for every $\mu\in\mathbb{R}$, it holds that

$\big{\|}f\big{\|}_{\mathcal{B}^{s,\mu}_{p,q}}\leq\big{\|}f\big{\|}_{\mathcal{B}^{s_{0},\mu}_{p_{0},q_{0}}}^{1-\theta}\big{\|}f\big{\|}_{\mathcal{B}^{s_{1},\mu}_{p_{1},q_{1}}}^{\theta}\,.$
(2.12)

###### Proof.

The proof is achieved by standard Hölder-type inequalities. It is left to the
reader for the sake of conciseness.  ∎

We next move to an embedding property which will be an important step in our
analysis.

######  Lemma 2.12.

We consider six parameters $1<p<p^{\prime}<\infty$,
$-\infty<\kappa^{\prime}<\kappa\leq 1$ and
$0\leq\mu<\mu^{\prime}\leq\mu_{\ast}$. Assume that the following relation
holds true,

$\kappa=\kappa^{\prime}+d\bigg{(}\frac{1}{p}-\frac{1}{p^{\prime}}\bigg{)}\,,\quad\text{and}\quad\mu^{\prime}=\frac{p^{\prime}}{p}\mu\,.$
(2.13)

Then for the Besov spaces introduced in Definition 2.7, it holds that

$\big{\|}f\big{\|}_{\mathcal{B}^{\kappa^{\prime},\mu^{\prime}}_{p^{\prime},\infty}}\lesssim\big{\|}f\big{\|}_{\mathcal{B}^{\kappa,\mu}_{p,\infty}}\,,$
(2.14)

for some proportional constant that only depends on $\mu_{\ast}$ (as far as
weight parameters are concerned).

###### Proof.

Consider an integer $L\geq 1$, the exact value of which will be determined
later on (namely we will see that $L=L(\kappa,\kappa^{\prime})$ depends on
$(\kappa,\kappa^{\prime})$ only). Recall that we have defined our Besov spaces
based on a function $\varphi_{0}$ as in Notation 2.9. By Theorem A.1 in the
Appendix, we know that there exist $\psi_{0}\in\mathcal{C}_{c}^{\infty}$ and
$\psi\in\mathcal{D}_{L}$ such that

$\varphi_{j}\ast f=\sum_{\ell\geq
0}(\varphi_{j}\ast\psi_{\ell})\ast(\varphi_{\ell}\ast f)\,.$ (2.15)

Observe that we have assumed the supports of the $\varphi$ and $\psi$
functions to be subsets of $\mathbf{B}_{1}$, and so we have
$\text{Supp}\,(\varphi_{j}\ast\psi_{\ell})\subset\mathbf{B}_{2}$ for all
$j,\ell\geq 0$. Hence we can use (2.4) to assert that

$\big{\|}(\varphi_{j}\ast\psi_{\ell})\ast(\varphi_{\ell}\ast
f)\big{\|}_{L^{p^{\prime}}_{\mu^{\prime}}}\lesssim\big{\|}|\varphi_{j}\ast\psi_{\ell}|\ast|w_{\mu}^{\frac{1}{p}}(\varphi_{\ell}\ast
f)|\big{\|}_{L^{p^{\prime}}}\,,$

for some proportional constant that only depends on $\mu_{\ast}$. We can then
invoke the classical Young inequality for convolution products to derive

$\big{\|}(\varphi_{j}\ast\psi_{\ell})\ast(\varphi_{\ell}\ast
f)\big{\|}_{L^{p^{\prime}}_{\mu^{\prime}}}\lesssim\big{\|}\varphi_{j}\ast\psi_{\ell}\big{\|}_{L^{r}}\big{\|}w_{\mu}^{\frac{1}{p}}(\varphi_{\ell}\ast
f)\big{\|}_{L^{p}}\lesssim\big{\|}\varphi_{j}\ast\psi_{\ell}\big{\|}_{L^{r}}\big{\|}\varphi_{\ell}\ast
f\big{\|}_{L^{p}_{\mu}}$ (2.16)

where $r$ is defined through the relation
$\frac{1}{r}+\frac{1}{p}=1+\frac{1}{p^{\prime}}$. Going back to (2.15), we
deduce that for every $j\geq 0$,

$\displaystyle\big{\|}\varphi_{j}\ast
f\big{\|}_{L^{p^{\prime}}_{\mu^{\prime}}}$
$\displaystyle\lesssim\sum_{\ell\geq
0}\big{\|}\varphi_{j}\ast\psi_{\ell}\big{\|}_{L^{r}}\big{\|}\varphi_{\ell}\ast
f\big{\|}_{L^{p}_{\mu}}\lesssim\|f\|_{\mathcal{B}^{\kappa,\mu}_{p,\infty}}\sum_{\ell\geq
0}2^{-\kappa\ell}\big{\|}\varphi_{j}\ast\psi_{\ell}\big{\|}_{L^{r}}\,.$

We now split the sum above into $\ell\leq j$ and $\ell>j$. We also assume
$\varphi\in\mathcal{D}_{L}$ in Definition 2.7. Then we apply Lemma 2.5 in two
different ways according to those two cases (recall that (2.7) is valid for
$j\leq\ell$). This yields,

$\displaystyle\big{\|}\varphi_{j}\ast
f\big{\|}_{L^{p^{\prime}}_{\mu^{\prime}}}$
$\displaystyle\lesssim\|f\|_{\mathcal{B}^{\kappa,\mu}_{p,\infty}}\bigg{[}\sum_{0\leq\ell\leq
j}2^{-\kappa\ell}\big{\|}\varphi_{j}\ast\psi_{\ell}\big{\|}_{L^{r}}+\sum_{\ell>j}2^{-\kappa\ell}\big{\|}\varphi_{j}\ast\psi_{\ell}\big{\|}_{L^{r}}\bigg{]}\,,$
$\displaystyle\lesssim\|f\|_{\mathcal{B}^{\kappa,\mu}_{p,\infty}}\bigg{[}\sum_{0\leq\ell\leq
j}2^{-\kappa\ell}2^{d\ell(1-\frac{1}{r})}2^{-L(j-\ell)}+2^{dj(1-\frac{1}{r})}\sum_{\ell>j}2^{-\kappa\ell}2^{-L(\ell-j)}\bigg{]}\,.$

Performing the change of variables $j-\ell\to\ell$ and $\ell-j\to\ell$ in the
two sums above, we thus get

$\displaystyle\big{\|}\varphi_{j}\ast
f\big{\|}_{L^{p^{\prime}}_{\mu^{\prime}}}$
$\displaystyle\lesssim\|f\|_{\mathcal{B}^{\kappa,\mu}_{p,\infty}}\bigg{[}2^{-j(\kappa-d(1-\frac{1}{r}))}\sum_{0\leq\ell\leq
j}2^{-(L-\kappa+d(1-\frac{1}{r}))\ell}+2^{-j(\kappa-d(1-\frac{1}{r}))}\sum_{\ell>0}2^{-(L+\kappa)\ell}\bigg{]}\,.$

Now remember that $1-\frac{1}{r}=\frac{1}{p}-\frac{1}{p^{\prime}}$, so that
$\kappa-d(1-\frac{1}{r})=\kappa^{\prime}$ according to (2.13). We thus have
obtained that

$2^{\kappa^{\prime}j}\big{\|}\varphi_{j}\ast
f\big{\|}_{L^{p^{\prime}}_{\mu^{\prime}}}\lesssim\|f\|_{\mathcal{B}^{\kappa,\mu}_{p,\infty}}\bigg{[}\sum_{0\leq\ell\leq
j}2^{-(L-\kappa^{\prime})\ell}+\sum_{\ell>0}2^{-(L+\kappa)\ell}\bigg{]}\,.$
(2.17)

Let us now pick $L>\max(-\kappa,\kappa^{\prime})$ in (2.17). We have obtained

$2^{\kappa^{\prime}j}\big{\|}\varphi_{j}\ast
f\big{\|}_{L^{p^{\prime}}_{\mu^{\prime}}}\lesssim\|f\|_{\mathcal{B}^{\kappa,\mu}_{p,\infty}}\,,$

where the propotional constant depends on $(L,\kappa,\kappa^{\prime})$ only.
To conclude, we simply recall that

$\|f\|_{\mathcal{B}^{\kappa^{\prime},\mu^{\prime}}_{p^{\prime},\infty}}=\sup_{j\geq
0}2^{\kappa^{\prime}j}\big{\|}\varphi_{j}\ast
f\big{\|}_{L^{p^{\prime}}_{\mu^{\prime}}}\,,$

from which our claim is easily deduced. ∎

In order to handle the wave equation (1.6), we will have to multiply the
candidate solution $u$ with the distribution-valued noise $\dot{W}$. The lemma
below ensures a proper definition of this kind of product in the Young sense.
Its proof is deferred to the Appendix (Section A.2).

######  Lemma 2.13.

Fix $\mu_{\ast}>0$. Let $\alpha,\beta\in(0,1]$, $p,p_{1},p_{2}\in(1,\infty)$
and $\mu,\mu_{1},\mu_{2}\in[0,\mu_{\ast}]$ be such that

$\alpha<\beta,\qquad\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}},\quad\text{and}\quad\frac{\mu}{p}=\frac{\mu_{1}}{p_{1}}+\frac{\mu_{2}}{p_{2}}\,.$
(2.18)

Consider $f\in\mathcal{B}_{p_{1},\infty}^{-\alpha,\mu_{1}}$ and
$g\in\mathcal{B}_{p_{2},\infty}^{\beta,\mu_{2}}$. Then it holds that

$\big{\|}f\cdot
g\big{\|}_{\mathcal{B}^{-\alpha,\mu}_{p,\infty}}\lesssim\big{\|}f\big{\|}_{\mathcal{B}^{-\alpha,\mu_{1}}_{p_{1},\infty}}\big{\|}g\big{\|}_{\mathcal{B}^{\beta,\mu_{2}}_{p_{2},\infty}}\,,$
(2.19)

for some proportional constant that only depends on $\mu_{\ast}$.

We close this section by rephrasing relation (2.19) in a way which turns out
to be more convenient for our later computations.

######  Corollary 2.14.

Fix $\mu_{\ast}>0$. Consider some parameters
$\mu,\nu\in[0,\mu_{\ast}],\alpha,\kappa\in(0,1]$ and $p>2$ such that

$0<\mu-\nu\leq\frac{2\mu_{\ast}}{p}\,,\qquad\nu\leq\mu_{\ast}\Big{(}1-\frac{2}{p}\Big{)}\,,\qquad\kappa>\alpha+\frac{d}{p}\,.$

Then setting $\varepsilon:=\frac{p}{2}(\mu-\nu)\in(0,\mu_{\ast}]$ , we have

$\big{\|}f\cdot
g\big{\|}_{\mathcal{B}^{-\alpha,\mu}_{2,\infty}}\lesssim\big{\|}f\big{\|}_{\mathcal{B}^{\kappa,\nu}_{2,\infty}}\big{\|}g\big{\|}_{\mathcal{B}^{-\alpha,\varepsilon}_{p,\infty}}\,,$
(2.20)

for some proportional constant that only depends on $\mu_{\ast}$.

###### Proof.

Set $\bar{\kappa}:=\kappa-\frac{d}{p}$, and consider $r>2$ such that
$\frac{1}{r}+\frac{1}{p}=\frac{1}{2}$. We also set
$\bar{\nu}:=\frac{r}{2}\nu\leq\mu_{\ast}$. Since $\bar{\kappa}>\alpha$, we can
first apply Lemma 2.13 to assert that

$\big{\|}f\cdot
g\big{\|}_{\mathcal{B}^{-\alpha,\mu}_{2,\infty}}\lesssim\big{\|}f\big{\|}_{\mathcal{B}^{\bar{\kappa},\bar{\nu}}_{r,\infty}}\big{\|}g\big{\|}_{\mathcal{B}^{-\alpha,\varepsilon}_{p,\infty}}\,.$

It now only remains to apply Lemma 2.12, which yields
$\|f\|_{\mathcal{B}^{\bar{\kappa},\bar{\nu}}_{r,\infty}}\lesssim\|f\|_{\mathcal{B}^{\kappa,\nu}_{2,\infty}}$.
∎

##  3\. Smoothing effect of the wave kernel

The wave kernel plays a fundamental role in the mild formulation of (1.6). In
this section, we investigate Strichartz type inequalities for this kernel in
the weighted Besov spaces defined in Section 2.

###  3.1. Definition of the wave kernel

In the sequel, we will write $\mathcal{G}_{t}$ for the wave operator on
$\mathbb{R}^{d}$, generated by $\partial^{2}_{tt}-\Delta$. The corresponding
kernel will be denoted by $G_{t}$. For $d=1,2$, the kernel $G_{t}$ has the
following explicit expression:

$G_{t}(x)=\left\\{\begin{array}[]{lll}\displaystyle\frac{1}{2}{\bf
1}_{[|x|<t]}&\text{ if }d=1,\vspace{0.2cm}\\\
\displaystyle\frac{1}{2\pi}\frac{1}{\sqrt{t^{2}-|x|^{2}}}{\bf
1}_{[|x|<t]}&\text{ if }d=2.\end{array}\right.$ (3.1)

It is often easier to express $G_{t}$ in Fourier modes. Namely for a function
$g(t,x)$ we set $\mathcal{F}g(t,\xi)$ or $\hat{g}(t,\xi)$ for the spatial
Fourier transform, defined by

$\hat{g}(t,\xi)=\int_{\mathbb{R}^{d}}e^{-\imath x\cdot\xi}g(t,x)dt\,.$

Then the expression for the Fourier transform of $G_{t}$ is

$\hat{G}_{t}(\xi)=\frac{\sin(t|\xi|)}{|\xi|}\,,\xi\in\mathbb{R}^{d}\,.$ (3.2)

###  3.2. Strichartz type estimates

The aim of this section is to quantify the smoothing effects of the operator
introduced in Section 3.1. We start by two propositions giving some
information about the behavior in time. Note that we are restricted to a
spatial dimension $d=1,2$ for this section. We first state a regularity
estimate in dimension 1.

######  Proposition 3.1.

Assume $d=1$. Let $\mu_{\ast}>0$ be a given constant, and recall that
$\mathcal{G}_{t}$ designates the wave operator. Consider
$\mu\in[0,\mu_{\ast}]$ and $0\leq s<t\leq 1$. Then, for all $f\in
L_{\mu}^{p}$, we have

$\|\\{\mathcal{G}_{t}-\mathcal{G}_{s}\\}f\|_{L^{p}_{\mu}}\lesssim|t-s|\|f\|_{L^{p}_{\mu}}\,,$
(3.3)

where the proportional constant only depends on $\mu_{\ast}$ and $p$. As a
consequence, for all $-\infty<\alpha\leq 1$ and $1<p,q\leq\infty$,

$\|\\{\mathcal{G}_{t}-\mathcal{G}_{s}\\}f\|_{\mathcal{B}^{\alpha,\mu}_{p,q}}\lesssim|t-s|\|f\|_{\mathcal{B}^{\alpha,\mu}_{p,q}}\,,$
(3.4)

where the proportional constant only depends on $\mu_{\ast}$, $p$ and $q$.

###### Proof.

As in the proof Lemma 2.4, we write Definition 2.2 for the $L^{p}$ norm and
recall that $\mathcal{G}_{t}$ admits the kernel $G_{t}$ given by (3.1). We get

$\|\\{\mathcal{G}_{t}-\mathcal{G}_{s}\\}f\|_{L^{p}_{\mu}}^{p}=\int_{\mathbb{R}}dx\,w_{\mu}(x)\bigg{|}\int_{\mathbb{R}}dy\,\\{G_{t}-G_{s}\\}(x-y)f(y)\bigg{|}^{p}.$

Still like in Lemma 2.4, we insert the weight $w_{\mu}(y)$ and invoke the fact
that $\text{Supp}\,(G_{t}-G_{s})\subset[-2,2]$. This yields

$\displaystyle\|\\{\mathcal{G}_{t}-\mathcal{G}_{s}\\}f\|_{L^{p}_{\mu}}^{p}$
$\displaystyle=\int_{\mathbb{R}}dx\,\bigg{|}\int_{\mathbb{R}}dy\,\bigg{(}\frac{w_{\mu}(x)}{w_{\mu}(y)}\bigg{)}^{\frac{1}{p}}\\{G_{t}-G_{s}\\}(x-y)\big{[}w_{\mu}(y)^{\frac{1}{p}}f(y)\big{]}\bigg{|}^{p}$
$\displaystyle\lesssim\int_{\mathbb{R}}dx\,\bigg{|}\int_{\mathbb{R}}dy\,\big{|}\\{G_{t}-G_{s}\\}(x-y)\big{|}\big{|}w_{\mu}(y)^{\frac{1}{p}}f(y)\big{|}\bigg{|}^{p}\,,$

where the proportional constant only depends on $\mu_{\ast}$ and $p$.

Taking into account the expression (3.1) for $d=1$, it is readily checked that

$\|G_{t}-G_{s}\|_{L^{1}(\mathbb{R})}\leq|t-s|\,.$ (3.5)

Therefore, a direct application of Young’s inequality (see also relation
(2.5)) leads to

$\|\\{\mathcal{G}_{t}-\mathcal{G}_{s}\\}f\|_{L^{p}_{\mu}}^{p}\lesssim\|G_{t}-G_{s}\|_{L^{1}}^{p}\|f\|_{L^{p}_{\mu}}^{p}\lesssim|t-s|^{p}\|f\|_{L^{p}_{\mu}}^{p}\,,$
(3.6)

which proves our claim (3.3).

As for (3.4), it now suffices to observe that for every $\ell\geq 0$,

$2^{\ell\alpha}\|\boldsymbol{\varphi}_{\ell}\ast\\{\mathcal{G}_{t}-\mathcal{G}_{s}\\}f\|_{L^{p}_{\mu}}=2^{\ell\alpha}\|\\{\mathcal{G}_{t}-\mathcal{G}_{s}\\}(\boldsymbol{\varphi}_{\ell}\ast
f)\|_{L^{p}_{\mu}}\lesssim
2^{\ell\alpha}|t-s|\|\boldsymbol{\varphi}_{\ell}\ast f\|_{L^{p}_{\mu}}\,.$

Inserting this expression into the definition (2.10) of Besov norm, the
desired inequality is achieved. ∎

We now turn to the corresponding property in the 2-dimensional case. We start
with a lemma which summarizes the distinction between the 1-d and the 2-d
cases. To get a quick understanding of this, recall from (3.5) that for $d=1$

$\|G_{t}-G_{s}\|_{L^{1}(\mathbb{R})}\leq|t-s|\,,$

while the same result for $d=2$ reads

######  Lemma 3.2.

Assume $d=2$ and consider the kernel $G_{t}$ defined by (3.1). Then for all
$0\leq s<t\leq 1$, we have

$\big{\|}G_{t}-G_{s}\big{\|}_{L^{1}}\lesssim|t-s|^{\frac{1}{2}}\,.$ (3.7)

###### Proof.

According to relation (3.1), for $d=2$ we have

$\displaystyle\big{\|}G_{t}-G_{s}\big{\|}_{L^{1}}$
$\displaystyle=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}dx\,\bigg{|}\frac{1}{\sqrt{t^{2}-|x|^{2}}}{\bf
1}_{\\{[x|<t\\}}-\frac{1}{\sqrt{s^{2}-|x|^{2}}}{\bf 1}_{\\{[x|<s\\}}\bigg{|}$
$\displaystyle\lesssim\int_{s<|x|<t}\frac{dx}{\sqrt{t^{2}-|x|^{2}}}+\int_{|x|<s}dx\,\bigg{|}\frac{1}{\sqrt{t^{2}-|x|^{2}}}-\frac{1}{\sqrt{s^{2}-|x|^{2}}}\bigg{|}\,.$

Next we use a polar change of coordinates and set $r:=r^{2}$ in order to get

$\displaystyle\big{\|}G_{t}-G_{s}\big{\|}_{L^{1}}$
$\displaystyle\lesssim\int_{s}^{t}\frac{r\,dr}{\sqrt{t^{2}-r^{2}}}+\int_{0}^{s}dr\,r\,\bigg{|}\frac{1}{\sqrt{t^{2}-r^{2}}}-\frac{1}{\sqrt{s^{2}-r^{2}}}\bigg{|}$
$\displaystyle\lesssim\int_{s^{2}}^{t^{2}}\frac{dr}{\sqrt{t^{2}-r}}+\int_{0}^{s^{2}}dr\,\bigg{|}\frac{1}{\sqrt{t^{2}-r}}-\frac{1}{\sqrt{s^{2}-r}}\bigg{|}\,.$

Furthermore, since $s<t$ we have $(s^{2}-r)^{-1/2}>(t^{2}-r)^{-1/2}$ for
$r\in[0,s^{2}]$. This yields

$\displaystyle\big{\|}G_{t}-G_{s}\big{\|}_{L^{1}}$
$\displaystyle\lesssim\int_{s^{2}}^{t^{2}}\frac{dr}{\sqrt{t^{2}-r}}+\int_{0}^{s^{2}}dr\,\bigg{(}\frac{1}{\sqrt{s^{2}-r}}-\frac{1}{\sqrt{t^{2}-r}}\bigg{)}\,.$

We now simply integrate the right-hand side above in order to get (recall that
we assume $s,t\in[0,1]$ for this lemma)

$\|G_{t}-G_{s}\|_{L^{1}}\lesssim\Big{(}(t+s)^{\frac{1}{2}}-(t-s)^{\frac{1}{2}}\Big{)}(t-s)^{\frac{1}{2}}\lesssim(t-s)^{\frac{1}{2}}\,,$

which proves our claim (3.7). ∎

As a direct consequence, we have the following result parallel to Proposition
3.1.

######  Proposition 3.3.

Assume $d=2$. Let $\mu_{\ast}>0$ be a given constant, and recall that
$\mathcal{G}_{t}$ designates the wave operator. Consider
$\mu\in[0,\mu_{\ast}]$ and $0\leq s<t\leq 1$. Then, for all $f\in
L_{\mu}^{p}$, we have

$\|\\{\mathcal{G}_{t}-\mathcal{G}_{s}\\}f\|_{L^{p}_{\mu}}\lesssim|t-s|^{\frac{1}{2}}\|f\|_{L^{p}_{\mu}}\,,$
(3.8)

where the proportional constant only depends on $\mu_{\ast}$ and $p$. As a
consequence, for all $-\infty<\alpha\leq 1$ and $1<p,q\leq\infty$,

$\|\\{\mathcal{G}_{t}-\mathcal{G}_{s}\\}f\|_{\mathcal{B}^{\alpha,\mu}_{p,q}}\lesssim|t-s|^{\frac{1}{2}}\|f\|_{\mathcal{B}^{\alpha,\mu}_{p,q}}\,,$
(3.9)

where the proportional constant only depends on $\mu_{\ast}$, $p$ and $q$.

###### Proof.

The proof follows exactly that of Proposition 3.1, except for replacing (3.5)
by relation (3.7). ∎

We now present our main result for this section, quantifying the smoothing
effect of $\mathcal{G}_{t}$ in our Besov scale.

######  Proposition 3.4.

In this proposition the dimension parameter takes the values $1$ or $2$. As in
Proposition 3.1, we consider a constant $\mu_{\ast}>0$ and the wave operator
$\mathcal{G}_{t}$. Let $\alpha,\mu,p,q$ be four parameters with

$-\infty<\alpha\leq 0,\qquad 0\leq\mu\leq\mu_{\ast},\qquad 1<p<\infty,\qquad
1<q\leq\infty.$ (3.10)

Recall that the weighted Besov spaces are introduced in Definition 2.7. Then
for all $t\in[0,1]$, it holds that

$\big{\|}\mathcal{G}_{t}f\big{\|}_{\mathcal{B}_{p,q}^{\alpha+\rho_{d},\mu}}\lesssim\|f\|_{\mathcal{B}_{p,q}^{\alpha,\mu}}\,,\text{
with }\rho_{d}:=\begin{cases}1&\text{if}\ d=1\\\ \frac{1}{2}&\text{if}\
d=2\end{cases}\,,$ (3.11)

and where the proportional constant only depends on $p$, $q$ and $\mu_{\ast}$.

Before starting the proof of Proposition 3.4, we label a technical lemma about
integrals of the wave kernel.

######  Lemma 3.5.

In the setting of Proposition 3.4, and for all $t\in[0,1]$, $j\geq 0$,
$y\in\mathbb{R}^{d}$, define the quantity

$\mathcal{K}^{(d)}_{t,j}(y):=2^{j\rho_{d}}\int_{\mathbf{B}_{2}}dz\,\big{|}G_{t}(y-2^{-j}z)-G_{t}(y)\big{|}\,.$
(3.12)

Then it holds that

$\text{Supp}\,\mathcal{K}^{(d)}_{t,j}\subset\mathbf{B}_{t+2},\quad\text{and}\quad\sup_{j\geq
0}\sup_{t\in[0,1]}\big{\|}\mathcal{K}^{(d)}_{t,j}\big{\|}_{L^{1}}\,<\infty.$
(3.13)

###### Proof.

See Appendix. ∎

######  Remark 3.6.

The presence of a $2^{j\rho_{d}}$ term in front of the integral in (3.12) is
what will guarantee a smoothing effect from $\mathcal{B}^{\alpha}$ to a
$\mathcal{B}^{\alpha+\rho_{d}}$ space.

###### Proof of Proposition 3.4.

We begin this proof like in Lemma 2.12. Namely we introduce a parameter $L\geq
0$ whose exact value is to be determined later on. Then we assume that the
construction of our weighted Besov spaces hinges on a function $\varphi_{0}$
such that $\varphi\in\mathcal{D}_{L}$ (where $\varphi$ is defined along
Notation 2.6).

Thanks to Theorem A.1 in the Appendix, we know that there exist two functions
$\psi_{0},\psi\in\mathcal{C}_{c}^{\infty}$ such that $\psi\in\mathcal{D}_{L}$
and for every $\ell\geq 1$ we have

$\varphi_{\ell}\ast\mathcal{G}_{t}f=\varphi_{\ell}\ast G_{t}\ast f=\sum_{j\geq
0}(G_{t}\ast\varphi_{\ell}\ast\psi_{j})\ast(\varphi_{j}\ast f)\,,$ (3.14)

where we recall that $G_{t}$ is the wave kernel given by (3.1). In relation
(3.14), recall that we have assumed the supports of $\varphi_{0}$, $\varphi$,
$\psi$ and $\psi_{0}$ to be all included in the unit ball $\mathbf{B}_{1}$.
However, the arguments could be easily extended to any supporting ball
$\mathbf{B}_{K}$. Let us now split the above sum (3.14) into

$\sum_{j\geq 0}(G_{t}\ast\varphi_{\ell}\ast\psi_{j})\ast(\varphi_{j}\ast
f)=\sum_{0\leq
j<\ell}(G_{t}\ast\varphi_{\ell}\ast\psi_{j})\ast(\varphi_{j}\ast
f)+(G_{t}\ast\varphi_{\ell}\ast\psi_{\ell})\ast(\varphi_{\ell}\ast f)\\\
+\sum_{j>\ell}(G_{t}\ast\varphi_{\ell}\ast\psi_{j})\ast(\varphi_{j}\ast
f):=I_{\ell}+I\\!I_{\ell}+I\\!I\\!I_{\ell}\,.$ (3.15)

Our inequality (3.11) is now easily seen to be reduced to prove

$\bigg{(}\sum_{\ell\geq
0}2^{\ell(\alpha+\rho_{d})q}\left(\|I_{\ell}\|^{q}_{L_{\mu}^{p}}+\|I\\!I_{\ell}\|^{q}_{L_{\mu}^{p}}+\|I\\!I\\!I_{\ell}\|^{q}_{L_{\mu}^{p}}\right)\bigg{)}^{1/q}\lesssim\|f\|_{\mathcal{B}_{p,q}^{\alpha,\mu}}\,.$
(3.16)

We will handle the 3 terms on the left-hand side of (3.16) separately.

Step 1: Bound on $I_{\ell}$. With (3.11) and (3.16) in mind, our aim is to
prove an inequality of the form

$\bigg{(}\sum_{\ell\geq
0}2^{\ell(\alpha+\rho_{d})q}\|I_{\ell}\|_{L_{\mu}^{p}}^{q}\bigg{)}^{\frac{1}{q}}\lesssim\|f\|_{\mathcal{B}_{p,q}^{\alpha,\mu}}\,.$
(3.17)

Our first claim is that for $0\leq j<\ell$ we can write

$\varphi_{\ell}\ast\psi_{j}=(\bar{\psi}\ast\varphi_{\ell-j})_{j}\,,\quad\text{where}\quad\bar{\psi}:=\begin{cases}\psi_{0}&\text{if}\
j=0\\\ \psi&\text{if}\ j\geq 1\end{cases}\,.$ (3.18)

Indeed, relation (3.18) is trivial whenever $j=0$ (note that we have used the
convention $g_{0}=g$ for any given $g\in\mathcal{C}_{c}^{\infty}$ in the
right-hand side of (3.18)). For $j\geq 1$, the left-hand side of (3.18) can be
expressed as

$(\varphi_{\ell}\ast\psi_{j})(x)=2^{\ell+j}\int_{\mathbb{R}^{d}}\varphi(2^{\ell}(x-y))\psi(2^{j}y)dy\,,$
(3.19)

while the right-hand side of (3.18) has the form

$(\varphi_{\ell-j}\ast\psi)_{j}(x)=2^{\ell}\int_{\mathbb{R}^{d}}\varphi(2^{\ell}x-2^{\ell-j}y)\psi(y)dy\,.$
(3.20)

Then an elementary change of variable $z=2^{-j}y$ shows that (3.19) and (3.20)
are equal.

Invoking (3.18), together with the change of variable $2^{j}z\to z$, it is
then readily checked that for $0\leq j<\ell$ and $y\in\mathbb{R}$ the term
$G_{t}*\varphi_{\ell}*\psi_{j}$ in (3.15) can be decomposed as

$(G_{t}\ast\varphi_{\ell}\ast\psi_{j})(y)=\int_{\mathbb{R}^{d}}dz\,(\bar{\psi}\ast\varphi_{\ell-j})_{j}(z)G_{t}(y-z)=\int_{\mathbb{R}^{d}}dz\,(\bar{\psi}\ast\varphi_{\ell-j})(z)G_{t}(y-2^{-j}z)\,.$
(3.21)

Let us take advantage of some cancellations in the right-hand of (3.21).
Indeed, owing to the fact that $\int_{\mathbb{R}^{d}}\varphi_{k}(x)dx=0$ for
all $k>0$, we also have

$G_{t}(y)\int_{\mathbb{R}^{d}}dz\,(\bar{\psi}\ast\varphi_{k})(z)=0.$

Subtracting the above quantity to the right-hand side of (3.21), we get that
for all $\ell>j$,

$\displaystyle(G_{t}\ast\varphi_{\ell}\ast\psi_{j})(y)$
$\displaystyle=\int_{\mathbb{R}^{d}}dz\,(\bar{\psi}\ast\varphi_{\ell-j})(z)\big{\\{}G_{t}(y-2^{-j}z)-G_{t}(y)\big{\\}}.$

In addition, observing
$\text{Supp}\,(\bar{\psi}\ast\varphi_{\ell-j})\subset\mathbf{B}_{2}$, we have
that

$\displaystyle(G_{t}\ast\varphi_{\ell}\ast\psi_{j})(y)$
$\displaystyle=\int_{\mathbf{B}_{2}}dz\,(\bar{\psi}\ast\varphi_{\ell-j})(z)\big{\\{}G_{t}(y-2^{-j}z)-G_{t}(y)\big{\\}}\,.$
(3.22)

With (3.22) in hand, the key point now is that due to (2.3) one has easily,
for all $0\leq j<\ell$,

$\|\bar{\psi}\ast\varphi_{\ell-j}\|_{\infty}\lesssim 2^{-L(\ell-j)}\,,$ (3.23)

which follows from Lemma 2.5 directly. Therefore, using the notation
introduced in (3.12), we have obtained that

$|(G_{t}*\varphi_{\ell}*\psi_{j})(y)|\leq
2^{-L(\ell-j)}2^{-j\rho_{d}}\mathcal{K}^{(d)}_{t,j}(y)\,.$ (3.24)

Let us insert (3.24) in $I_{\ell}$ in order to upper bound this term. That is
according to the definition (3.15) of $I_{\ell}$, it is obvious that

$\displaystyle|I_{\ell}(x)|$ $\displaystyle\leq\sum_{0\leq
j<\ell}\int_{\mathbb{R}^{d}}dy\,|(G_{t}\ast\varphi_{\ell}\ast\psi_{j})(y)||(\varphi_{j}\ast
f)(x-y)|$ $\displaystyle\leq\sum_{0\leq
j<\ell}2^{-L(\ell-j)}2^{-j\rho_{d}}\int_{\mathbb{R}}dy\,\mathcal{K}^{(d)}_{t,j}(y)|(\varphi_{j}\ast
f)(x-y)|$ $\displaystyle\leq\sum_{0\leq
j<\ell}2^{-L(\ell-j)}2^{-j\rho_{d}}\big{[}\mathcal{K}^{(d)}_{t,j}\ast|\varphi_{j}\ast
f|\big{]}(x).$ (3.25)

We are now ready to analyze the weighted $L^{p}$ norms of the function
$I_{\ell}$. To this aim, we multiply (3.25) by $2^{\ell(\alpha+\rho_{d})}$ and
compute the $L^{p}_{\mu}$-norm given in Definition 2.2 for
$\mu\leq\mu_{\ast}$. We get

$\displaystyle 2^{\ell(\alpha+\rho_{d})}\|I_{\ell}\|_{L^{p}_{\mu}}$
$\displaystyle\lesssim\sum_{0\leq
j<\ell}2^{\ell(\alpha+\rho_{d})}2^{-L(\ell-j)}2^{-j\rho_{d}}2^{-j\alpha}\Big{[}2^{j\alpha}\big{\|}\mathcal{K}^{(d)}_{t,j}\ast|\varphi_{j}\ast
f|\big{\|}_{L^{p}_{\mu}}\Big{]}$ $\displaystyle\lesssim\sum_{0\leq
j<\ell}2^{(\ell-j)(\alpha+\rho_{d}-L)}\Big{[}2^{j\alpha}\big{\|}\mathcal{K}^{(d)}_{t,j}\ast|\varphi_{j}\ast
f|\big{\|}_{L^{p}_{\mu}}\Big{]}.$ (3.26)

At this point, we can combine the results of Lemma 2.4, Lemma 3.5 and Young’s
inequality to assert that

$\displaystyle\big{\|}\mathcal{K}^{(d)}_{t,j}\ast|\varphi_{j}\ast
f|\big{\|}_{L^{p}_{\mu}}$
$\displaystyle\lesssim\big{\|}|\mathcal{K}^{(d)}_{t,j}|\ast
w_{\mu}^{1/p}|\varphi_{j}\ast f|\big{\|}_{L^{p}}$
$\displaystyle\leq\big{\|}\mathcal{K}^{(d)}_{t,j}\big{\|}_{L^{1}}\big{\|}w_{\mu}^{1/p}|\varphi_{j}\ast
f|\big{\|}_{L^{p}}$ $\displaystyle\lesssim\big{\|}\varphi_{j}\ast
f\big{\|}_{L^{p}_{\mu}}\,,$ (3.27)

for some proportional constant that depends neither on $j$ nor on $t$.
Injecting this uniform bound into (3.26), we deduce

$2^{\ell(\alpha+\rho_{d})}\|I_{\ell}\|_{L^{p}_{\mu}}\lesssim\sum_{0\leq
j<\ell}2^{(\ell-j)(\alpha+\rho_{d}-L)}\Big{[}2^{j\alpha}\big{\|}\varphi_{j}\ast
f\big{\|}_{L^{p}_{\mu}}\Big{]}\,.$ (3.28)

We shall recast the right-hand side of (3.28) as a discrete convolution.
Namely, if we set

$u_{j}:=2^{j(\alpha+\rho_{d}-L)}{\bf 1}_{\\{j\geq 1\\}}\,,\quad\text{and}\quad
v_{j}:=2^{j\alpha}\|\varphi_{j}\ast f\|_{L^{p}_{\mu}}{\bf 1}_{\\{j\geq
0\\}}\,,$

and if $\ast_{\mathbb{Z}}$ refers to the discrete convolution in $\mathbb{Z}$,
then (3.28) can be written as

$2^{\ell(\alpha+\rho_{d})}\|I_{\ell}\|_{L^{p}_{\mu}}\lesssim(u\ast_{\mathbb{Z}}v)_{\ell}\,.$
(3.29)

The inequality above enables the application of Young’s inequality, similarly
to what we did in (2.16). We end up with

$\displaystyle\big{\|}\ell\mapsto
2^{\ell(\alpha+\rho_{d})}\|I_{\ell}\|_{L^{p}_{\mu}}\big{\|}_{\ell^{q}(\mathbb{N})}$
$\displaystyle\lesssim\big{\|}j\mapsto
2^{j(\alpha+\rho_{d}-L)}\big{\|}_{\ell^{1}(\mathbb{N})}\,\big{\|}j\mapsto
2^{j\alpha}\|\varphi_{j}\ast
f\|_{L^{p}_{\mu}}\big{\|}_{\ell^{q}(\mathbb{N})}\,.$ (3.30)

Let us finally pick $L>\rho_{d}+\alpha$, so that the above estimate yields

$\bigg{(}\sum_{\ell\geq
0}2^{\ell(\alpha+\rho_{d})q}\|I_{\ell}\|^{q}_{L_{\mu}^{p}}\bigg{)}^{1/q}\lesssim\|f\|_{\mathcal{B}^{\alpha,\mu}_{p,q}}\,,$
(3.31)

which is our claim (3.17) and the first part of relation (3.16).

Step 2: Bound on $I\\!I_{\ell}$. Recall from (3.15) that
$I\\!I_{\ell}=(G_{t}*\varphi_{\ell}*\psi_{\ell})*(\varphi_{\ell}*f)$. Our aim
for this step is to obtain inequality (3.31) for the sequence
$\\{I\\!I_{\ell},\ell\geq 0\\}$. Whenever $\ell=0$, we appeal again to Lemma
2.4 in order to write

$\displaystyle\|I\\!I_{0}\|_{L^{p}_{\mu}}$
$\displaystyle\lesssim\|G_{t}\ast\varphi_{0}\ast\psi_{0}\|_{L^{1}}\|\varphi_{0}\ast
f\|_{L^{p}_{\mu}}\lesssim\|G_{t}\|_{L^{1}}\|\varphi_{0}\|_{L^{1}}\|\psi_{0}\|_{L^{1}}\|\varphi_{0}\ast
f\|_{L^{p}_{\mu}}\,.$

Furthermore, $\|\varphi_{0}\|_{L^{1}},\|\psi_{0}\|_{L^{1}}$ above are finite
quantities. Since $t\leq 1$, the norm $\|G_{t}\|_{L^{1}}$ is also bounded,
along the same lines as for (3.3). We thus get

$\displaystyle\|I\\!I_{0}\|_{L^{p}_{\mu}}\lesssim\|\varphi_{0}\ast
f\|_{L^{p}_{\mu}}\,,$ (3.32)

where the proportional constant only depends on $p$ and $\mu_{\ast}$.

For $\ell\geq 1$, observe that similarly to (3.18) we have
$\varphi_{\ell}\ast\psi_{\ell}=(\varphi\ast\psi)_{\ell}$. We can then use the
same arguments as in (3.21)-(3.24) (replace $\varphi_{\ell-j}$ with $\varphi$,
$\bar{\psi}$ with $\psi$ and $j$ with $\ell$) to derive that for all
$y\in\mathbb{R}$,

$\big{|}(G_{t}\ast\varphi_{\ell}\ast\psi_{\ell})(y)\big{|}\lesssim\|\varphi\ast\psi\|_{\infty}2^{-\ell\rho_{d}}\mathcal{K}^{(d)}_{t,\ell}(y)\,.$

Just as in Step 1 (see (3.26)-(3.2)), this entails that for $\ell\geq 1$,

$2^{\ell(\alpha+\rho_{d})}\|I\\!I_{\ell}\|_{L^{p}_{\mu}}\lesssim
2^{\ell\alpha}\|\mathcal{K}^{(d)}_{t,\ell}\|_{L^{1}}\|\varphi_{\ell}\ast
f\|_{L^{p}_{\mu}}\lesssim 2^{\ell\alpha}\|\varphi_{\ell}\ast
f\|_{L^{p}_{\mu}}$

for some proportional constant that only depends on $p$ and $\mu_{\ast}$.
Summing the above inequality over $\ell$ and applying Definition 2.7, we have
shown the second part of relation (3.16), that is

$\bigg{(}\sum_{\ell\geq
0}2^{\ell(\alpha+\rho_{d})q}\|I\\!I_{\ell}\|^{q}_{L_{\mu}^{p}}\bigg{)}^{1/q}\lesssim\bigg{(}\sum_{\ell\geq
0}2^{\ell\alpha q}\|\varphi_{\ell}\ast
f\|^{q}_{L_{\mu}^{p}}\bigg{)}^{1/q}\lesssim\|f\|_{\mathcal{B}_{p,q}^{\alpha,\mu}}\,.$
(3.33)

Step 3: Bound on $I\\!I\\!I_{\ell}$. We now treat the term $I\\!I\\!I_{\ell}$
in (3.15). This will be done along the same lines as for the other steps, and
we thus skip some details of computation. First, since $I\\!I\\!I_{\ell}$
involves a sum over $j>\ell$, we replace (3.18) by

$\varphi_{\ell}\ast\psi_{j}=(\bar{\varphi}\ast\psi_{j-\ell})_{\ell}\,,\
\text{where}\ \bar{\varphi}:=\begin{cases}\varphi_{0}&\text{if}\ \ell=0\\\
\varphi&\text{if}\ \ell\geq 1\end{cases}\,.$

Thus, for all $j>\ell$ and $y\in\mathbb{R}$, the equivalent of (3.21) is

$(G_{t}\ast\varphi_{\ell}\ast\psi_{j})(y)=\int_{\mathbb{R}^{d}}dz\,(\bar{\varphi}\ast\psi_{j-\ell})_{\ell}(z)G_{t}(y-z)=\int_{\mathbb{R}^{d}}dz\,(\bar{\varphi}\ast\psi_{j-\ell})(z)G_{t}(y-2^{-\ell}z)\,.$

Just as in Step 1, it holds that
$\int_{\mathbb{R}}dz\,(\bar{\varphi}\ast\psi_{j-\ell})(z)=0$ and
$\text{Supp}\,(\bar{\varphi}\ast\psi_{j-\ell})\subset\mathbf{B}_{2}$ for all
$j>\ell$. Accordingly, relation (3.22) for the case $j>\ell$ becomes

$\displaystyle(G_{t}\ast\varphi_{\ell}\ast\psi_{j})(y)$
$\displaystyle=\int_{\mathbf{B}_{2}}dz\,(\bar{\varphi}\ast\psi_{j-\ell})(z)\big{\\{}G_{t}(y-2^{-\ell}z)-G_{t}(y)\big{\\}}\,.$

Next using the notation in (3.12), we obtain that for all $j>\ell$ and
$y\in\mathbb{R}$,

$\big{|}(G_{t}\ast\varphi_{\ell}\ast\psi_{j})(y)\big{|}\lesssim\|\bar{\varphi}\ast\psi_{j-\ell}\|_{\infty}2^{-\ell\rho_{d}}\mathcal{K}^{(d)}_{t,\ell}(y)\,.$

In addition for the same reasons as for (3.23) and owing to the fact that
$\psi\in\mathcal{D}_{L}$, we have

$\|\bar{\varphi}\ast\psi_{j-\ell}\|_{\infty}\lesssim 2^{-L(j-\ell)}\,.$

Following the arguments in (3.26)-(3.28), we derive

$\displaystyle 2^{\ell(\alpha+\rho_{d})}\|I\\!I\\!I_{\ell}\|_{L^{p}_{\mu}}$
$\displaystyle\lesssim\sum_{j>\ell}2^{\ell(\alpha+\rho_{d})}2^{-L(j-\ell)}2^{-\ell\rho_{d}}2^{-j\alpha}\Big{[}2^{j\alpha}\big{\|}\mathcal{K}^{(d)}_{t,\ell}\ast|\varphi_{j}\ast
f|\big{\|}_{L^{p}_{\mu}}\Big{]}$
$\displaystyle\lesssim\sum_{j>\ell}2^{-(L+\alpha)(j-\ell)}\Big{[}2^{j\alpha}\big{\|}\varphi_{j}\ast
f\big{\|}_{L^{p}_{\mu}}\Big{]}\lesssim\sum_{j\geq
1}2^{-(L+\alpha)j}\Big{[}2^{(j+\ell)\alpha}\big{\|}\varphi_{j+\ell}\ast
f\big{\|}_{L^{p}_{\mu}}\Big{]}\,.$ (3.34)

At this point, remember that the pair $(L,\varphi_{0})$ has been arbitrarily
chosen. Therefore, we can a posteriori pick $(L,\varphi_{0})$ such that
$L>\max(-\alpha,\rho_{d}+\alpha)$ (which is also consistent with the
conditions exhibited in Step 1). Going back to (3.34), this puts us in a
position to apply Jensen inequality and assert that

$\displaystyle\sum_{\ell\geq
0}2^{\ell(\alpha+\rho_{d})q}\|I\\!I\\!I_{\ell}\|^{q}_{L_{\mu}^{p}}$
$\displaystyle\lesssim\sum_{\ell\geq 0}\bigg{(}\sum_{j\geq
1}2^{-(L+\alpha)j}\Big{[}2^{(j+\ell)\alpha}\big{\|}\varphi_{j+\ell}\ast
f\big{\|}_{L^{p}_{\mu}}\Big{]}\bigg{)}^{q}$
$\displaystyle\lesssim\sum_{\ell\geq 0}\sum_{j\geq
1}2^{-(L+\alpha)j}\Big{[}2^{(j+\ell)\alpha}\big{\|}\varphi_{j+\ell}\ast
f\big{\|}_{L^{p}_{\mu}}\Big{]}^{q}$ $\displaystyle\lesssim\sum_{\ell\geq
0}\Big{[}2^{\ell\alpha}\big{\|}\varphi_{\ell}\ast
f\big{\|}_{L^{p}_{\mu}}\Big{]}^{q}\lesssim\|f\|_{\mathcal{B}^{\alpha,\mu}_{p,q}}^{q}.$

Accordingly, we have obtained

$\bigg{(}\sum_{\ell\geq
0}2^{\ell(\alpha+\rho_{d})q}\|I\\!I\\!I_{\ell}\|^{q}_{L_{\mu}^{p}}\bigg{)}^{1/q}\lesssim\|f\|_{\mathcal{B}^{\alpha,\mu}_{p,q}}.$
(3.35)

Step 4: Conclusion. By injecting (3.31), (3.33) and (3.35) into (3.14)-(3.15),
we deduce that for some suitable $\varphi_{0}\in\mathcal{C}_{c}^{\infty}$
satisfying Assumption ($\mathcal{A}_{\varphi}$),

$\big{\|}\ell\mapsto
2^{\ell(\alpha+\rho_{d})}\|\varphi_{\ell}\ast\mathcal{G}_{t}f\|_{L^{p}_{\mu}}\big{\|}_{\ell^{q}(\mathbb{N})}\lesssim\|f\|_{\mathcal{B}^{\alpha,\mu}_{p,q}}\,.$

We are therefore in a position to apply the definition (2.10) of Besov norm
and assert that

$\|\mathcal{G}_{t}f\|_{B^{\alpha+\rho_{d},\mu}_{p,q}(\varphi_{0})}=\big{\|}\ell\mapsto
2^{\ell(\alpha+\rho_{d})}\|\varphi_{\ell}\ast\mathcal{G}_{t}f\|_{L^{p}_{\mu}}\big{\|}_{\ell^{q}(\mathbb{N})}\lesssim\|f\|_{B^{\alpha,\mu}_{p,q}(\varphi_{0})}\,.$

The desired inequality (3.11) is now a straightforward consequence of the
equivalence property stated in Proposition 2.8. ∎

We close this section by combining the previous results in order to obtain an
appropriate regularity result for the operator $\mathcal{G}_{t}$.

######  Corollary 3.7.

We work under the same conditions as for Proposition 3.4. Namely, consider the
wave operator $\mathcal{G}_{t}$, a constant $\mu_{\ast}>0$, and set

$\rho_{d}:=\begin{cases}1&\text{if}\ d=1\\\ \frac{1}{2}&\text{if}\
d=2.\end{cases}$

Let $\alpha,\mu,p,q$ be parameters such that (3.10) is fulfilled. Let $\kappa$
be an additional coefficient in $[0,\rho_{d}]$. Then it holds that for all
$0\leq s<t\leq 1$,

$\|\\{\mathcal{G}_{t}-\mathcal{G}_{s}\\}f\|_{\mathcal{B}^{\alpha+\kappa,\mu}_{p,q}}\lesssim|t-s|^{\rho_{d}-\kappa}\|f\|_{\mathcal{B}^{\alpha,\mu}_{p,q}}\,,$
(3.36)

for some proportional constant that depends only on $\mu_{\ast}$.

###### Proof.

The proof of this result, which is a standard application of interpolation
properties, is included here for the sake of completeness. Indeed, write
$\alpha+\kappa=(1-\frac{\kappa}{\rho_{d}})\alpha+\frac{\kappa}{\rho_{d}}(\alpha+\rho_{d})$.
Then apply inequality (2.12) with
$s_{0}=\alpha,s_{1}=\alpha+\rho_{d},\theta=\frac{\kappa}{\rho_{d}},p_{0}=p_{1}=p,q_{0}=q_{1}=q.$
We obtain

$\|(\mathcal{G}_{t}-\mathcal{G}_{s})f\|_{\mathcal{B}^{\alpha+\kappa,\mu}_{p,q}}\leq\|(\mathcal{G}_{t}-\mathcal{G}_{s})f\|_{\mathcal{B}^{\alpha,\mu}_{p,q}}^{1-\frac{\kappa}{\rho_{d}}}\|(\mathcal{G}_{t}-\mathcal{G}_{s})f\|_{\mathcal{B}^{\alpha+\rho_{d},\mu}_{p,q}}^{\frac{\kappa}{\rho_{d}}}\,.$
(3.37)

Next, for the first term on the right-hand side of (3.37) we apply (3.4) (for
$d=1$) or (3.8) (for $d=2$) , while for the second term we invoke (3.11) to
write

$\|(\mathcal{G}_{t}-\mathcal{G}_{s})f\|_{\mathcal{B}_{p,q}^{\alpha+\rho_{d},\mu}}\leq\|\mathcal{G}_{t}f\|_{\mathcal{B}_{p,q}^{\alpha+\rho_{d},\mu}}+\|\mathcal{G}_{s}f\|_{\mathcal{B}_{p,q}^{\alpha+\rho_{d},\mu}}\lesssim\|f\|_{\mathcal{B}^{\alpha,\mu}_{p,q}}\,.$

Going back to (3.37), our conclusion easily follows. ∎

##  4\. Young wave equation

This section aims to develop the Young-type integration theory that will allow
us to interpret and solve (1.6) pathwisely in Section 5. For this goal, we
will introduce an appropriate space of processes and define the Young wave
integral in Section 4.1. Then we will prove the wellposedness of the
corresponding Young wave equation in Section 4.2.

###  4.1. Young wave integral

This subsection is devoted to introducing a convenient space of processes $u$,
allowing the definition of noisy integrals weighted by the wave operator. We
start by giving a new piece of notation.

######  Notation 4.1.

Throughout the section we fix two parameters $a\geq 0$ and $b>0$. We then set
$\mu_{t}=a+bt$.

Next we introduce the set of space-time functions which will allow a proper
integration theory in our noisy context.

######  Definition 4.2.

Fix $T>0$ and let $(\mu_{t})_{t\in[0,T]}$ be a time-dependent weight as in
Notation 4.1. Consider two parameters $\gamma,\kappa\in[0,1]$. Then we define
$\mathcal{E}^{\gamma,\kappa}_{2,\infty}(T)$ as the set of functions
$u:[0,T]\times\mathbb{R}\to\mathbb{R}$ for which the following norm is finite:

$\|u\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(T)}:=\sup_{s\in[0,T]}\|u_{s}\|_{\mathcal{B}^{\kappa,\mu_{s}}_{2,\infty}}+\sup_{0\leq
s<t\leq
T}\frac{\|u_{t}-u_{s}\|_{\mathcal{B}^{\kappa,\mu_{t}}_{2,\infty}}}{|t-s|^{\gamma}}\,.$
(4.1)

For processes $u$ in a space of the form
$\mathcal{E}^{\gamma,\kappa}_{2,\infty}$, let us define some Riemann sum
approximation of the integral
$\int_{0}^{t}\int_{\mathbb{R}}G_{t-s}(x,y)u_{s}(y)\dot{W}(ds,dy).$ This
definition is given below.

######  Definition 4.3.

Let $P(x)=(1+|x|^{2})^{-1}$ be the polynomial given in (2.2) for $d=1$. We
consider two regularity parameters $\alpha,\theta\in(0,1)$ and an
integrability parameter $p>2.$ Let $\dot{W}$ be a noisy input which is
$\theta$-Hölder continuous in time with values in the weighted space
$\mathcal{B}_{p,\infty}^{-\alpha,P}$; that is
$\dot{W}\in\mathcal{C}^{\theta}([0,T];\mathcal{B}^{-\alpha,P}_{p,\infty}).$ In
addition, let $\Pi^{n}$ be the regular dyadic partition of $[0,T]$, whose
generic element is $t_{m}^{n}=mT/2^{n}$ for all $0\leq m\leq 2^{n}$. For $u$
regular enough we define the following Riemann sum based on $\Pi^{n}$,

$\mathcal{J}^{(n)}_{t}:=\sum_{k=0}^{m-1}\mathcal{G}_{t-t^{n}_{k}}(u_{t^{n}_{k}}\,\delta\dot{W}_{t^{n}_{k}t^{n}_{k+1}}),~{}\mbox{
for }t\in(t^{n}_{m-1},t^{n}_{m}]\,,$ (4.2)

where we recall that $\delta\dot{W}_{t^{n}_{k}t^{n}_{k+1}}$ stands for
$\dot{W}_{t^{n}_{k+1}}-\dot{W}_{t^{n}_{k}}$.

We now state a proposition giving natural conditions such that the Riemann
sums (4.2) converge in the Young sense. This result will be the basic brick
towards a proper wellposedness of equation (1.6). For this statement we set,
just as in Corollary 3.7,

$\rho_{d}:=\begin{cases}1&\text{if}\ d=1\\\ \frac{1}{2}&\text{if}\
d=2.\end{cases}$ (4.3)

######  Proposition 4.4.

Fix two times $0<T\leq T_{0}$ . Let $\gamma,\theta$ be two time regularity
parameters and $\kappa,\alpha$ be two space regularity parameters. Also
consider $p>d+1$. We assume the following conditions.

  1. (i)

The coefficients $\gamma,\theta,\kappa,\alpha$ all sit in the interval
$[0,1]$, and we have

$\kappa+\alpha+\gamma+(1-\rho_{d})<\theta\,,\quad\gamma+\theta>1\,,\quad\kappa>\alpha+\frac{d}{p}\,,\quad\gamma<1-\frac{d+1}{p}\,.$
(4.4)

  2. (ii)

The process $u$ is an element of $\mathcal{E}^{\gamma,\kappa}_{2,\infty}(T)$
as introduced in Definition 4.2, and $\dot{W}$ belongs to
$\mathcal{C}^{\theta}([0,T];\mathcal{B}^{-\alpha,P}_{p,\infty})$.

Let now $\\{\mathcal{J}^{(n)};n\geq 1\\}$ be the sequence defined by (4.2).
Then we have that $\mathcal{J}^{(n)}$ converges in
$\mathcal{E}_{2,\infty}^{\gamma,\kappa}(T)$. We denote

$\lim_{n\to\infty}\mathcal{J}_{t}^{(n)}=:\int_{0}^{t}\mathcal{G}_{t-r}(u_{r}\,d\dot{W}_{r})\,.$
(4.5)

Moreover, the Young integral (4.5) verifies

$\bigg{\|}\int_{0}^{.}\mathcal{G}_{.-r}(u_{r}\,d\dot{W}_{r})\bigg{\|}_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(T)}\leq\big{\|}\mathcal{G}_{.}(u_{0}\,(\dot{W}_{T}-\dot{W}_{0}))\big{\|}_{\mathcal{E}_{2,\infty}^{\gamma,\kappa}(T)}+c_{T_{0}}\|\dot{W}\|_{\mathcal{C}^{\theta}([0,T];\mathcal{B}^{-\alpha,P}_{p,\infty})}\|u\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(T)}\,,$
(4.6)

where $c_{T_{0}}>0$ does not depend on $T$, $u$ and $\dot{W}$.

######  Remark 4.5.

As the reader will see, we will in fact prove a slightly stronger version of
relation (4.6): there exists a finite constant
$q=q(\gamma,\kappa,\theta,\alpha,p)\geq 1$ such that

$\begin{split}&\bigg{\|}\int_{0}^{.}\mathcal{G}_{.-r}(u_{r}\,d\dot{W}_{r})\bigg{\|}_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(T)}\\\
&\leq\big{\|}\mathcal{G}_{.}(u_{0}\,(\dot{W}_{T}-\dot{W}_{0}))\big{\|}_{\mathcal{E}_{2,\infty}^{\gamma,\kappa}(T)}+c_{T_{0}}\|\dot{W}\|_{\mathcal{C}^{\theta}([0,T];\mathcal{B}^{-\alpha,P}_{p,\infty})}\Big{(}\int_{0}^{T}\|u\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(r)}^{q}\,dr\Big{)}^{\frac{1}{q}}\,,\end{split}$
(4.7)

where $c_{T_{0}}>0$ does not depend on $T$, $u$ and $\dot{W}$. Inequality
(4.7) will be crucial in our fixed point argument for the existence-uniqueness
of a solution to (1.6).

###### Proof of Proposition 4.4.

We shall prove the convergence of $\mathcal{J}^{(n)}$ thanks to an upper bound
on the difference $\mathcal{J}^{(n+1)}-\mathcal{J}^{(n)}$. To this aim, let us
denote $\delta f_{st}$ for the increment $f_{t}-f_{s}$ of a function $f$.
Furthermore, for the sake of clarity, consider two dyadic points
$s=t^{n}_{\ell}$ and $t=t^{n}_{m}$ for $\ell\leq m$. Then an elementary
manipulation on the expression (4.2) for $\mathcal{J}^{(n)}$ reveals that

$\delta\mathcal{J}_{st}^{(n+1)}-\delta\mathcal{J}_{st}^{(n)}=I^{(n)}_{st}+I\\!I^{(n)}_{st}+I\\!I\\!I^{(n)}_{st}+I\\!V^{(n)}_{st}\,,$
(4.8)

where the terms $I^{(n)}_{st},\dots,I\\!V^{(n)}_{st}$ are defined by (writing
$t_{j}=t_{j}^{n+1}$ for notational sake),

$\displaystyle I^{(n)}_{st}$
$\displaystyle=\sum_{k=\ell}^{m-1}\mathcal{G}_{t-t_{2k}}\big{(}\delta
u_{t_{2k}t_{2k+1}}\,\delta\dot{W}_{t_{2k+1}t_{2k+2}}\big{)};$ $\displaystyle
I\\!I^{(n)}_{st}$
$\displaystyle=\sum_{k=\ell}^{m-1}\big{\\{}\mathcal{G}_{t-t_{2k+1}}-\mathcal{G}_{t-t_{2k}}\big{\\}}\big{(}u_{t_{2k+1}}\,\delta\dot{W}_{t_{2k+1}t_{2k+2}}\big{)};$
$\displaystyle I\\!I\\!I_{st}^{(n)}$
$\displaystyle=\sum_{k=0}^{\ell-1}\big{\\{}\mathcal{G}_{t-t_{2k}}-\mathcal{G}_{s-t_{2k}}\big{\\}}\big{(}\delta
u_{t_{2k}t_{2k+1}}\,\delta\dot{W}_{t_{2k+1}t_{2k+2}}\big{)};$ $\displaystyle
I\\!V_{st}^{(n)}$
$\displaystyle=\sum_{k=0}^{\ell-1}\big{\\{}\mathcal{G}_{t-t_{2k+1}}-\mathcal{G}_{t-t_{2k}}-\mathcal{G}_{s-t_{2k+1}}+\mathcal{G}_{s-t_{2k}}\big{\\}}\big{(}u_{t_{2k+1}}\,\delta\dot{W}_{t_{2k+1}t_{2k+2}}\big{)}.$

We now upper bound the four terms above.

Step 0: A general bound on products. Recall that the (fixed) parameters $a$
and $b$ have been introduced in Notation 4.1. Setting

$\mu_{\ast}=\mu_{\ast}(T_{0},p):=\max\bigg{(}\frac{a+bT_{0}}{1-\frac{2}{p}},\frac{pbT_{0}}{2}\bigg{)},$

it is readily checked that for all $0\leq s<t\leq T$,

$\mu_{s}\leq\mu_{\ast}\Big{(}1-\frac{2}{p}\Big{)}\quad\text{and}\quad\frac{p}{2}(\mu_{t}-\mu_{s})\leq\mu_{\ast}.$
(4.9)

Consider our noisy input $\dot{W}$ and a function
$f\in\mathcal{B}_{2,\infty}^{\kappa,\mu_{s}}$ for $0\leq s<t<T$. Then we claim
that for $s\leq r<v\leq t$ we have

$\displaystyle\big{\|}f\cdot\delta\dot{W}_{rv}\big{\|}_{\mathcal{B}^{-\alpha,\mu_{t}}_{2,\infty}}$
$\displaystyle\lesssim\big{\|}f\big{\|}_{\mathcal{B}^{\kappa,\mu_{s}}_{2,\infty}}\frac{|v-r|^{\theta}}{|t-s|^{\frac{d+1}{p}}}\big{\|}\dot{W}\big{\|}_{\mathcal{C}^{\theta}([0,T];\mathcal{B}^{-\alpha,P}_{p,\infty})}\,,$
(4.10)

for some proportional constant that depends only on $\mu_{\ast}$, that is only
on $T_{0}$ and $p$ (note that we will no longer explicitly indicate such a
dependence on $(T_{0},p)$ for the constants arising in the rest of the proof).

To prove (4.10), observe first that, owing to (4.9), we are in a position to
apply our product rule (2.20) and assert that

$\big{\|}f\cdot\delta\dot{W}_{rv}\big{\|}_{\mathcal{B}^{-\alpha,\mu_{t}}_{2,\infty}}\lesssim\big{\|}f\big{\|}_{\mathcal{B}^{\kappa,\mu_{s}}_{2,\infty}}\big{\|}\delta\dot{W}_{rv}\big{\|}_{\mathcal{B}^{-\alpha,\varepsilon_{st}}_{p,\infty}}\,,$

where we have set
$\varepsilon_{st}=\frac{p}{2}(\mu_{t}-\mu_{s})\leq\mu_{\ast}$. Next we invoke
Lemma 2.10 for the right-hand side above, which allows to write

$\big{\|}f\cdot\delta\dot{W}_{rv}\big{\|}_{\mathcal{B}^{-\alpha,\mu_{t}}_{2,\infty}}\lesssim\big{\|}f\big{\|}_{\mathcal{B}^{\kappa,\mu_{s}}_{2,\infty}}\frac{1}{(\mu_{t}-\mu_{s})^{\frac{d+1}{p}}}\big{\|}\delta\dot{W}_{rv}\big{\|}_{\mathcal{B}^{-\alpha,P}_{p,\infty}}\,,$

from which (4.10) is easily deduced.

Step 1: Bound for $I_{st}^{(n)}$. For the term $I_{st}^{(n)}$ in the right-
hand side of (4.8), let us first simply write

$\big{\|}I^{(n)}_{st}\big{\|}_{\mathcal{B}^{\kappa,\mu_{t}}_{2,\infty}}\leq\sum_{k=\ell}^{m-1}\big{\|}\mathcal{G}_{t-t_{2k}}\big{(}\delta
u_{t_{2k}t_{2k+1}}\,\delta\dot{W}_{t_{2k+1}t_{2k+2}}\big{)}\big{\|}_{\mathcal{B}^{\kappa,\mu_{t}}_{2,\infty}}\,.$
(4.11)

Then we apply (3.36), interpolating from a regularity $(-\alpha)$ to a
regularity $\kappa$ and considering the times $t:=t-t_{2k}$ and $s:=0$ (recall
that $G_{0}=0$ owing to (3.1)). This yields

$\big{\|}I^{(n)}_{st}\big{\|}_{\mathcal{B}^{\kappa,\mu_{t}}_{2,\infty}}\lesssim\sum_{k=\ell}^{m-1}|t-t_{2k}|^{\rho_{d}-(\kappa+\alpha)}\big{\|}\delta
u_{t_{2k}t_{2k+1}}\,\delta\dot{W}_{t_{2k+1}t_{2k+2}}\big{\|}_{\mathcal{B}^{-\alpha,\mu_{t}}_{2,\infty}}\,.$
(4.12)

We are now in a position to resort to our general estimate (4.10), which gives

$\big{\|}I^{(n)}_{st}\big{\|}_{\mathcal{B}^{\kappa,\mu_{t}}_{2,\infty}}\lesssim\sum_{k=\ell}^{m-1}|t-t_{2k+1}|^{\rho_{d}-(\kappa+\alpha)}\big{\|}\delta
u_{t_{2k}t_{2k+1}}\big{\|}_{\mathcal{B}^{\kappa,\mu_{t_{2k+1}}}_{2,\infty}}\frac{2^{-n\theta}}{|t-t_{2k+1}|^{\frac{d+1}{p}}}\,\|\dot{W}\|_{\mathcal{C}^{\theta}([0,T];\mathcal{B}^{-\alpha,P}_{p,\infty})}\,.$
(4.13)

Invoking the Definition (4.1) for the norm in
$\mathcal{E}_{2,\infty}^{\gamma,\mu}$ and rearranging terms, we thus get

$\big{\|}I^{(n)}_{st}\big{\|}_{\mathcal{B}^{\kappa,\mu_{t}}_{2,\infty}}\lesssim
2^{-n(\gamma+\theta-1)}\|\dot{W}\|_{\mathcal{C}^{\theta}([0,T];\mathcal{B}^{-\alpha,P}_{p,\infty})}\bigg{(}2^{-n}\sum_{k=\ell}^{m-1}|t-t_{2k+1}|^{\rho_{d}-(\kappa+\alpha)-\frac{d+1}{p}}\|u\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(t_{2k+1})}\bigg{)}\,.$
(4.14)

The last expression in the right-hand side above can be upper bounded by the
continuous integral

$\int_{s}^{t}(t-v)^{\rho_{d}-(\kappa+\alpha)-\frac{d+1}{p}}\|u\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(v)}\,dv.$
(4.15)

Therefore (4.14) can be recast as

$\big{\|}I^{(n)}_{st}\big{\|}_{\mathcal{B}^{\kappa,\mu_{t}}_{2,\infty}}\lesssim
2^{-n(\gamma+\theta-1)}\|\dot{W}\|_{\mathcal{C}^{\theta}([0,T];\mathcal{B}^{-\alpha,P}_{p,\infty})}\,\int_{s}^{t}(t-v)^{\rho_{d}-(\kappa+\alpha)-\frac{d+1}{p}}\|u\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(v)}\,dv\,.$
(4.16)

We now wish to apply Hölder’s inequality to the integral term (4.15) above.
Namely, for $r_{1},q_{1}$ such that $\frac{1}{r_{1}}+\frac{1}{q_{1}}=1$, we
write

$\int_{s}^{t}(t-v)^{\rho_{d}-(\kappa+\alpha)-\frac{d+1}{p}}\|u\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(v)}\,dv\,\\\
\leq\bigg{(}\int_{s}^{t}(t-v)^{r_{1}(\rho_{d}-(\kappa+\alpha)-\frac{d+1}{p})}\,dv\bigg{)}^{\frac{1}{r_{1}}}\bigg{(}\int_{s}^{t}\|u\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(v)}^{q_{1}}\,dv\bigg{)}^{\frac{1}{q_{1}}}\,.$
(4.17)

In order to make the right-hand side of (4.17) finite, we need to have
$r_{1}(\rho_{d}-(\kappa+\alpha)-\frac{d+1}{p})>-1$. Now observe that thanks to
the conditions in (4.4) we can successively guarantee that

$\gamma-\rho_{d}+\kappa+\alpha+\frac{d+1}{p}<\theta-\left(1-\frac{d+1}{p}\right)<\theta-\gamma\leq
1.$

As a result, we can pick $r_{1}>1$ such that

$\gamma-\rho_{d}+\kappa+\alpha+\frac{d+1}{p}<\frac{1}{r_{1}}\quad\Longrightarrow\quad
r_{1}\left(\rho_{d}-(\kappa+\alpha)-\frac{d+1}{p}\right)>-1+r_{1}\gamma.$
(4.18)

With this set of parameters in hand, inequality (4.17) reads

$\displaystyle\int_{s}^{t}(t-v)^{\rho_{d}-(\kappa+\alpha)-\frac{d+1}{p}}\|u\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(v)}\,dv$
$\displaystyle\lesssim$
$\displaystyle|t-s|^{\rho_{d}-(\kappa+\alpha)-\frac{d+1}{p}+\frac{1}{r_{1}}}\bigg{(}\int_{0}^{T}\|u\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(v)}^{q_{1}}\,dv\bigg{)}^{\frac{1}{q_{1}}}$
$\displaystyle\lesssim$
$\displaystyle|t-s|^{\gamma}\bigg{(}\int_{0}^{T}\|u\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(v)}^{q_{1}}\,dv\bigg{)}^{\frac{1}{q_{1}}}\,,$

where we have invoked (4.18) to derive the second step above. Plugging this
inequality into (4.16) and setting $\varepsilon_{1}:=\gamma+\theta-1>0$
(thanks to (4.4)), we end up with

$\big{\|}I^{(n)}_{st}\big{\|}_{\mathcal{B}^{\kappa,\mu_{t}}_{2,\infty}}\lesssim
2^{-n\varepsilon_{1}}|t-s|^{\gamma}\|\dot{W}\|_{\mathcal{C}^{\theta}([0,T];\mathcal{B}^{-\alpha,P}_{p,\infty})}\,\bigg{(}\int_{0}^{T}\|u\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(v)}^{q_{1}}\,dv\bigg{)}^{\frac{1}{q_{1}}}\,.$
(4.19)

Step 2: Bound for $I\\!I_{st}^{(n)}$. The computations are similar to those of
Step 1, and we will just highlight the main differences. First the equivalent
of (4.11) is

$\big{\|}I\\!I^{(n)}_{st}\big{\|}_{\mathcal{B}^{\kappa,\mu_{t}}_{2,\infty}}\leq\sum_{k=\ell}^{m-1}\big{\|}\big{\\{}\mathcal{G}_{t-t_{2k+1}}-\mathcal{G}_{t-t_{2k}}\big{\\}}\big{(}u_{t_{2k+1}}\,\delta\dot{W}_{t_{2k+1}t_{2k+2}}\big{)}\big{\|}_{\mathcal{B}^{\kappa,\mu_{t}}_{2,\infty}}\,.$

Then we invoke the regularity property (3.36) and the fact that
$t_{2k+1}-t_{2k}=2^{-(n+1)}$ in order to get

$\big{\|}I\\!I^{(n)}_{st}\big{\|}_{\mathcal{B}^{\kappa,\mu_{t}}_{2,\infty}}\lesssim
2^{-n(\rho_{d}-(\kappa+\alpha))}\sum_{k=\ell}^{m-1}\big{\|}u_{t_{2k+1}}\,\delta\dot{W}_{t_{2k+1}t_{2k+2}}\big{\|}_{\mathcal{B}^{-\alpha,\mu_{t}}_{2,\infty}}\,,$

which is parallel to (4.12). We now repeat the arguments in (4.13)-(4.14),
which yields

$\big{\|}I\\!I^{(n)}_{st}\big{\|}_{\mathcal{B}^{\kappa,\mu_{t}}_{2,\infty}}\\\
\lesssim
2^{-n(\theta+\rho_{d}-1-(\kappa+\alpha))}\|\dot{W}\|_{\mathcal{C}^{\theta}([0,T];\mathcal{B}^{-\alpha,P}_{p,\infty})}\bigg{(}2^{-n}\sum_{k=\ell}^{m-1}\frac{1}{|t-t_{2k+1}|^{\frac{d+1}{p}}}\|u\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(t_{2k+1})}\bigg{)}\,.$
(4.20)

The Riemann type sum in (4.20) can be upper bounded by
$\int_{s}^{t}(t-v)^{-\frac{d+1}{p}}\|u\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(v)}\,dv$.
Besides, recall from (4.4) that we have assumed $1-\frac{d+1}{p}>\gamma$, and
so there exists $r_{2}>1$ such that $\frac{1}{r_{2}}-\frac{d+1}{p}>\gamma$.
Denoting by $q_{2}$ the (finite) conjugate of $r_{2}$, we can go back to
(4.20) and deduce

$\displaystyle\big{\|}I\\!I^{(n)}_{st}\big{\|}_{\mathcal{B}^{\kappa,\mu_{t}}_{2,\infty}}\lesssim
2^{-n(\theta+\rho_{d}-1-(\kappa+\alpha))}\|\dot{W}\|_{\mathcal{C}^{\theta}([0,T];\mathcal{B}^{-\alpha,P}_{p,\infty})}\int_{s}^{t}(t-v)^{-\frac{d+1}{2}}\|u\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(v)}\,dv$
$\displaystyle\lesssim
2^{-n(\theta+\rho_{d}-1-(\kappa+\alpha))}\|\dot{W}\|_{\mathcal{C}^{\theta}([0,T];\mathcal{B}^{-\alpha,P}_{p,\infty})}\bigg{(}\int_{s}^{t}(t-v)^{-\frac{r_{2}(d+1)}{p}}\,dv\bigg{)}^{\frac{1}{r_{2}}}\bigg{(}\int_{s}^{t}\|u\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(v)}^{q_{2}}\,dv\bigg{)}^{\frac{1}{q_{2}}}$
$\displaystyle\lesssim
2^{-n(\theta+\rho_{d}-1-(\kappa+\alpha))}\|\dot{W}\|_{\mathcal{C}^{\theta}([0,T];\mathcal{B}^{-\alpha,P}_{p,\infty})}|t-s|^{\frac{1}{r_{2}}-\frac{d+1}{p}}\bigg{(}\int_{0}^{T}\|u\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(v)}^{q_{2}}\,dv\bigg{)}^{\frac{1}{q_{2}}},$

which finally yields

$\big{\|}I\\!I^{(n)}_{st}\big{\|}_{\mathcal{B}^{\kappa,\mu_{t}}_{2,\infty}}\lesssim
2^{-n\varepsilon_{2}}\|u\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}}\|\dot{W}\|_{\mathcal{C}^{\theta}([0,T];\mathcal{B}^{-\alpha,P}_{p,\infty})}|t-s|^{\gamma}\bigg{(}\int_{0}^{T}\|u\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(v)}^{q_{2}}\,dv\bigg{)}^{\frac{1}{q_{2}}},$
(4.21)

where $\varepsilon_{2}:=\theta+\rho_{d}-1-(\kappa+\alpha)>\gamma\geq 0$
(thanks to (4.4)).

Step 3: Bound for $I\\!I\\!I_{st}^{(n)}$. The term $I\\!I\\!I_{st}^{(n)}$ in
the right-hand side of (4.8) is treated again as in Step 1 and Step 2. Namely
applying once again (3.36) we get

$\big{\|}I\\!I\\!I^{(n)}_{st}\big{\|}_{\mathcal{B}^{\kappa,\mu_{t}}_{2,\infty}}\lesssim|t-s|^{\rho_{d}-(\kappa+\alpha)}\sum_{k=0}^{\ell-1}\big{\|}\delta
u_{t_{2k}t_{2k+1}}\,\delta\dot{W}_{t_{2k+1}t_{2k+2}}\big{\|}_{\mathcal{B}^{-\alpha,\mu_{t}}_{2,\infty}}\,.$

Then we invoke the fact that $\mu_{t}<\mu_{t_{2k+1}}$ for all $k<\ell-1$ and
we take advantage of the time regularity of $u$ and $\dot{W}$, mimicking again
(4.13)-(4.14). This allows to write

$\big{\|}I\\!I\\!I^{(n)}_{st}\big{\|}_{\mathcal{B}^{\kappa,\mu_{t}}_{2,\infty}}\\\
\lesssim
2^{-n(\theta+\gamma-1)}\|\dot{W}\|_{\mathcal{C}^{\theta}([0,T];\mathcal{B}^{-\alpha,P}_{p,\infty})}|t-s|^{\rho_{d}-(\kappa+\alpha)}\bigg{(}2^{-n}\sum_{k=0}^{\ell-1}\frac{1}{|t-t_{2k+1}|^{\frac{d+1}{p}}}\|u\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(t_{2k+1})}\bigg{)}\,.$

Proceeding as in Step 2, we can upper bound the quantity into brackets by

$\displaystyle\int_{0}^{t}(t-r)^{-\frac{d+1}{2}}\|u\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(r)}dr$
$\displaystyle\leq
T^{\frac{1}{r_{2}}-\frac{d+1}{p}}\bigg{(}\int_{0}^{T}\|u\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(v)}^{q_{3}}\,dv\bigg{)}^{\frac{1}{q_{3}}}$

where $q_{3}=q_{2}$. We end up with

$\big{\|}I\\!I\\!I^{(n)}_{st}\big{\|}_{\mathcal{B}^{\kappa,\mu_{t}}_{2,\infty}}\lesssim
2^{-n\varepsilon_{3}}|t-s|^{\gamma}\|\dot{W}\|_{\mathcal{C}^{\theta}([0,T];\mathcal{B}^{-\alpha,P}_{p,\infty})}\bigg{(}\int_{0}^{T}\|u\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(v)}^{q_{3}}\,dv\bigg{)}^{\frac{1}{q_{3}}}\,,$
(4.22)

where $\varepsilon_{3}:=\gamma+\theta-1>0$.

Step 4: Bound for $I\\!V_{st}^{(n)}$. For the term $I\\!V_{st}^{(n)}$, we will
bound the increment of $\mathcal{G}$ in two different ways. Specifically,
alleviate notations by setting

$\mathcal{V}_{st}^{k}:=\big{\\{}\mathcal{G}_{t-t_{2k+1}}-\mathcal{G}_{t-t_{2k}}-\mathcal{G}_{s-t_{2k+1}}+\mathcal{G}_{s-t_{2k}}\big{\\}}\big{(}u_{t_{2k+1}}\,\delta\dot{W}_{t_{2k+1}t_{2k+2}}\big{)}\,.$

Then on the one hand we have

$\displaystyle\big{\|}\mathcal{V}^{k}_{st}\big{\|}_{\mathcal{B}^{\kappa,\mu_{t}}_{2,\infty}}$
$\displaystyle\leq\big{\|}\big{\\{}\mathcal{G}_{t-t_{2k+1}}-\mathcal{G}_{s-t_{2k+1}}\big{\\}}\big{(}u_{t_{2k+1}}\,\delta\dot{W}_{t_{2k+1}t_{2k+2}}\big{)}\big{\|}_{\mathcal{B}^{\kappa,\mu_{t}}_{2,\infty}}$
$\displaystyle\qquad+\big{\|}\big{\\{}\mathcal{G}_{t-t_{2k}}-\mathcal{G}_{s-t_{2k}}\big{\\}}\big{(}u_{t_{2k+1}}\,\delta\dot{W}_{t_{2k+1}t_{2k+2}}\big{)}\big{\|}_{\mathcal{B}^{\kappa,\mu_{t}}_{2,\infty}}$
$\displaystyle\lesssim|t-s|^{\rho_{d}-(\kappa+\alpha)}\|u_{t_{2k+1}}\,\delta\dot{W}_{t_{2k+1}t_{2k+2}}\big{\|}_{\mathcal{B}^{-\alpha,\mu_{t}}_{2,\infty}}\,,$
(4.23)

where we have invoked (3.36) for the second inequality. On the other hand,
still using (3.36) we also have

$\displaystyle\big{\|}\mathcal{V}^{k}_{st}\big{\|}_{\mathcal{B}^{\kappa,\mu_{t}}_{2,\infty}}$
$\displaystyle\leq\big{\|}\big{\\{}\mathcal{G}_{t-t_{2k+1}}-\mathcal{G}_{t-t_{2k}}\big{\\}}\big{(}u_{t_{2k+1}}\,\delta\dot{W}_{t_{2k+1}t_{2k+2}}\big{)}\big{\|}_{\mathcal{B}^{\kappa,\mu_{t}}_{2,\infty}}$
$\displaystyle\qquad+\big{\|}\big{\\{}\mathcal{G}_{s-t_{2k+1}}-\mathcal{G}_{s-t_{2k}}\big{\\}}\big{(}u_{t_{2k+1}}\,\delta\dot{W}_{t_{2k+1}t_{2k+2}}\big{)}\big{\|}_{\mathcal{B}^{\kappa,\mu_{t}}_{2,\infty}}$
$\displaystyle\lesssim
2^{-n(\rho_{d}-(\kappa+\alpha))}\|u_{t_{2k+1}}\,\delta\dot{W}_{t_{2k+1}t_{2k+2}}\big{\|}_{\mathcal{B}^{-\alpha,\mu_{t}}_{2,\infty}}\,.$
(4.24)

We now introduce an additional parameter $\beta\in(0,1)$, whose exact value
will be specified later on. Combining (4.1) and (4.1), we get

$\big{\|}I\\!V^{(n)}_{st}\big{\|}_{\mathcal{B}^{\kappa,\mu_{t}}_{2,\infty}}\lesssim\sum_{k=0}^{\ell-1}\big{[}|t-s|^{\rho_{d}-(\kappa+\alpha)}\big{]}^{\beta}\big{[}2^{-n(\rho_{d}-(\kappa+\alpha))}\big{]}^{1-\beta}\big{\|}u_{t_{2k+1}}\,\delta\dot{W}_{t_{2k+1}t_{2k+2}}\big{\|}_{\mathcal{B}^{-\alpha,\mu_{t}}_{2,\infty}}\,.$

Along the same lines as for (4.13)-(4.14), we then let the reader check that
we have

$\big{\|}I\\!V^{(n)}_{st}\big{\|}_{\mathcal{B}^{\kappa,\mu_{t}}_{2,\infty}}\lesssim
2^{-n(\rho_{d}-(\kappa+\alpha))(1-\beta)}|t-s|^{(\rho_{d}-(\kappa+\alpha))\beta}\\\
\times\|\dot{W}\|_{\mathcal{C}^{\theta}([0,T];\mathcal{B}^{-\alpha,P}_{p,\infty})}2^{-n(\theta-1)}\bigg{(}2^{-n}\sum_{k=0}^{\ell-1}\frac{1}{|t-t_{2k+1}|^{\frac{d+1}{p}}}\|u\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(t_{2k+1})}\bigg{)}\,.$
(4.25)

We choose $\beta\in(0,1)$ such that $(\rho_{d}-(\kappa+\alpha))\beta=\gamma$,
which is compatible with our assumption (4.4). Bounding the Riemann sum in
(4.25) just as in Step 3, that is by

$\int_{0}^{t}(t-r)^{-\frac{d+1}{2}}\|u\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(r)}dr\leq
T^{\frac{1}{r_{2}}-\frac{d+1}{p}}\bigg{(}\int_{0}^{T}\|u\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(v)}^{q_{4}}\,dv\bigg{)}^{\frac{1}{q_{4}}},$

where $q_{4}=q_{3}$, we obtain

$\big{\|}I\\!V^{(n)}_{st}\big{\|}_{\mathcal{B}^{\kappa,\mu_{t}}_{2,\infty}}\lesssim
2^{-n\varepsilon_{4}}|t-s|^{\gamma}\|\dot{W}\|_{\mathcal{C}^{\theta}([0,T];\mathcal{B}^{-\alpha,P}_{p,\infty})}\bigg{(}\int_{0}^{T}\|u\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(v)}^{q_{4}}\,dv\bigg{)}^{\frac{1}{q_{4}}}\,,$
(4.26)

with the notation $\varepsilon_{4}=\rho_{d}-(\kappa+\alpha)-\gamma+\theta-1$.
Notice that $\varepsilon_{4}>0$ thanks to (4.4).

Step 5: Conclusion. Recall the decomposition (4.8). Then owing to (4.19),
(4.21), (4.22) and (4.26), we have obtained that for all $0\leq s<t\leq T$,

$\Big{\|}\delta\mathcal{J}_{st}^{(n+1)}-\delta\mathcal{J}_{st}^{(n)}\Big{\|}_{\mathcal{B}^{\kappa,\mu_{t}}_{2,\infty}}\lesssim
2^{-n\varepsilon}|t-s|^{\gamma}\|\dot{W}\|_{\mathcal{C}^{\theta}([0,T];\mathcal{B}^{-\alpha,P}_{p,\infty})}\bigg{(}\int_{0}^{T}\|u\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(v)}^{q}\,dv\bigg{)}^{\frac{1}{q}}\,,$
(4.27)

where $\varepsilon:=\min_{i=1,\ldots,4}\varepsilon_{i}>0$ and
$q:=\max_{i=1,\ldots,4}q_{i}$. Using the fact that $\mathcal{J}^{(n)}_{0}=0$,
the bound (4.27) also allows us to assert that for all $0\leq s\leq T$,

$\Big{\|}\mathcal{J}_{s}^{(n+1)}-\mathcal{J}_{s}^{(n)}\Big{\|}_{\mathcal{B}^{\kappa,\mu_{s}}_{2,\infty}}\lesssim
2^{-n\varepsilon}\|\dot{W}\|_{\mathcal{C}^{\theta}([0,T];\mathcal{B}^{-\alpha,P}_{p,\infty})}\bigg{(}\int_{0}^{T}\|u\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(v)}^{q}\,dv\bigg{)}^{\frac{1}{q}}\,,$

Recalling the definition (4.1) of the norm
$\|\cdot\|_{\mathcal{E}_{2,\infty}^{\gamma,\kappa}(T)}$, we have thus
established that

$\|\mathcal{J}^{(n+1)}-\mathcal{J}^{(n)}\|_{\mathcal{E}_{2,\infty}^{\gamma,\kappa}(T)}\lesssim
2^{-n\varepsilon}\|\dot{W}\|_{\mathcal{C}^{\theta}([0,T];\mathcal{B}^{-\alpha,P}_{p,\infty})}\bigg{(}\int_{0}^{T}\|u\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(v)}^{q}\,dv\bigg{)}^{\frac{1}{q}}\,.$
(4.28)

This shows that $\mathcal{J}^{(n)}$ is a Cauchy sequence in
$\mathcal{E}^{\gamma,\kappa}_{2,\infty}(T)$, which concludes the proof of the
convergence statement.

Moreover, denoting (temporarily) the limit of $\mathcal{J}^{(n)}$ by
$\mathcal{J}$, we can use again (4.28) to deduce

$\|\mathcal{J}\|_{\mathcal{E}_{2,\infty}^{\gamma,\kappa}(T)}\leq\|\mathcal{J}^{(0)}\|_{\mathcal{E}_{2,\infty}^{\gamma,\kappa}(T)}+c_{T_{0}}\,\|\dot{W}\|_{\mathcal{C}^{\theta}([0,T];\mathcal{B}^{-\alpha,P}_{p,\infty})}\bigg{(}\int_{0}^{T}\|u\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(v)}^{q}\,dv\bigg{)}^{\frac{1}{q}}\,,$

for some constant $c_{T_{0}}>0$, which exactly corresponds to the desired
inequality (4.7). Also remember that (4.7) is stronger than (4.6), and thus
(4.6) holds as well. This finishes the proof. ∎

###  4.2. Wellposedness of the Young wave equation for $\mathbf{d=1,2}$

Recall that the wave operator $\mathcal{G}_{t}$ has been introduced in Section
3.1. With the preliminary results of Section 4.1 in hand, equation (1.6) with
initial conditions $u_{0},u_{1}$ will now be written in the following mild
sense:

$u_{t}:=(\partial_{t}\mathcal{G})_{t}u_{0}+\mathcal{G}_{t}u_{1}+\int_{0}^{t}\mathcal{G}_{t-r}(u_{r}\,d\dot{W}_{r}),$
(4.29)

In (4.29) the integral is interpreted in the Young sense, that is via
Proposition 4.4.

The initial conditions in (4.29) have to satisfy some standard smoothness
conditions. Those conditions will be expressed in the usual Sobolev scale, for
which we introduce a new notation.

######  Notation 4.6.

For every $s\geq 0$, we denote by
$\mathcal{H}^{s}=\mathcal{H}^{s}(\mathbb{R}^{d})$ the usual Sobolev space of
order $s$, that is the set of functions $f\in L^{2}(\mathbb{R}^{d})$ such that

$\|f\|_{\mathcal{H}^{s}}^{2}:=\int_{\mathbb{R}^{d}}d\xi\,\\{1+|\xi|^{2}\\}^{s}|\hat{f}(\xi)|^{2}\,<\,\infty.$

We now turn to our main abstract existence and uniqueness result. Let us
recall that we have set

$\rho_{d}:=\begin{cases}1&\text{if}\ d=1\\\ \frac{1}{2}&\text{if}\
d=2.\end{cases}$

######  Theorem 4.7.

Assume $d\in\\{1,2\\}$. Fix an arbitrary time $T>0$, as well as parameters
$\gamma,\theta,\kappa,\alpha\in[0,1]$ and $p>2$ satisfying the conditions in
(4.4). Assume that $\dot{W}$ belongs to
$\mathcal{C}^{\theta}([0,T];\mathcal{B}^{-\alpha,P}_{p,\infty})$ and pick
$(u_{0},u_{1})\in\mathcal{H}^{1+\kappa}\times\mathcal{H}^{\kappa}$. Then
equation (4.29) admits a unique solution $u$ in the space
$\mathcal{E}^{\gamma,\kappa}_{2,\infty}(T)$ introduced in Definition 4.2.

Before proving Theorem 4.7, we start with a technical lemma giving a control
on the terms related to the initial conditions.

######  Lemma 4.8.

Under the assumptions of Theorem 4.7 and for all $t\in[0,T]$, set

$u^{(0)}_{t}:=(\partial_{t}\mathcal{G})_{t}u_{0}+\mathcal{G}_{t}u_{1}\ .$
(4.30)

Then $u^{(0)}$ is an element of $\mathcal{E}^{\gamma,\kappa}_{2,\infty}(T)$
and we have

$\big{\|}u^{(0)}\big{\|}_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(T)}\lesssim\big{\|}u_{0}\big{\|}_{\mathcal{H}^{1+\kappa}}+\big{\|}u_{1}\big{\|}_{\mathcal{H}^{\kappa}}.$
(4.31)

###### Proof.

Let $\varphi\in\mathcal{D}_{1}$ denote the function involved in our definition
of the spaces $\mathcal{B}^{\kappa,\mu}_{p,q}$ (see Definition 2.7 and
Notation 2.9). As a preliminary step, observe that owing to assumption
($\mathcal{A}_{\varphi}$) and the fact that $\varphi$ has compact support, one
has

$|\widehat{\varphi}(\xi)|=\bigg{|}\int_{\mathbb{R}^{d}}dx\,e^{-\imath\xi\cdot
x}\varphi(x)\bigg{|}=\bigg{|}\int_{\mathbb{R}^{d}}dx\,\\{e^{-\imath\xi\cdot
x}-1\\}\varphi(x)\bigg{|}\lesssim|\xi|^{\kappa},$

where we have used the fact that $\kappa\in(0,1)$. Hence for every $j\geq 1$,

$|\widehat{\varphi_{j}}(\xi)|=|\widehat{\varphi}(2^{-j}\xi)|\lesssim
2^{-j\kappa}|\xi|^{\kappa}.$ (4.32)

Now the weight $w=e^{-\mu|x|}$ is smaller than 1. Thus one can trivially bound
the norm in $L^{2}_{\mu_{t}}$ by the usual norm in $L^{2}(\mathbb{R}^{d})$.
Hence invoking (3.2) we get

$\displaystyle\big{\|}\varphi_{j}\ast\\{(\partial_{t}\mathcal{G})_{t}-(\partial_{t}\mathcal{G})_{s}\\}u_{0}\big{\|}_{L^{2}_{\mu_{t}}}$
$\displaystyle\lesssim\big{\|}\big{(}\varphi_{j}\ast(\\{(\partial_{t}G)_{t}-(\partial_{t}G)_{s}\\}\ast
u_{0}\big{)}\big{\|}_{L^{2}}$
$\displaystyle\lesssim\bigg{(}\int_{\mathbb{R}^{d}}d\xi\,\big{|}\widehat{\varphi_{j}}(\xi)\big{|}^{2}\big{|}\cos(t|\xi|)-\cos(s|\xi|)\big{|}^{2}\big{|}\widehat{u_{0}}(\xi)\big{|}^{2}\bigg{)}^{\frac{1}{2}}\,.$

Combining this inequality with (4.32), this yields

$\displaystyle\big{\|}\varphi_{j}\ast\\{(\partial_{t}\mathcal{G})_{t}-(\partial_{t}\mathcal{G})_{s}\\}u_{0}\big{\|}_{L^{2}_{\mu_{t}}}$
$\displaystyle\lesssim$ $\displaystyle
2^{-j\kappa}|t-s|\bigg{(}\int_{\mathbb{R}^{d}}d\xi\,|\xi|^{2+2\kappa}\big{|}\widehat{u_{0}}(\xi)\big{|}^{2}\bigg{)}^{\frac{1}{2}}$
(4.33) $\displaystyle\lesssim$ $\displaystyle
2^{-j\kappa}|t-s|^{\gamma}\|u_{0}\|_{\mathcal{H}^{1+\kappa}}\,.$

Plugging (4.33) into the definition (2.10) of Besov norm, we end up with

$\big{\|}\\{(\partial_{t}\mathcal{G})_{t}-(\partial_{t}\mathcal{G})_{s}\\}u_{0}\big{\|}_{\mathcal{B}^{\kappa,\mu_{t}}_{2,\infty}}\lesssim|t-s|^{\gamma}\|u_{0}\|_{\mathcal{H}^{1+\kappa}}\,.$
(4.34)

In the same way, we have

$\displaystyle\big{\|}\varphi_{j}\ast(\partial_{t}\mathcal{G})_{s}u_{0}\big{\|}_{L^{2}_{\mu_{s}}}$
$\displaystyle\lesssim\bigg{(}\int_{\mathbb{R}^{d}}d\xi\,\big{|}\widehat{\varphi_{j}}(\xi)\big{|}^{2}\big{|}\cos(s|\xi|)\big{|}^{2}\big{|}\widehat{u_{0}}(\xi)\big{|}^{2}\bigg{)}^{\frac{1}{2}}\lesssim
2^{-j\kappa}\bigg{(}\int_{\mathbb{R}^{d}}d\xi\,|\xi|^{2\kappa}\big{|}\widehat{u_{0}}(\xi)\big{|}^{2}\bigg{)}^{\frac{1}{2}}\lesssim
2^{-j\kappa}\|u_{0}\|_{\mathcal{H}^{1+\kappa}}.$

Therefore, resorting to (2.10) again, we get
$\sup_{s\in[0,T]}\big{\|}(\partial_{t}\mathcal{G})_{s}u_{0}\big{\|}_{\mathcal{B}^{\kappa,\mu_{s}}_{2,\infty}}\lesssim\|u_{0}\|_{\mathcal{H}^{1+\kappa}}$.
Gathering this inequality with (4.34) we have thus shown that

$\big{\|}(\partial_{t}\mathcal{G})u_{0}\big{\|}_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(T)}\lesssim\big{\|}u_{0}\big{\|}_{\mathcal{H}^{1+\kappa}}.$
(4.35)

Let us now handle the term $\mathcal{G}_{t}u_{1}$ in (4.30). It is treated
similarly to $(\partial_{t}\mathcal{G})_{t}u_{0}$ in (4.33)-(4.35), and we
omit the details for sake of conciseness. Let us just mention that (4.33) is
replaced by

$\displaystyle\big{\|}\varphi_{j}\ast\\{\mathcal{G}_{t}-\mathcal{G}_{s}\\}u_{1}\big{\|}_{L^{2}_{\mu_{t}}}$
$\displaystyle\lesssim\bigg{(}\int_{\mathbb{R}^{d}}d\xi\,\big{|}\widehat{\varphi_{j}}(\xi)\big{|}^{2}\frac{|\sin(t|\xi|)-\sin(s|\xi|)|^{2}}{|\xi|^{2}}\big{|}\widehat{u_{1}}(\xi)\big{|}^{2}\bigg{)}^{\frac{1}{2}}$
$\displaystyle\lesssim
2^{-j\kappa}|t-s|\bigg{(}\int_{\mathbb{R}^{d}}d\xi\,|\xi|^{2\kappa}\big{|}\widehat{u_{1}}(\xi)\big{|}^{2}\bigg{)}^{\frac{1}{2}}.$

Following the same steps as before, we then get

$\big{\|}\mathcal{G}u_{1}\big{\|}_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(T)}\lesssim\big{\|}u_{1}\big{\|}_{\mathcal{H}^{\kappa}}\,.$
(4.36)

Plugging together (4.35) and (4.36), we have thus proved our claim (4.31). ∎

With Lemma 4.8 in hand, we now turn to the proof of our existence-uniqueness
result.

###### Proof of Theorem 4.7.

Due to the expression (4.31) for the initial condition $u^{(0)}$, we will not
use a standard argument based on patching solutions defined on small
intervals. Instead of that, our method will be based on Picard iterations. In
other words, we consider the sequence of processes $(u^{(\ell)})_{\ell\geq 0}$
defined as: for every $t\in[0,T]$,
$u^{(0)}_{t}:=(\partial_{t}\mathcal{G})_{t}u_{0}+\mathcal{G}_{t}u_{1}$ and

$u^{(\ell+1)}_{t}:=(\partial_{t}\mathcal{G})_{t}u_{0}+\mathcal{G}_{t}u_{1}+\int_{0}^{t}\mathcal{G}_{t-r}(u^{(\ell)}_{r}\,d\dot{W}_{r}).$
(4.37)

By combining the results of Proposition 4.4 and Lemma 4.8, we can immediately
guarantee that $u^{(\ell)}$ is well defined in the space
$\mathcal{E}^{\gamma,\kappa}_{2,\infty}(T)$, for every $\ell\geq 0$. We now
divide our proof into the existence and the uniqueness parts.

Existence. In order to show the convergence of $u^{(\ell)}$ in
$\mathcal{E}_{2,\infty}^{\gamma,\mu}(T)$, let us consider the difference
$v^{(\ell)}:=u^{(\ell+1)}-u^{(\ell)}$. Owing to (4.37), it is clear that for
every $\ell\geq 0$ we have $v_{0}^{(\ell)}=0$ and $v^{(\ell)}$ satisfies

$v^{(\ell)}_{t}:=\int_{0}^{t}\mathcal{G}_{t-r}(v^{(\ell-1)}_{r}\,d\dot{W}_{r}).$
(4.38)

We are here in a position to apply the estimate (4.7) and assert that, for all
$t\in[0,T]$

$\|v^{(\ell)}\|^{q}_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(t)}\leq
c_{T}\int_{0}^{t}\|v^{(\ell-1)}\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(t_{1})}^{q}\,dt_{1}\,,$

where $c_{T}$ only depends on $T$ and
$\|\dot{W}\|_{\mathcal{C}^{\theta}([0,T];\mathcal{B}^{-\alpha,P}_{p,\infty})}$,
and $q$ is the (finite) parameter provided by Proposition 4.4. Iterating this
inequality, we deduce

$\|v^{(\ell)}\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(T)}^{q}\leq(c_{T})^{\ell}\int_{0}^{T}\int_{0}^{t_{1}}\ldots\int_{0}^{t_{\ell}}\|v^{(0)}\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(t_{\ell})}^{q}\,dt_{1}\cdots
dt_{\ell}\leq\|v^{(0)}\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(T)}^{q}\frac{(c_{T}T)^{\ell}}{\ell!},$

which can naturally be recast as

$\|u^{(\ell+1)}-u^{(\ell)}\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(T)}\leq\|u^{(1)}-u^{(0)}\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(T)}\frac{\left(c_{T}T\right)^{\frac{\ell}{q}}}{(\ell!)^{\frac{1}{q}}}.$

This proves that $(u^{(\ell)})$ is a Cauchy sequence in
$\mathcal{E}^{\gamma,\kappa}_{2,\infty}(T)$, and accordingly it converges to
some limit $u$. Letting $\ell$ tend to infinity in (4.37), a standard
procedure reveals that $u$ is the desired solution to equation (4.29).

Uniqueness. The argument is essentially the same as above. Namely, if $u,v$
are two solutions, we have for all $\ell\geq 1$,

$\|u-v\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(T)}^{q}\leq(c_{T})^{\ell}\int_{0}^{T}\int_{0}^{t_{1}}\ldots\int_{0}^{t_{\ell}}\|u-v\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(t_{\ell})}^{q}\,dt_{1}\cdots
dt_{\ell}\leq\|u-v\|_{\mathcal{E}^{\gamma,\kappa}_{2,\infty}(T)}^{q}\frac{(c_{T}T)^{\ell}}{\ell!},$

and we get the conclusion by letting $\ell$ tend to infinity. ∎

##  5\. Application to Gaussian noises

In this section, we prove that the Gaussian noise whose covariance is defined
by (1.2) is an appropriate input for the wave equation. This amounts to prove
that $\dot{W}$ sits in a space of the form
$C^{\theta}([0,T],\mathcal{B}_{p,\infty}^{-\alpha,P})$ as in Proposition 4.4.
Before proceeding to the proof, we first characterize the space
$\mathcal{B}_{p,\infty}^{-\alpha,P}$ thanks to simple properties of the
Fourier transform. Let us start by defining a useful Fourier type operator.

######  Definition 5.1.

Let $f$ be a smooth enough function defined on $\mathbb{R}^{d}$, and recall
that its Fourier transform is denoted by $\mathcal{F}f$. For an arbitrary
constant $c>0$ and $s\in(-\infty,1]$, we set

$\mathcal{J}^{s}_{c}f:=\mathcal{F}^{-1}\big{(}\\{1+c\,|\cdot|^{2}\\}^{\frac{s}{2}}\mathcal{F}f\big{)}\,.$

We now upper bound norms in some weighted Besov spaces $\mathcal{B}$ thanks to
the Fourier operator $\mathcal{J}$. This is summarized in the following Lemma.

######  Lemma 5.2.

Consider $p\geq 2$ and $s\in(-\infty,1]$. Recall that the spaces $L_{w}^{p}$
are given in Definition 2.2 and the weighted Besov spaces
$\mathcal{B}_{p,q}^{s,\mu}$ are introduced in Definition 2.7. Let us also
recall that we are working with the polynomial weight
$P(x)=(1+|x|^{1+d})^{-1}$ and the related Besov spaces
$\mathcal{B}_{p,q}^{s,P}$ as in Lemma 2.10. Then there exists a constant
$c_{p}>0$ depending on $p$ and $d$ only such that

$\|f\|_{\mathcal{B}^{s,P}_{p,\infty}}\lesssim\left\|\mathcal{J}^{s}_{c_{p}}f\right\|_{L^{p}_{P}}\,,$
(5.1)

where the operator $\mathcal{J}_{c}^{s}$ has been introduced in Definition
5.1.

###### Proof.

The proof of this Lemma is basically borrowed from [9], to which we will refer
for further details. First going back to the definition (2.10) of Besov norms
and invoking the fact that
$\|\cdot\|_{\ell^{\infty}(\mathbb{N})}\leq\|\cdot\|_{\ell^{p}(\mathbb{N})}$
for $1\leq p<\infty$, we easily get that

$\|f\|_{\mathcal{B}_{p,\infty}^{s,P}}\lesssim\|f\|_{\mathcal{B}_{p,p}^{s,P}}\,.$

Next, write equation (2.10), which gives

$\displaystyle\|f\|_{\mathcal{B}_{p,p}^{s,P}}^{p}=\sum_{j\geq
0}2^{sjp}\int_{\mathbb{R}^{d}}dx\,P(x)|(\boldsymbol{\varphi}_{j}\ast
f)(x)|^{p}$ $\displaystyle=\int_{\mathbb{R}^{d}}dx\,P(x)\bigg{(}\sum_{j\geq
0}2^{sjp}|(\boldsymbol{\varphi}_{j}\ast
f)(x)|^{p}\bigg{)}=\int_{\mathbb{R}^{d}}dx\,P(x)\left\|j\mapsto
2^{sj}|(\boldsymbol{\varphi}_{j}\ast
f)(x)|\right\|^{p}_{\ell^{p}(\mathbb{N})}\,.$

Owing to the fact that
$\|\cdot\|_{\ell^{p}(\mathbb{N})}\leq\|\cdot\|_{\ell^{2}(\mathbb{N})}$ for
$p\geq 2$, we obtain

$\|f\|_{\mathcal{B}_{p,p}^{s,P}}^{p}\leq\int_{\mathbb{R}^{d}}dx\,P(x)\bigg{(}\sum_{j\geq
0}2^{2sj}|(\boldsymbol{\varphi}_{j}\ast f)(x)|^{2}\bigg{)}^{\frac{p}{2}}\,.$
(5.2)

Note that the right-hand side of (5.2) is exactly $\|f\|^{p}_{F_{p,2}^{s,P}}$
(see (2.2) in [9] for the definition of $\|\cdot\|_{F_{p,q}^{s,P}}$). Hence
(5.2) can be rephrased as

$\|f\|_{\mathcal{B}^{s,P}_{p,p}}\leq\|f\|_{F^{s,P}_{p,2}}\,.$

The estimate (5.1) now stems from the combination of [9, Theorem 2.18] and [9,
Theorem 1.10], which allows us to assert that for some suitable $c_{p}>0$,

$\big{\|}f\big{\|}_{F^{s,P}_{p,2}}\lesssim\big{\|}\mathcal{J}^{s}_{c_{p}}f\big{\|}_{L^{p}_{P}}\,,$

for some proportional constant that only depends on $p$ and $d$ . ∎

We now turn to a definition of our Gaussian noise allowing some proper
couplings for approximations. Namely we will define $\dot{W}$ through an
harmonizable representation of the form

$\dot{W}_{t}(x):=c_{a_{0}}\int_{\lambda\in\mathbb{R}}\int_{\beta\in\mathbb{R}^{d}}\,\frac{e^{\imath\lambda
t}-1}{|\lambda|^{\frac{3-a_{0}}{2}}}\,e^{\imath\beta\cdot
x}\,\widehat{B}(d\lambda,d\beta)\,.$ (5.3)

Here we use a parameter $a_{0}\in(0,2)$, $c_{a_{0}}$ is some proper positive
constant, and $\widehat{B}$ is a complex-valued Gaussian random measure on
$\mathcal{B}(\mathbb{R}\times\mathbb{R}^{d})$ such that for any
$I_{1},I_{2}\in\mathcal{B}(\mathbb{R})$ and
$A_{1},A_{2}\in\mathcal{B}(\mathbb{R}^{d})$,

$\mathbb{E}[\hat{B}(I_{1}\times A_{1})]=0,\quad\mathbb{E}[\hat{B}(I_{1}\times
A_{1})\overline{\hat{B}(I_{2}\times A_{2})}]=m(I_{1}\cap J_{1})\mu(A_{1}\cap
A_{2}),$ (5.4)

where $m$ is the Lebesgue measure on $\mathbb{R}$ and $\mu(d\beta)$ is the
spectral measure of a nonnegative definite (generalized) function $\gamma(x)$.
In order to get a real-valued noise, we also assume that the Gaussian measure
$\widehat{B}$ is such that

$\overline{\hat{B}(I_{1}\times A_{1})}=\hat{B}\left(-\left(I_{1}\times
A_{1}\right)\right)\,.$ (5.5)

It is readily checked from (5.4)-(5.5) that for an appropriate value of
$c_{a_{0}}$, the covariance of $\dot{W}$ is (formally) given by

$\mathbb{E}[\dot{W}_{t}(x)\dot{W}_{s}(y)]=R_{a_{0}}(t,s)\gamma(x-y),$ (5.6)

with

$R_{a_{0}}(t,s):=\frac{1}{2}\big{\\{}|t|^{2-a_{0}}+|s|^{2-a_{0}}-|t-s|^{2-a_{0}}\big{\\}}.$

The definition (5.3) of $\dot{W}$ comes with a natural approximation by smooth
functions. Namely for $n\geq 1$ we define

$\dot{W}^{n}_{t}(x):=c_{a_{0}}\int_{\lambda\in\mathbb{R}}\int_{\beta\in\mathbf{B}_{2^{n}}}\widehat{B}(d\lambda,d\beta)\,\frac{e^{\imath\lambda
t}-1}{|\lambda|^{\frac{3-a_{0}}{2}}}\,e^{\imath\beta\cdot x}\,,$ (5.7)

where we recall that the balls $\mathbf{B}_{R}$ are introduced in Notation
1.2. We shall show that $\dot{W}^{n}$ converges to $\dot{W}_{t}(x)$ given in
(5.3) in a proper space.

######  Theorem 5.3.

Let $a_{0}\in(0,2)$ and consider $\theta\in(0,1-\frac{a_{0}}{2})$. Let
$\\{\dot{W}^{n},t\in[0,T],x\in\mathbb{R}\\}$ be the field defined by (5.7). We
assume the existence of a threshold parameter $\alpha_{d}>0$ such that the
measure $\mu$ satisfies

$\int_{\mathbb{R}^{d}}\left(\frac{1}{1+|\beta|}\right)^{2\alpha_{d}+\varepsilon}\mu(d\beta)<\infty\,,~{}\text{
for every }~{}\varepsilon>0\,.$ (5.8)

Then for all $\alpha>\alpha_{d}$ and $p$ large enough, $\\{\dot{W}^{n},n\geq
1\\}$ forms a Cauchy sequence in the space

$L^{p}\left(\Omega;\mathcal{C}^{\theta}\left([0,T],\mathcal{B}^{-\alpha,P}_{p,\infty}\right)\right),$

with the limit denoted by $\dot{W}$. The noisy input $\dot{W}$ satisfies the
following properties:

  1. (i)

Almost surely we have
$\dot{W}\in\mathcal{C}^{\theta}([0,T],\mathcal{B}_{p,\infty}^{-\alpha,P})$.

  2. (ii)

The covariance function of $\dot{W}$ is given by (5.6).

###### Proof.

In the sequel we will write
$\delta\dot{W}_{st}^{n}=\dot{W}_{t}^{n}-\dot{W}_{s}^{n}$ to alleviate
notations. We will reduce our proof to moment computations thanks to two
standard steps:
(1) Invoking a telescoping sum argument, the convergence of the sequence
$\\{\dot{W}^{n};n\geq 1\\}$ is implied by the convergence of the series

$\sum_{n=1}^{\infty}\left(\mathbb{E}\left[\big{\|}\dot{W}^{n+1}-\dot{W}^{n}\big{\|}^{p}_{\mathcal{C}^{\theta}([0,T],\mathcal{B}^{-\alpha,P}_{p,\infty})}\right]\right)^{1/p}\,.$
(5.9)

(2) Owing to a standard application of Garsia’s lemma the following holds
true: if we assume that for each summand in (5.9) and for
$p\geq\frac{1}{1-\frac{a_{0}}{2}-\theta}$, there exists $\varepsilon>0$ such
that

$\mathbb{E}\left[\left\|\delta\dot{W}_{st}^{n+1}-\delta\dot{W}_{st}^{n}\right\|^{p}_{\mathcal{B}_{p,\infty}^{-\alpha,P}}\right]\lesssim\frac{|t-s|^{(1-\frac{a_{0}}{2})p}}{2^{\varepsilon
np}}\,,$ (5.10)

then we also have

$\mathbb{E}\left[\big{\|}\dot{W}^{n+1}-\dot{W}^{n}\big{\|}^{p}_{\mathcal{C}^{\theta}([0,T],\mathcal{B}^{-\alpha,P}_{p,\infty})}\right]\lesssim\frac{1}{2^{\varepsilon
np}}\,.$ (5.11)

Putting together (5.9) and (5.11), we are now reduced to prove (5.10).

In order to upper bound the right-hand side of (5.10), we invoke (5.1) which
can be read

$\mathbb{E}\left[\left\|\delta\dot{W}_{st}^{n+1}-\delta\dot{W}_{st}^{n}\right\|^{p}_{\mathcal{B}_{p,\infty}^{-\alpha,P}}\right]\\\
\lesssim\int_{\mathbb{R}^{d+1}}\frac{dx}{1+|x|^{1+d}}\,\mathbb{E}\Big{[}\Big{|}\mathcal{F}^{-1}\Big{(}\\{1+|\cdot|^{2}\\}^{-\frac{\alpha}{2}}\mathcal{F}\big{(}\delta\dot{W}_{st}^{n+1}-\delta\dot{W}_{st}^{n}\big{)}\Big{)}(x)\Big{|}^{p}\Big{]}\,.$
(5.12)

In addition, $\dot{W}^{n+1}-\dot{W}^{n}$ is a Gaussian process and
$\mathcal{F}$ is a linear transform. Hence we get

$\mathbb{E}\left[\left\|\delta\dot{W}_{st}^{n+1}-\delta\dot{W}_{st}^{n}\right\|^{p}_{\mathcal{B}_{p,\infty}^{-\alpha,P}}\right]\lesssim\int_{\mathbb{R}^{d}}\frac{dx}{1+|x|^{1+d}}\,\big{(}Q_{st}^{n}(x)\big{)}^{p/2}\,,$
(5.13)

where we have set

$Q_{st}^{n}(x):=\mathbb{E}\Big{[}\Big{|}\mathcal{F}^{-1}\Big{(}\\{1+|\cdot|^{2}\\}^{-\frac{\alpha}{2}}\mathcal{F}\big{(}\delta\dot{W}^{n+1}_{st}-\delta\dot{W}^{n}_{st}\big{)}\Big{)}(x)\Big{|}^{2}\Big{]}\,.$

Let us focus our attention on the evaluation of $Q^{n}$ above, defined for
$0\leq s<t\leq T$ and $x\in\mathbb{R}^{d}$. To this aim, by simply writing the
definition of Fourier transform we get

$Q^{n}_{st}(x)=\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}d\xi
d\tilde{\xi}\,e^{\imath
x\cdot(\xi-\tilde{\xi})}\\{1+|\xi|^{2}\\}^{-\frac{\alpha}{2}}\\{1+|\tilde{\xi}|^{2}\\}^{-\frac{\alpha}{2}}\\\
\times\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}dyd\tilde{y}\,e^{-\imath\xi\cdot
y}e^{\imath\tilde{\xi}\cdot\tilde{y}}\,\mathbb{E}\Big{[}\big{\\{}\delta\dot{W}^{n+1}_{st}-\delta\dot{W}^{n}_{st}\big{\\}}(y)\overline{\big{\\{}\delta\dot{W}^{n+1}_{st}-\delta\dot{W}^{n}_{st}\big{\\}}(\tilde{y})}\Big{]}\,.$
(5.14)

In addition, invoking relation (5.7) and the covariance structure of a complex
valued white noise, we obtain

$\mathbb{E}\Big{[}\big{\\{}\delta\dot{W}^{n+1}_{st}-\delta\dot{W}^{n}_{st}\big{\\}}(y)\overline{\big{\\{}\delta\dot{W}^{n+1}_{st}-\delta\dot{W}^{n}_{st}\big{\\}}(\tilde{y})}\Big{]}=c\,|t-s|^{2-a_{0}}\int_{\mathbf{B}_{2^{n+1}}\backslash\mathbf{B}_{2^{n}}}e^{\imath\beta\cdot(y-\tilde{y})}\mu(d\beta)\,,$

for some constant $c$. Plugging this inequality into (5.14) and carefully
computing the Fourier transforms reveals that

$\displaystyle Q^{n}_{st}(x)$
$\displaystyle=c\,|t-s|^{2-a_{0}}\int_{\mathbf{B}_{2^{n+1}}\backslash\mathbf{B}_{2^{n}}}\frac{\mu(d\beta)}{\\{1+|\beta|^{2}\\}^{\alpha}}\,.$
(5.15)

We now bound the right-hand side of (5.15): by setting
$\varepsilon:=\alpha-\alpha_{d}>0$, we get

$\displaystyle\int_{\mathbf{B}_{2^{n+1}}\backslash\mathbf{B}_{2^{n}}}\frac{\mu(d\beta)}{\\{1+|\beta|^{2}\\}^{\alpha}}$
$\displaystyle=\int_{\mathbf{B}_{2^{n+1}}\backslash\mathbf{B}_{2^{n}}}\\{1+|\beta|^{2}\\}^{-\frac{\varepsilon}{2}}\frac{\mu(d\beta)}{\\{1+|\beta|^{2}\\}^{\alpha_{d}+\frac{\varepsilon}{2}}}\lesssim
2^{-n\varepsilon},$ (5.16)

where the inequality follows from the assumption (5.8). Reporting (5.16) into
(5.15), we have thus obtained

$Q_{st}^{n}(x)\lesssim 2^{-n\varepsilon}|t-s|^{2-a_{0}}\,.$

Plugging this information into (5.13), this trivially yields our claim (5.10).
Since we have argued that the convergence of $\dot{W}^{n}$ could be reduced to
(5.10), we have proved that $\dot{W}^{n}$ converges in
$L^{p}(\Omega;\mathcal{C}^{\theta}([0,T],\mathcal{B}^{-\alpha,P}_{p,\infty}))$
for $p\geq\frac{1}{1-\frac{a_{0}}{2}-\theta}$. In particular, this proves item
$(i)$ in our Theorem.

Finally, item $(ii)$ in our Theorem can be proved by a standard procedure on
covariance functions from (5.7) to (5.6). The proof is concluded. ∎

######  Remark 5.4.

For the special cases $\gamma(x)=\prod_{i=1}^{d}|x_{i}|^{-a_{i}}$ and
$\gamma(x)=|x|^{-a}$ with $a_{i}\in(0,2),a\in(0,2d)$,111Here we use the
convention that for $x\in\mathbb{R}$, $|x|^{-1}:=\delta(x)$ where $\delta(x)$
is the Dirac delta function and $|x|^{-a}:=(|x|^{-a+2})^{\prime\prime}$ for
$a\in(1,2)$ where the second derivative is taken in the distributional sense.
Similar convention also applies for $x\in\mathbb{R}^{d}$. the spectral
measures are $\mu(d\beta)=\prod_{i=1}^{d}|\beta_{i}|^{-(1-a_{i})}\,d\beta$ and
$\mu(d\beta)=|\beta|^{-(d-a)}\,d\beta$, respectively. If we use the notation
$a=\sum_{i=1}^{d}a_{i}$, condition (5.8) is verified for the optimal threshold
$\alpha_{d}=\frac{a}{2}$.

######  Remark 5.5.

In the current paper, we have chosen the harmonic representation (5.7) to
approximate the Gaussian noise, which is convenient in our setting. We could
also have used, similarly to what is done in [2], a sequence of mollifications
$\dot{W}^{n}=\varphi_{n}*\dot{W}$. This second approach might be closer in
spirit to our previous work [4], but is slightly more cumbersome.

Let us recall that we have denoted by $\rho_{d}$ the regularization parameter
exhibited in Proposition 3.4, with explicit value

$\rho_{d}:=\begin{cases}1&\text{if}\ d=1\\\ \frac{1}{2}&\text{if}\
d=2.\end{cases}$

######  Proposition 5.6.

Assume that $d\in\\{1,2\\}$. Under the same setting as for Theorem 5.3, assume
that $a_{0}<\rho_{d}$ and that condition (5.8) is satisfied for $\alpha_{d}$
such that

$0<\alpha_{d}<\frac{1}{2}(\rho_{d}-a_{0}).$ (5.17)

Then there exist coefficients $0<\kappa,\alpha,\gamma,\theta<1$ and $p>d+1$
that satisfy (4.4) and such that
$\dot{W}\in\mathcal{C}^{\theta}([0,T];\mathcal{B}^{-\alpha,P}_{p,\infty})$
almost surely. In particular Theorem 4.7 can be applied to the noise $\dot{W}$
and equation (4.29) admits a unique solution in
$\mathcal{E}_{2,\infty}^{\gamma,\kappa}(T)$.

###### Proof.

Thanks to condition (5.17), we can fix $\varepsilon>0$ small enough so that
$\alpha_{d}<\frac{1}{2}(\rho_{d}-a_{0})-2\varepsilon$,
$\theta:=1-\frac{a_{0}}{2}-\varepsilon\in[0,1]$ and
$\alpha:=\alpha_{d}+\varepsilon\in[0,1]$. By Theorem 5.3, we know that
$\dot{W}\in C^{\theta}([0,T];\mathcal{B}_{p,\infty}^{-\alpha,P})$. Besides,
observe that

$\theta-\alpha=1-\frac{a_{0}}{2}-\alpha_{d}-2\varepsilon>1-\frac{\rho_{d}}{2}\,.$

We can thus pick two parameters $p$ large enough and $\delta>0$ small enough
so that

$\theta-\alpha>1-\frac{\rho_{d}}{2}+\frac{d}{2p}+\delta\,.$ (5.18)

We now consider $\kappa,\gamma$ such that the second and third conditions in
(4.4) are fulfilled, of the form

$\kappa=\alpha+\frac{d}{p}+\delta,\mbox{ and }\gamma=1-\theta+\delta\,,$
(5.19)

for $\delta>0$ small enough and $p\geq 1$ large enough. Notice that since we
have $\alpha,\theta\in[0,1]$, we also have $\kappa,\gamma\in[0,1]$ whenever
$p$ is large and $\delta$ is small. Moreover, for $\gamma$ as in (5.19), the
last condition in (4.4) is verified:

$1-\frac{d+1}{p}>\gamma\,.$

It remains to prove that the first condition in (4.4) holds true. Towards this
aim, consider $\kappa,\gamma$ as in (5.19) and compute

$\kappa+\alpha+\gamma+(1-\rho_{d})=2\alpha+\frac{d}{p}+1-\theta+2\delta+(1-\rho_{d})=2-\rho_{d}-2(\theta-\alpha)+\frac{d}{p}+2\delta+\theta\,.$
(5.20)

In addition, resorting to (5.18), we have
$2(\theta-\alpha)>2-\rho_{d}+\frac{d}{p}+2\delta$. Plugging this relation into
(5.20), we get

$\kappa+\alpha+\gamma+(1-\rho_{d})<\theta\,.$

This shows that the set of conditions (4.4) is satisfied and that $\dot{W}$ is
an element of $C^{\theta}([0,T];\mathcal{B}_{p,\infty}^{-\alpha,P})$. Our
proof is achieved. ∎

Our final result draws on Proposition 5.6 and gives conditions on $a_{0},a$
when the covariance function $\gamma(x)$ satisfies the scaling property (1.4).

######  Corollary 5.7.

Assume that $d\in\\{1,2\\}$. Under the same setting as in Remark 5.4, assume
that

$a_{0}+a<\rho_{d}\,.$ (5.21)

Then there exist coefficients $0<\kappa,\alpha,\gamma,\theta<1$ and $p>d+1$
that satisfy (4.4) and such that
$\dot{W}\in\mathcal{C}^{\theta}([0,T];\mathcal{B}^{-\alpha,P}_{p,\infty})$
almost surely. In particular Theorem 4.7 can be applied to the noise $\dot{W}$
and equation (4.29) admits a unique solution in
$\mathcal{E}_{2,\infty}^{\gamma,\kappa}(T)$.

###### Proof.

As noticed in Remark (5.4), condition (5.8) is satisfied by
$\alpha_{d}=\frac{a}{2}$ in this situation. The desired result then follows
from a straightforward verification that (5.21) is equivalent to (5.17) when
$\alpha_{d}=\frac{a}{2}$. ∎

## Appendix A

This Appendix summarizes some technical computations which play a prominent
role in our Besov space considerations.

###  A.1. A local reproducing formula

We start by labelling a Theorem borrowed from [9, Theorem 1.6]:

######  Theorem A.1 (Rychkov).

Let $\varphi_{0}\in\mathcal{C}_{c}^{\infty}$ be such that
$\int_{\mathbb{R}^{d}}\varphi_{0}(x)\,dx\neq 0$. Then for any integer $L\geq
0$, there exist two functions $\psi_{0},\psi\in\mathcal{C}_{c}^{\infty}$ such
that:

  1. (i)

The function $\psi$ is an element of $\mathcal{D}_{L}$, where we recall that
$\mathcal{D}_{L}$ is introduced in Definition 2.3

  2. (ii)

For every distribution $f\in\mathcal{D}^{\prime}$, the following decomposition
holds true in $\mathcal{D}^{\prime}$:

$f=\sum_{j\geq 0}\varphi_{j}\ast\psi_{j}\ast f\,.$

###  A.2. Proof of Lemma 2.13

Recall that $\boldsymbol{\varphi}_{0},\boldsymbol{\varphi}_{j}$ stand for the
test-functions introduced in Notation 2.9, and consider an arbitrary number
$L\geq 2$. By Theorem A.1, there exist two functions
$\psi_{0},\psi\in\mathcal{C}_{c}^{\infty}$ such that $\psi\in\mathcal{D}_{L}$
and for every $h\in\mathcal{D}^{\prime}$,

$h=\sum_{\ell\geq 0}\psi_{\ell}\ast\boldsymbol{\varphi}_{\ell}\ast h\,.$ (A.1)

As in Lemma 2.5, for the sake of clarity, we will assume in the sequel that
the supports of $\boldsymbol{\varphi}_{0}$, $\boldsymbol{\varphi}$, $\psi$ and
$\psi_{0}$ are all included in the interval $\mathbf{B}_{1}$, but the
arguments could be easily extended to any supporting interval
$\mathbf{B}_{K}$.

For all $\ell,k\geq 0$, let us set
$\delta_{\ell}f:=\psi_{\ell}\ast\boldsymbol{\varphi}_{\ell}\ast f$ and
$\delta_{k}g:=\psi_{k}\ast\boldsymbol{\varphi}_{k}\ast g$. Owing to formula
(A.1), one can proceed similarly to (3.15) and split $\varphi_{j}*(f\cdot g)$
as

$\displaystyle\boldsymbol{\varphi}_{j}\ast(f\cdot g)$
$\displaystyle=\sum_{\ell,k\geq
0}\boldsymbol{\varphi}_{j}\ast(\delta_{\ell}f\cdot\delta_{k}g)=:\
I_{j}+I\\!I_{j}\,,$ (A.2)

where the terms $I_{j}$ and $I\\!I_{j}$ are defined by

$I_{j}=\sum_{\begin{subarray}{c}\ell,k\geq 0\\\
0\leq\ell\leq\max(j,k)\end{subarray}}\boldsymbol{\varphi}_{j}\ast(\delta_{\ell}f\cdot\delta_{k}g)\,,\quad
I\\!I_{j}=\sum_{\begin{subarray}{c}\ell,k\geq 0\\\
\ell>\max(j,k)\end{subarray}}\boldsymbol{\varphi}_{j}\ast(\delta_{\ell}f\cdot\delta_{k}g)\,.$
(A.3)

We now bound $I_{j}$ and $I\\!I_{j}$ in two different steps.

Treatment of $I_{j}$. We first bound each term
$\boldsymbol{\varphi}_{j}*(\delta_{\ell}f\cdot\delta_{k}g)$ in the sum
defining $I_{j}$ in (A.3). That is, resorting to (2.5) together with the fact
that $\boldsymbol{\varphi}_{j}$ is bounded in $L^{1}$, we get

$\|\boldsymbol{\varphi}_{j}*(\delta_{\ell}f\cdot\delta_{k}g)\|_{L_{\mu}^{p}}\lesssim\|\delta_{\ell}f\cdot\delta_{k}g\|_{L_{\mu}^{p}}\,.$

Let us now consider $p_{1},p_{2},\mu_{1},\mu_{2}$ as in (2.18). A standard
application of Hölder’s inequality shows that

$\|\delta_{\ell}f\cdot\delta_{k}g\|_{L_{\mu}^{p}}\leq\|\delta_{\ell}f\|_{L_{\mu_{1}}^{p_{1}}}\|\delta_{k}g\|_{L_{\mu_{2}}^{p_{2}}}\,.$

Next for each block $\delta_{\ell}f,\delta_{k}g$, we use the fact that
$\psi_{\ell}$ is bounded in $L^{1}$ (uniformly in $\ell$), the expression
(2.10) for the norm in Besov spaces, and Young’s inequality. We get

$\|\delta_{\ell}f\cdot\delta_{k}g\|_{L_{\mu}^{p}}\lesssim
2^{\alpha\ell}2^{-\beta
k}\|f\|_{\mathcal{B}^{-\alpha,\mu_{1}}_{p_{1},\infty}}\|g\|_{\mathcal{B}^{\beta,\mu_{2}}_{p_{2},\infty}}\,.$
(A.4)

Summarizing our considerations so far, we have obtained

$\|\boldsymbol{\varphi}_{j}*(\delta_{\ell}f\cdot\delta_{k}g)\|_{L_{\mu}^{p}}\lesssim
2^{\alpha\ell}2^{-\beta
k}\|f\|_{\mathcal{B}^{-\alpha,\mu_{1}}_{p_{1},\infty}}\|g\|_{\mathcal{B}^{\beta,\mu_{2}}_{p_{2},\infty}}\,,$

for a proportional constant that only depends on $\mu_{\ast}$. Reporting this
estimate into the sum (A.3) defining $I_{j}$, we end up with

$\big{\|}I_{j}\big{\|}_{L^{p}_{\mu}}\lesssim\|f\|_{\mathcal{B}^{-\alpha,\mu_{1}}_{p_{1},\infty}}\|g\|_{\mathcal{B}^{\beta,\mu_{2}}_{p_{2},\infty}}\sum_{\begin{subarray}{c}\ell,k\geq
0\\\ 0\leq\ell\leq\max(j,k)\end{subarray}}2^{\alpha\ell}2^{-\beta k}\,.$ (A.5)

Let us upper bound the sum in the right-hand side of (A.5). Splitting the
summation in a straightforward way, we get

$\displaystyle\sum_{\begin{subarray}{c}\ell,k\geq 0\\\
0\leq\ell\leq\max(j,k)\end{subarray}}2^{\alpha\ell}2^{-\beta k}$
$\displaystyle\lesssim\sum_{0\leq k\leq j}2^{-\beta k}\sum_{0\leq\ell\leq
j}2^{\alpha\ell}+\sum_{k>j}2^{-\beta k}\sum_{0\leq\ell\leq k}2^{\alpha\ell}$
$\displaystyle\lesssim 2^{\alpha j}\sum_{k\geq 0}2^{-\beta k}+\sum_{k\geq
0}2^{-(\beta-\alpha)k}\lesssim 2^{\alpha j}+1\lesssim 2^{\alpha j}\,,$

where we have used the fact that $\beta>\alpha$ (see (2.18)). Going back to
(A.5), we have thus obtained

$\|I_{j}\|_{L_{\mu}^{p}}\lesssim 2^{\alpha
j}\|f\|_{\mathcal{B}^{-\alpha,\mu_{1}}_{p_{1},\infty}}\|g\|_{\mathcal{B}^{\beta,\mu_{2}}_{p_{2},\infty}}\,.$
(A.6)

Treatment of $I\\!I_{j}$. As in the previous step, we first analyze a single
term $\boldsymbol{\varphi}_{j}*(\delta_{\ell}f\cdot\delta_{k}g)$. In the
regime $\ell>\max(j,k)$, we write the definition of convolution product and
apply Fubini in order to get

$\displaystyle\big{(}\boldsymbol{\varphi}_{j}\ast\big{[}\delta_{\ell}f\cdot\delta_{k}g\big{]}\big{)}(x)$
$\displaystyle=\int
dy\,\boldsymbol{\varphi}_{j}(x-y)\big{[}\psi_{\ell}\ast(\boldsymbol{\varphi}_{\ell}\ast
f)\big{]}(y)\big{[}\psi_{k}\ast(\boldsymbol{\varphi}_{k}\ast g)\big{]}(y)$
$\displaystyle=\int dz\int dv\,(\boldsymbol{\varphi}_{\ell}\ast
f)(z)(\boldsymbol{\varphi}_{k}\ast g)(v)K_{jk\ell}(x,v,z)\,,$ (A.7)

where we have set

$K_{jk\ell}(x,v,z):=\int
dy\,\boldsymbol{\varphi}_{j}(x-y)\psi_{\ell}(y-z)\psi_{k}(y-v)\,.$ (A.8)

Let us further investigate the term $K_{jk\ell}$ defined in (A.8). Resorting
to a change of variable $y:=y-z$ and expressing the scalings in the functions
$\boldsymbol{\varphi}$ and $\psi$, we have

$\displaystyle K_{jk\ell}(x,v,z)$ $\displaystyle=\int
dy\,\psi_{\ell}(y)\boldsymbol{\varphi}_{j}((x-z)-y)\psi_{k}(y+(z-v))$
$\displaystyle=2^{d(\ell+j+k)}\int
dy\,\psi(2^{\ell}y)\boldsymbol{\varphi}(2^{j}(x-z)-2^{j}y)\psi(2^{k}y+2^{k}(z-v))$
$\displaystyle=2^{d(j+k)}\int
dy\,\psi(y)\boldsymbol{\varphi}(2^{j}(x-z)-2^{-(\ell-j)}y)\psi(2^{-(\ell-k)}y+2^{k}(z-v))\,,$

where the last identity stems from another elementary change of variable
$y:=2^{\ell}y$. For $m\geq 0$, we now introduce the indicator function
$\chi_{m}:={\bf 1}_{\mathbf{B}_{2^{-m+1}}}$. Thanks to the fact that the
support of both $\boldsymbol{\varphi}$ and $\psi$ is a subset of
$\mathbf{B}_{1}$, for $\ell>\max(j,k)$, we end up with

$K_{jk\ell}(x,v,z)=2^{d(j+k)}\chi_{j}(x-z)\chi_{k}(z-v)\\\ \times\int
dy\,\psi(y)\boldsymbol{\varphi}(2^{j}(x-z)-2^{-(\ell-j)}y)\psi(2^{-(\ell-k)}y+2^{k}(z-v))\,.$
(A.9)

We now proceed as for (2.1)-(2.9), invoking the fact that
$\psi\in\mathcal{D}_{L}$ and using a Taylor expansion for
$y\mapsto\boldsymbol{\varphi}(2^{j}(x-z)-2^{-(\ell-j)}y)\psi(2^{k}(z-v)+2^{-(\ell-k)}y)$.
We let the reader check that we get

$\bigg{|}\int
dy\,\psi(y)\boldsymbol{\varphi}(2^{j}(x-z)-2^{-(\ell-j)}y)\psi(2^{k}(z-v)+2^{-(\ell-k)}y)\bigg{|}\lesssim
2^{-L(\ell-\max(j,k))}\,,$

for some proportional constant that does not depend on $j,k,\ell$. Taking the
above inequality into account in (A.9), this leads to

$|K_{jk\ell}(x,v,z)|\lesssim
2^{d(j+k)-L(\ell-\max(j,k))}\chi_{j}(x-z)\chi_{k}(z-v)\,.$ (A.10)

Let us go back to (A.7) with (A.10) in hand. This allows to write

$\Big{|}\big{(}\boldsymbol{\varphi}_{j}\ast\big{[}\delta_{\ell}f\cdot\delta_{k}g\big{]}\big{)}(x)\Big{|}\lesssim
2^{-L(\ell-\max(j,k))+d(j+k)}\tilde{K}_{jk\ell}(x)\,,$ (A.11)

where the function $\tilde{K}_{jk\ell}$ can be expressed as

$\tilde{K}_{jk\ell}(x)=\int dz\int dv\,|(\boldsymbol{\varphi}_{\ell}\ast
f)(z)||(\boldsymbol{\varphi}_{k}\ast g)(v)|\chi_{j}(x-z)\chi_{k}(z-v)\,.$

We now express $\tilde{K}_{jk\ell}$ as a convolution product thanks to
elementary changes of variable. Specifically, setting $z:=x-z$ and $v=x-z-v$,
we get

$\tilde{K}_{jk\ell}(x)=\int dz\,|(\boldsymbol{\varphi}_{\ell}\ast
f)(x-z)|\chi_{j}(z)\int dv\,|(\boldsymbol{\varphi}_{k}\ast
g)(x-z-v)|\chi_{k}(v)\,.$

Hence it is easily seen that

$\displaystyle\tilde{K}_{jk\ell}(x)$ $\displaystyle=\int
dz\,|(\boldsymbol{\varphi}_{\ell}\ast
f)(x-z)|\chi_{j}(z)\big{[}\chi_{k}\ast|\boldsymbol{\varphi}_{k}\ast
g|\big{]}(x-z)$
$\displaystyle=\bigg{[}\chi_{j}\ast\Big{[}|\boldsymbol{\varphi}_{\ell}\ast
f|\cdot\big{[}\chi_{k}\ast|\boldsymbol{\varphi}_{k}\ast
g|\big{]}\Big{]}\bigg{]}(x)\,.$

We plug the above formula into (A.11). Then our expression with convolutions
allows the application of (2.5), as in the previous step. We first get

$\big{\|}\boldsymbol{\varphi}_{j}\ast\big{[}\delta_{\ell}f\cdot\delta_{k}g\big{]}\big{\|}_{L^{p}_{\mu}}\lesssim
2^{-L(\ell-\max(j,k))}\big{(}2^{dj}\|\chi_{j}\|_{L^{1}}\big{)}\,2^{dk}\Big{\|}|\boldsymbol{\varphi}_{\ell}\ast
f|\cdot\big{[}\chi_{k}\ast|\boldsymbol{\varphi}_{k}\ast
g|\big{]}\Big{\|}_{L^{p}_{\mu}}\,.$

Then thanks to Hölder’s inequality, we obtain

$\big{\|}\boldsymbol{\varphi}_{j}\ast\big{[}\delta_{\ell}f\cdot\delta_{k}g\big{]}\big{\|}_{L^{p}_{\mu}}\lesssim
2^{-L(\ell-\max(j,k))}\big{\|}\boldsymbol{\varphi}_{\ell}\ast
f\big{\|}_{L^{p_{1}}_{\mu_{1}}}\big{(}2^{dk}\big{\|}\chi_{k}\ast|\boldsymbol{\varphi}_{k}\ast
g|\big{\|}_{L^{p_{2}}_{\mu_{2}}}\big{)}\,.$

Then owing to (2.5) again and similarly to (A.4), this yields

$\displaystyle\big{\|}\boldsymbol{\varphi}_{j}\ast\big{[}\delta_{\ell}f\cdot\delta_{k}g\big{]}\big{\|}_{L^{p}_{\mu}}\lesssim
2^{-L(\ell-\max(j,k))}2^{\alpha\ell}2^{-\beta
k}\|f\|_{\mathcal{B}^{-\alpha,\mu_{1}}_{p_{1},\infty}}\|g\|_{\mathcal{B}^{\beta,\mu_{2}}_{p_{2},\infty}}\,,$
(A.12)

where the proportional constants only depend on $\mu_{\ast}$.

We can now proceed as in (A.5)-(A.6). That is we plug (A.12) into the
definition (A.3) of $I\\!I_{j}$. This yields

$\|I\\!I_{j}\|_{L_{\mu}^{p}}\lesssim\|f\|_{\mathcal{B}^{-\alpha,\mu_{1}}_{p_{1},\infty}}\|g\|_{\mathcal{B}^{\beta,\mu_{2}}_{p_{2},\infty}}\sum_{\begin{subarray}{c}\ell,k\geq
0\\\
\ell>\max(j,k)\end{subarray}}2^{-L(\ell-\max(j,k))}2^{\alpha\ell}2^{-\beta
k}\,.$ (A.13)

Then we split the sum in the right-hand side above and take into account the
fact that $\beta>\alpha,L>\alpha$ in order to obtain

$\displaystyle\sum_{\begin{subarray}{c}\ell,k\geq 0\\\
\ell>\max(j,k)\end{subarray}}2^{-L(\ell-\max(j,k))}2^{\alpha\ell}2^{-\beta k}$
$\displaystyle=\sum_{0\leq k\leq j}2^{-\beta
k}\sum_{\ell>j}2^{-L(\ell-j)}2^{\alpha\ell}+\sum_{k>j}2^{-\beta
k}\sum_{\ell>k}2^{-L(\ell-k)}2^{\alpha\ell}$ $\displaystyle=2^{\alpha
j}\sum_{0\leq k\leq j}2^{-\beta
k}\sum_{\ell>0}2^{-(L-\alpha)\ell}+\sum_{k>j}2^{-(\beta-\alpha)k}\sum_{\ell>0}2^{-(L-\alpha)\ell}$
$\displaystyle\lesssim 2^{\alpha j}+2^{-(\beta-\alpha)j}\lesssim 2^{\alpha
j}\,.$

Resorting to this inequality in (A.13), we have obtained

$\|I\\!I_{j}\|_{L_{\mu}^{p}}\lesssim 2^{\alpha
j}\|f\|_{\mathcal{B}^{-\alpha,\mu_{1}}_{p_{1},\infty}}\|g\|_{\mathcal{B}^{\beta,\mu_{2}}_{p_{2},\infty}}\,.$
(A.14)

Conclusion. We simply gather (A.6) and (A.14) into (A.3) and (A.2). This
yields

$\|\varphi_{j}*(f\cdot g)\|_{L_{\mu}^{p}}\lesssim 2^{\alpha
j}\|f\|_{\mathcal{B}^{-\alpha,\mu_{1}}_{p_{1},\infty}}\|g\|_{\mathcal{B}^{\beta,\mu_{2}}_{p_{2},\infty}}\,.$

Owing to the definition (2.10) of norms in Besov spaces, we thus easily get
the fact that $f\cdot g$ sits in $\mathcal{B}_{p,\infty}^{-\alpha,\mu}$ and
that (2.19) holds. This finishes the proof. ∎

###  A.3. Proof of Lemma 3.5

Case $d=1$. Invoking the definition (3.1), one has here

$\displaystyle G_{t}(y-2^{-j}z)-G_{t}(y)=\frac{1}{2}\big{\\{}{\bf
1}_{\\{|y-2^{-j}z|\leq t\\}}-{\bf 1}_{\\{|y|\leq t\\}}\big{\\}}\,.$ (A.15)

Whenever $|z|\leq\min\\{2^{j+1}t,2\\}$, a careful analysis of the intervals at
stake reveals that

$\displaystyle\big{|}{\bf 1}_{\\{|y-2^{-j}z|\leq t\\}}-{\bf 1}_{\\{|y|\leq
t\\}}\big{|}$ $\displaystyle\lesssim$ $\displaystyle{\bf 1}_{\\{|y-t|\leq
2^{-j}|z|\\}}+{\bf 1}_{\\{|y+t|\leq 2^{-j}|z|\\}}$ (A.16)
$\displaystyle\lesssim$ $\displaystyle{\bf 1}_{\\{|y-t|\leq 2^{-j+1}\\}}+{\bf
1}_{\\{|y+t|\leq 2^{-j+1}\\}}\,.$

Furthermore, if $2^{j+1}t<|z|<2$, then $t\leq 2^{-j}$ and

$\big{|}{\bf 1}_{\\{|y-2^{-j}z|\leq t\\}}-{\bf 1}_{\\{|y|\leq
t\\}}\big{|}\lesssim{\bf 1}_{\\{|y-2^{-j}z|\leq t\\}}+{\bf 1}_{\\{|y|\leq
t\\}}\lesssim{\bf 1}_{\\{|y|\leq 2^{-j+2}\\}}\,.$ (A.17)

Combining the bounds (A.16)-(A.17) with the expression (A.15), we obtain

$\displaystyle\big{\|}\mathcal{K}^{(1)}_{t,j}\big{\|}_{L^{1}}$
$\displaystyle\leq
2^{j}\int_{\mathbb{R}}dy\int_{|z|<2}dz\,\big{|}G_{t}(y-2^{-j}z)-G_{t}(y)\big{|}$
$\displaystyle\lesssim 2^{j}\int_{\mathbb{R}}dy\,\big{\\{}{\bf
1}_{\\{|y-t|\leq 2^{-j+1}\\}}+{\bf 1}_{\\{|y+t|\leq 2^{-j+1}\\}}+{\bf
1}_{\\{|y|\leq 2^{-j+2}\\}}\big{\\}}$ $\displaystyle\lesssim
2^{j}\int_{\mathbb{R}}dy\,{\bf 1}_{\\{|y|\leq 2^{-j+2}\\}}\lesssim
2^{j}\int_{\mathbb{R}}dy\,{\bf 1}_{\\{|2^{j}y|\leq
4\\}}\lesssim\int_{\mathbb{R}}dy\,{\bf 1}_{\\{|y|\leq 4\\}},$

which naturally corresponds to the desired uniform bound.

Case $d=2$. Let us denote the difference within the integral as

$\displaystyle A_{t,j,z}(y)$ $\displaystyle:=G_{t}(y-2^{-j}z)-G_{t}(y)$
$\displaystyle=\frac{1}{\left(t^{2}-|y-2^{-j}z|^{2}\right)^{1/2}}{\bf
1}_{\\{t-|y-2^{-j}z|>0\\}}-\frac{1}{\left(t^{2}-|y|^{2}\right)^{1/2}}{\bf
1}_{\\{t-|y|>0\\}}\,.$ (A.18)

For every $z\in\mathbf{B}_{2}$, we can write

$\displaystyle\int_{\mathbb{R}^{2}}dy\,|A_{t,j,z}(y)|$
$\displaystyle=\int_{\\{|y-2^{-j}z|\leq|y|\\}}dy\,|A_{t,j,z}(y)|+\int_{\\{|y|\leq|y-2^{-j}z|\\}}dy\,|A_{t,j,z}(y)|$
$\displaystyle=\int_{\\{|y-2^{-j}z|\leq|y|\\}}dy\,|A_{t,j,z}(y)|+\int_{\\{|y+2^{-j}z|\leq|y|\\}}dy\,|A_{t,j,z}(y+2^{-j}z)|$
$\displaystyle=\mathcal{I}_{t,j}(z)+\mathcal{I}_{t,j}(-z),$ (A.19)

where we have set

$\mathcal{I}_{t,j}(z):=\int_{\\{|y-2^{-j}z|\leq|y|\\}}dy\,|A_{t,j,z}(y)|.$

Let us split the latter expression into

$\displaystyle\mathcal{I}_{t,j}(z)$
$\displaystyle=\int_{\\{t\leq|y-2^{-j}z|\leq|y|\\}}dy\,|A_{t,j,z}(y)|$
$\displaystyle\hskip
28.45274pt+\int_{\\{|y-2^{-j}z|<t\leq|y|\\}}dy\,|A_{t,j,z}(y)|+\int_{\\{|y-2^{-j}z|\leq|y|<t\\}}dy\,|A_{t,j,z}(y)|$
$\displaystyle=:\mathcal{I}^{(1)}_{t,j}(z)+\mathcal{I}^{(2)}_{t,j}(z)+\mathcal{I}^{(3)}_{t,j}(z).$
(A.20)

We will now treat those three terms separately.

Given the expression (A.18) of $A_{t,j,z}(y)$, it is readily checked that
$\mathcal{I}^{(1)}_{t,j}(z)=0$. Next for the term
$\mathcal{I}^{(2)}_{t,j}(z)$, we notice that for every $z\in\mathbf{B}_{2}$,

$\displaystyle\mathcal{I}^{(2)}_{t,j}(z)$
$\displaystyle=\int_{\\{|y-2^{-j}z|<t\leq|y|\\}}\frac{dy}{\left(t^{2}-|y-2^{-j}z|^{2}\right)^{1/2}}=\int_{\\{|y|<t\leq|y+2^{-j}z|\\}}\frac{dy}{\left(t^{2}-|y|^{2}\right)^{1/2}}$
(A.21)
$\displaystyle\leq\int_{\\{\max(0,t-2^{-j+1})<|y|<t\\}}\frac{dy}{\left(t^{2}-|y|^{2}\right)^{1/2}}$
$\displaystyle\lesssim{\bf 1}_{\\{0\leq t\leq
2^{-j+1}\\}}\int_{0}^{t}\frac{r\,dr}{\left(t^{2}-r^{2}\right)^{1/2}}+{\bf
1}_{\\{t>2^{-j+1}\\}}\int_{t-2^{-j+1}}^{t}\frac{r\,dr}{\left(t^{2}-r^{2}\right)^{1/2}}$
$\displaystyle\lesssim{\bf 1}_{\\{0\leq t\leq
2^{-j+1}\\}}\int_{0}^{t^{2}}\frac{ds}{\left(t^{2}-s\right)^{1/2}}+{\bf
1}_{\\{t>2^{-j+1}\\}}\int_{(t-2^{-j+1})^{2}}^{t^{2}}\frac{ds}{\left(t^{2}-s\right)^{1/2}}$
$\displaystyle\lesssim{\bf 1}_{\\{0\leq t\leq 2^{-j+1}\\}}t+{\bf
1}_{\\{t>2^{-j+1}\\}}(t^{2}-(t-2^{-j+1})^{2})^{1/2}\lesssim
2^{-j+1}+(2^{-j+1}t)^{1/2},$

which immediately entails that

$\sup_{z\in\mathbf{B}_{2}}\sup_{t\in[0,1]}\mathcal{I}^{(2)}_{t,j}(z)\lesssim
2^{-\frac{j}{2}}.$ (A.22)

Finally, for every $z\in\mathbf{B}_{2}$,

$\displaystyle\mathcal{I}^{(3)}_{t,j}(z)$
$\displaystyle=\int_{\\{|y-2^{-j}z|<|y|<t\\}}dy\,\bigg{|}\frac{1}{\left(t^{2}-|y-2^{-j}z|^{2}\right)^{1/2}}-\frac{1}{\left(t^{2}-|y|^{2}\right)^{1/2}}\bigg{|}$
$\displaystyle=\int_{\\{|y-2^{-j}z|<|y|<t\\}}dy\,\bigg{[}\frac{1}{\left(t^{2}-|y|^{2}\right)^{1/2}}-\frac{1}{\left(t^{2}-|y-2^{-j}z|^{2}\right)^{1/2}}\bigg{]}$
$\displaystyle=\int_{\\{|y-2^{-j}z|<|y|<t\\}}\frac{dy}{\left(t^{2}-|y|^{2}\right)^{1/2}}-\int_{\\{|y|<|y+2^{-j}z|<t\\}}\frac{dy}{\left(t^{2}-|y|^{2}\right)^{1/2}}$
$\displaystyle=\int_{\mathbb{R}^{2}}\frac{dy}{\left(t^{2}-|y|^{2}\right)^{1/2}}\big{[}{\bf
1}_{\\{|y-2^{-j}z|<|y|<t\\}}-{\bf 1}_{\\{|y|<|y+2^{-j}z|<t\\}}\big{]}.$ (A.23)

At this point, the key observation is that

${\bf 1}_{\\{|y-2^{-j}z|<|y|<t\\}}-{\bf 1}_{\\{|y|<|y+2^{-j}z|<t\\}}\leq{\bf
1}_{\\{|y|<t\leq|y+2^{-j}z|\\}}.$ (A.24)

Indeed, the difference ${\bf 1}_{\\{|y-2^{-j}z|<|y|<t\\}}-{\bf
1}_{\\{|y|<|y+2^{-j}z|<t\\}}$ is strictly positive if and only if

$|y-2^{-j}z|<|y|<t\quad\text{and}\quad\big{(}|y|\geq|y+2^{-j}z|\quad\text{or}\quad|y+2^{-j}z|\geq
t\big{)}$

It turns out that we cannot have simultaneously $|y-2^{-j}z|<|y|$ and
$|y|\geq|y+2^{-j}z|$, since it would imply that $-2^{-j+1}y\cdot
z+2^{-2j}|z|^{2}<0$ and $2^{-j+1}y\cdot z+2^{-2j}|z|^{2}\leq 0$, leading to an
immediate contradiction. Thus, the difference ${\bf
1}_{\\{|y-2^{-j}z|<|y|<t\\}}-{\bf 1}_{\\{|y|<|y+2^{-j}z|<t\\}}$ is strictly
positive if and only if

$|y-2^{-j}z|<|y|<t\quad\text{and}\quad|y+2^{-j}z|\geq t,$

which immediately yields (A.24).

Injecting (A.24) into (A.23), we obtain for every $z\in\mathbf{B}_{2}$,

$\displaystyle\mathcal{I}^{(3)}_{t,j}(z)$
$\displaystyle\leq\int_{\mathbb{R}^{2}}\frac{dy}{\left(t^{2}-|y|^{2}\right)^{1/2}}{\bf
1}_{\\{|y|<t\leq|y+2^{-j}z|\\}}.$

We are here in the same position as in (A.21), and so we can use the same
estimates as above to derive that uniformly over $z\in\mathbf{B}_{2}$ and
$t\in[0,1]$,

$\mathcal{I}^{(3)}_{t,j}(z)\lesssim 2^{-\frac{j}{2}}.$ (A.25)

Plugging (A.22) and (A.25) into (A.20), we see that

$\sup_{z\in\mathbf{B}_{2}}\sup_{t\in[0,1]}\mathcal{I}_{t,j}(z)\lesssim
2^{-\frac{j}{2}},$

which, going back to decomposition (A.19), easily leads us to the assertion
(3.13). ∎

#### Acknowledgment.

X. Chen is partially supported by the Simons Foundation grant 585506. J. Song
is partially supported by Shandong University grant 11140089963041 and
National Natural Science Foundation of China grant 12071256. S. Tindel is
partially supported by the NSF grant DMS-1952966.

## References

  * [1] R. M. Balan. Stratonovich Solution for the Wave Equation. Journal of Theoretical Probability (2022).
  * [2] X. Chen, A, Deya, C, Ouyang, and S. Tindel. A $K$-rough path above the space-time fractional Brownian motion. Stochastics and Partial Differential Equations: Analysis and Computations (2021) Vol.31, 819-866.
  * [3] X. Chen, A, Deya, C, Ouyang, and S. Tindel. Moment estimates for some renormalized parabolic Anderson models. The Annals of Probability (2021) Vol.49, 2599-2636.
  * [4] X. Chen, A. Deya, J. Song and S. Tindel: Solving the hyperbolic Anderson model 1: Skorohod setting
  * [5] J.-C. Mourrat and H. Weber: Global well-posedness of the dynamic $\Phi^{4}$ model in the plane. Ann. Probab. 45 (2017), no. 4.
  * [6] A. Deya. A nonlinear wave equation with fractional perturbation. Ann. Probab. 47 (2019), no. 3, 1775-1810.
  * [7] J. Ginibre and G. Velo. Generalized Strichartz inequalities for the wave equation. Journal of Functional Analysis 133 (1995), 50-68.
  * [8] M. Gubinelli, H. Koch, and T. Oh. Renormalization of the two-dimensional stochastic nonlinear wave equations. Trans. Amer. Math. Soc. 370 (2018), 7335-7359.
  * [9] V. S. Rychkov: Littlewood-Paley theory and function spaces with $A_{p}^{\textnormal{loc}}$ weights. Math. Nachr., 224 (2001), no. 1.
  * [10] L. Quer-Sardanyons and S. Tindel: The 1-d stochastic wave equation driven by a fractional Brownian sheet. Stochastic Processes and their Applications 117 (2007), 1448-1472.
  * [11] G. Samorodnitsky and M.S. Taqqu: Stable non-gaussian random processes. Chapman and Hall, Boca Raton (1994)
  * [12] E.M. Stein: Harmonic Analysis: Real – Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton, NJ, 1993