A generalized Cauchy-Lipschitz theorem in low regularity spaces

TL;DR

This paper establishes well-posedness for a class of differential equations in low regularity spaces, extending classical results to include integro-differential operators with controlled derivative loss.

Contribution

It generalizes the Cauchy-Lipschitz theorem to low regularity spaces and includes integro-differential operators with limited derivative loss.

Findings

Proves well-posedness for abstract differential equations with low regularity.

Extends classical Lipschitz-based results to more general operators.

Identifies conditions on derivative loss for operator dominance.

Abstract

We prove well-posedness for some abstract differential equations of the first order. Our result covers the usual case of Lipschitz composition operators. It also contains the case of some integro-differential operators acting on spaces with low regularity indexes. The loss of derivatives induced by such operators has to be lower than one, in order to be dominated by the first order derivative involved in the problem.

Equations

$$\{\begin{array}{l}{u}^{\prime }={\mathcal{H}}_{t_0}(u)\\ u(0)=u_0\end{array}$$

$$<u,v{>}_{\sigma ,p,q,I}:=\sum _{|k^{\prime }-k|\le 1}{\int }_{\mathbb{R}}{\Delta }_k(P_{0,I}(u))(t).{\Delta }_{k^{\prime }}(P_{0,I}(v))(t)dt$$

$$B_{t,\alpha }(u_0{1}_{]0,t[},\rho )=\{u\in {\mathcal{B}}_{t,\alpha }(u_0{1}_{]0,t[},\rho )\text{ with }u(0)=u_0\}$$

$$u_{\ast }{|}_{]0,t_1[}=u-u_0$$

$$\Vert u_{\ast }{\Vert }_{{\mathrm{B}}_{\mathrm{p},\mathrm{q}}^{1/p+\alpha }(]0,t_2[,\mathrm{E})}\le \Vert u_{\ast }{\Vert }_{B_{p,q}^{1/p+\alpha }(\mathbb{R},E)}\le \Vert u-u_0{\Vert }_{B_{p,q}^{1/p+\alpha }(]0,t_1[,E)}+ϵ<\rho$$

$${\mathcal{H}}_t:{\mathcal{B}}_{t,\alpha }(u_0,R)\to B_{p,q}^{-1/p^{\prime }+\eta }(]0,t[,E)$$

$${\mathcal{H}}_t(u)={\mathcal{H}}_T(U){|}_{]0,t[}$$

$$\{\begin{array}{l}{u}^{\prime }={\mathcal{H}}_{t_0}(u)\\ u(0)=u_0\end{array}$$

$$\displaystyle\|P_{0,t}u\|_{B_{p,q}^{s}(\mathbb{R},E)}\leq C\big{(}\|\chi_{]0,T[}\|_{B_{p^{\prime},\infty}^{1/p^{\prime}}(\mathbb{R},E)}+\|\chi_{]0,T[}\|_{L^{\infty}(\mathbb{R},E)}\big{)}\|\phi\|_{B_{p,q}^{s}(\mathbb{R},E)}$$

$$\Vert P_{0,t}u{\Vert }_{B_{p,q}^s(\mathbb{R},E)}\le C_T\Vert u{\Vert }_{B_{p,q}^s(]0,t[,E)}$$

$$\{\begin{array}{l}{u}^{\prime }=\varphi \\ u(0)=u_0\end{array}$$

$$u(t)=u_0+<\varphi {|}_{]0,t[},1_{]0,t[}>$$

$$\Vert u(t)-u_n(t){\Vert }_E$$

$$\Vert u-u_n{\Vert }_{L^{\infty }(]0,T[,E)}\le C_T\Vert \varphi -{\varphi }_n{\Vert }_{B_{p,q}^{s-1}(]0,T[,E)}$$

$$\Vert {\chi }_{1/t}{\Vert }_{{\mathrm{B}}_{m,\infty }^{1/m-ϵ}(\mathbb{R})}\le C_T{t}^{ϵ}$$

$$\Vert {\chi }_{1/t}{\Vert }_{B_{m,\infty }^{1/m-ϵ}}=\Vert {\chi }_{1/T}(t^{-1}T.){\Vert }_{B_{m,\infty }^{1/m-ϵ}}\le C(T/t{)}^{1/m-ϵ-1/m}\Vert {\chi }_{1/T}{\Vert }_{B_{m,\infty }^{1/m-ϵ}}$$

$$\Vert \theta {\chi }_{1/t}{\Vert }_{B_{p,q}^s(\mathbb{R},E)}\le \mathrm{C}\Vert \theta {\Vert }_{B_{p,q}^{\sigma }(\mathbb{R},E)}{t}^{\sigma -s}$$

$$B_{p,q}^{\sigma }(\mathbb{R},E)\times B_{p,q}^{1/p-ϵ}(\mathbb{R})\hookrightarrow B_{\infty ,q}^{\sigma -1/p}(\mathbb{R},E)\times B_{p,\infty }^{1/p-ϵ}(\mathbb{R})\stackrel{\Pi }{\to }{B}_{p,q}^{\sigma -ϵ}(\mathbb{R},E)$$

$$B_{p,q}^{1/p-ϵ}(\mathbb{R})\times B_{p,q}^{\sigma }(\mathbb{R},E)\hookrightarrow B_{\infty ,\infty }^{-ϵ}(\mathbb{R})\times B_{p,q}^{\sigma }(\mathbb{R},E)\stackrel{\Pi }{\to }{\mathrm{B}}_{p,q}^{\sigma -ϵ}(\mathbb{R},E)$$

$${\mathrm{B}}_{p^{\prime },q}^{1/p^{\prime }-ϵ}(\mathbb{R})\times {\mathrm{B}}_{p,q}^{\sigma }(\mathbb{R},\mathrm{E})\stackrel{\mathfrak{R}}{\to }{\mathrm{B}}_{1,q}^{\sigma +1/p^{\prime }-ϵ}(\mathbb{R},\mathrm{E})\hookrightarrow {\mathrm{B}}_{p,q}^{\sigma -ϵ}(\mathbb{R},E)$$

$$\Vert u{\Vert }_{B_{p,q}^s(]0,t[,E)}\le \Vert \theta {\chi }_{1/t}{\Vert }_{B_{p,q}^s(]0,t[,E)}\le \mathrm{C}\Vert \theta {\Vert }_{B_{p,q}^{\sigma }(\mathbb{R},E)}{t}^{\sigma -s}$$

$$\Vert uo{I}_{\lambda }{\Vert }_{B_{p,q}^s(I_{1/\lambda }(\Omega ),E)}\le C{\lambda }^{s-1/p}\Vert u{\Vert }_{B_{p,q}^s(\Omega ,E)}$$

$$\Vert u{\Vert }_{B_{p,q}^s(I_{\lambda }(\Omega ),E)}\le C_{\Omega }\Vert u^{\prime }{\Vert }_{B_{p,q}^{s-1}(I_{\lambda }(\Omega ),E)}$$

$$\Vert u{\Vert }_{B_{p,q}^s(\Omega ,E)}$$

$$\Vert u{\Vert }_{B_{p,q}^{s-1}(\Omega ,E)}\le ϵ\Vert u{\Vert }_{B_{p,q}^s(\Omega ,E)}+C_{ϵ}\Vert u{\Vert }_{L^p(\Omega ,E)}$$

$$\Vert u{\Vert }_{B_{p,q}^s(\Omega ,E)}$$

$$\Vert u{\Vert }_{L^p(\Omega ,E)}\le C_T\Vert u^{\prime }{\Vert }_{B_{p,q}^{s-1}(\Omega ,E)}$$

$$\Vert u{\Vert }_{B_{p,q}^s(I_{\lambda }(\Omega ),E)}$$

$$\le C(1/\lambda {)}^{s-1/p}\Vert v{\Vert }_{B_{p,q}^s(\Omega ,E)}$$

$$\Vert v{\Vert }_{B_{p,q}^s(\Omega ,E)}$$

$$=C_{\Omega }\lambda \Vert u^{\prime }o{I}_{\lambda }{\Vert }_{B_{p,q}^{s-1}(\Omega ,E)}$$

Full text

1 Université de Lyon, Institut Camille Jordan et Insa Lyon, Bât. Leonard de Vinci No. 401, 21 Avenue Jean Capelle, F-69621, Villeurbanne, France.

Keywords: Differential equations, irregular coefficients, Poincaré inequality, well-posedness, Cauchy-Lipschitz theorem, Besov spaces.

1 Introduction.

The aim of this note is to prove an extended Cauchy-Lipschitz theorem for problems formally written as $u^{\prime}=\mathcal{H}_{T}(u)$ and $u(0)=u_{0}$. Here, $\mathcal{H}_{T}:U\subset B_{p,q}^{s}(]0,T[,E)\rightarrow B_{p,q}^{\sigma}(]0,T[,E)$ is a local operator loosing less than one derivative, i.e $\sigma>s-1$.

Theorem 1.1**.**

Let $T>0$, $R>0$, $1\leq p,q\leq\infty$, $1/p<s<\sigma+1<1/p+1$, and $u_{0}\in E$. Assume that $\mathcal{H}_{T}:B_{p,q}^{s}(]0,T[,E)\rightarrow B_{p,q}^{\sigma}(]0,T[,E)$ is a Lipschitz and local operator. Then, there exists $0<\mathscr{T}\leq T$ such that, for any $0<t_{0}\leq\mathscr{T}$ the problem: find $u\in B_{p,q}^{s}(]0,t_{0}[,E)$ with:

[TABLE]

admits exactly one distributional solution. This solution belongs to $B_{p,q}^{\sigma+1}(]0,t_{0}[,E)$.

See part 3 for a definition of a local operator (assumption L2). In the above statement, the microscopic q-index plays practically no role, and similar statements hold within the functional frame of Sobolev spaces $W^{s,p}$. Nevertheless, this microscopic index has some importance when dealing with critical spaces. Last, for other extensions of the ODE theory, see for instance [5], [6] and [8].

Notice that in the case of operators acting on smooth functions spaces, there’s no reason to work within $B_{p,q}^{\sigma}(]0,T[,E)\hookrightarrow B_{\infty,q}^{\sigma-1/p}(]0,T[,E)$, $\sigma-1/p<0$. Therefore, significant examples of applications of theorem 1.1 must be searched among irregular operators. Using Bony’s decomposition, a simple operator is given by $\mathcal{H}_{T}(u)=\psi u$ with $\psi\in B_{p,q}^{\sigma}(]0,T[,\mathscr{L}(E))$, $1/p<s<\sigma+1<1/p+1$, $s>1/2$ and $s+\sigma>0$. See section 7 for extensions and other examples.

Our proof makes use of Picard fixed point theorem. Since there’s some “residual” compactness for the above Cauchy problem ($s-\sigma<1=$ order of derivation), this proof should be routine. Nevertheless, some technical difficulties arise, due to the fractional feature of the problem. and the low regularity index of the space $B_{p,q}^{\sigma}(]0,T[,E)$. For such a space, multiplication, composition and, above all, restriction-extension operations, must be handled with care. In particular, all the constants of continuity have to be bounded when working on vanishing intervals $]0,t[$, $t\rightarrow 0$, even for equivalent norms. From this point of view, $B_{p,q}^{-1/p^{\prime}}(]0,T[,E)$ seems to be a critical space, and most of our proofs relies on the following simple fact: the family of zero-extension operators $P_{0,t}:B_{p,q}^{\sigma}(]0,t[,E)\rightarrow B_{p,q}^{\sigma}(\mathbb{R},E)$ $(0<t<T)$ is equicontinuous under the condition $1/p-1<\sigma<1/p$. Equivalently, for such indexes, the characteristic function of an interval is a multiplier for $B_{p,q}^{\sigma}(\mathbb{R},E)$.

The paper is organized as follows. In the second section, we recall some notations and basic results, merely a definition and some properties of the Besov spaces, the definition of the paraproduct and remainder, and also the definition of some duality brackets. The third part is devoted to the statement of the main theorem. The proof of uniform inequalities, essentially a uniform fractional Poincaré’s inequality and a fractional integration inequality, is given in a fourth part. The firth part contains the proof of the main theorem. The sixth part concerns some extensions of this theorem. We state a Peano’s type theorem, and also give a global existence result. Some examples are given in the last part.

2 Notations and classical

results.

Throughout this paper $\mathrm{E}$ and $\mathrm{F}$ are two Banach spaces, and $\mathscr{L}(E,F)$ is the space of continuous linear applications from $E$ to $F$ . In the sequel, we consider Banach-valued distributions, and generalize, often without comments, scalar results to that context. The reader is refered to [1], [2], but also to [12], [13], [14], [4], since the Banach-valued case follows from the scalar case by few additional arguments. 2. 2.

For $1\leq r\leq\infty$, we denote by $r^{\prime}$ its conjugate exponent i.e $r^{-1}+r^{{}^{\prime}-1}=1$. 3. 3.

The symbol $\hookrightarrow$ stands for classical continuous embeddings. 4. 4.

Let $1\leq p,q\leq\infty$ and $s\in\mathbb{R}$. The non-homogeneous Besov space $B_{p,q}^{s}(\mathbb{R}^{n},E)$ can be defined as the space of tempered distribution $f$ such that (see[4]) $\|f\|_{B_{p,q}^{s}(\mathbb{R}^{n},E)}:=\|(2^{js}\|\Delta_{j}f\|_{L^{p}(\mathbb{R}^{n},E)})_{j\in\mathbb{Z}}\|_{l^{q}(\mathbb{Z})}<\infty$. In the above writings, the analytic functions $\Delta_{j}f$ are defined by the standard dyadic procedure (se [4] p.99). In particular, for $j\leq-2$ we have $\Delta_{j}f=0$. For $j\in\mathbb{Z}$, set, for future reference $S_{j}f=\sum_{k\leq j-1}\Delta_{k}f$. 5. 5.

For $u\in\mathscr{S}^{\prime}(\mathbb{R}^{n},\mathscr{L}(E,F))$, $v\in\mathscr{S}^{\prime}(\mathbb{R}^{n},E)$, the usual paraproduct (case $E=F=\mathbb{R}$) generalizes immediately as: $\Pi(u,v)=\sum_{j\geq-1}S_{j-1}u.\Delta_{j}v$ , and for the remainder: $\mathfrak{R}(u,v)=\sum_{|j-k|\leq 1}\Delta_{j}u.\Delta_{k}v$. So that formally, we get the Bony decomposition $u.v=\Pi(u,v)+\Pi(v,u)+\mathfrak{R}(u,v)$. We shall use freely continuity results for the paradaproduct and remainder. See for instance [4] pp. 102-104 or [11] p.35. 6. 6.

For $t>0$, we denote by $\chi_{1/t}\in\mathrm{B}_{m,\infty}^{1/m}(\mathbb{R},\mathbb{R})$ ($1\leq m\leq\infty$) the characteristic function of $]0,t[$. We set $\chi=\chi_{1}$. Similarly, $\chi_{J}\in\mathrm{B}_{m,\infty}^{1/m}(\mathbb{R},\mathbb{R})$ stands for the characteristic function of the interval $J$. Last, $\mathbf{1}_{]0,t[}:]0,t[\rightarrow\mathbb{R}$ is the unit function of $]0,t[$. 7. 7.

Let $\Omega\subset\mathbb{R}^{n}$ be a smooth domain. The Besov space $B_{p,q}^{s}(\Omega,\mathrm{E})$ is defined as the restrictions of elements of $B^{s}_{p,q}(\mathbb{R}^{n},\mathrm{E})$ to $\Omega$. The space $B_{p,q}^{s}(\Omega,\mathrm{E})$ is endowed with the quotient norm $\|\mathrm{u}\|_{B_{p,q}^{s}(\Omega,\mathrm{E})}=\mathrm{inf}\|\mathrm{v}\|_{B_{p,q}^{s}(\mathbb{R}^{n},\mathrm{E})}$ , the inf being taken on all the extensions $\mathrm{v}\in B_{p,q}^{s}(\mathbb{R}^{n},\mathrm{E})$ of $\mathrm{u}$. 8. 8.

Let $\Omega$ be a (smooth) domain of $\mathbb{R}^{n}$. For any $A\subset\mathscr{D}^{\prime}(\Omega)$, the restriction of a distribution $T\in\mathscr{D}^{\prime}(\Omega)$ to a domain $\mathscr{O}\subset\Omega$ is denoted by $T|_{\mathscr{O}}$. The set $\mathrm{A}|_{\mathscr{O}}$ is the set of elements $T|_{\mathscr{O}}$ with $T\in\mathrm{A}$. For $\mathrm{u}\in B_{p,q}^{s}(\Omega,E)$, we write $\|u\|_{B_{p,q}^{s}(\mathscr{O},E)}:=\|u|_{\mathscr{O}}\|_{B_{p,q}^{s}(\mathscr{O},E)}$. 9. 9.

We define duality-like pairings. The construction is similar to the one given in [4], p.70 and p.101 for the duality bracket. We restrict to the case of an interval $I$, and assume that $1\leq p,q\leq\infty$, $-1/p^{\prime}<\sigma<1/p$, or equivalently $-1/p<-\sigma<1/p^{\prime}$. It follows that the extension by zero operators $P_{0,I}$ is continuous in both case:

• •

$P_{0,I}:B^{-\sigma}_{p^{\prime},q^{\prime}}(I,\mathscr{L}(E,F))\rightarrow B^{-\sigma}_{p^{\prime},q^{\prime}}(\mathbb{R},\mathscr{L}(E,F))$

• •

$P_{0,I}:B^{\sigma}_{p,q}(I,E)\rightarrow B^{\sigma}_{p,q}(\mathbb{R},E)$

where, as customary, we have denoted by the same letter the two operators. Hence, we define the pairing $<.,.>_{\sigma,p,q,I}:B^{-\sigma}_{p^{\prime},q^{\prime}}(I,\mathscr{L}(E,F))\times B^{\sigma}_{p,q}(I,E)\rightarrow F$ (or simply, $<.,.>_{I}$, or $<,>$) by

[TABLE]

Hence $<u,v>_{I}=<P_{0,I}(u),P_{0,I}(v)>_{\mathbb{R}}$. Function $<.,.>_{\sigma,p,q,I}$ ”extends” continuously the pairing of $L^{2}(I,\mathscr{L}(E,F))\times L^{2}(I,E)\rightarrow F$ given by $\int_{I}u(t)v(t)dt$.

Last, we will sometimes write $<v,u>$ in place of $<u,v>$.

3 Statement of the theorem.

In order to state the main theorem, we have to define restriction procedures for an operator denoted below by $\mathcal{H}_{T}$. Let $1\leq p,q\leq\infty$, $0<\alpha<1$. For any $t\in\mathbb{R}_{+}$, $\rho>0$, $u_{0}\in E$, define $\mathscr{B}_{t,\alpha}(u_{0},\rho)$ as the open ball of $B_{p,q}^{1/p+\alpha}(]0,t[,E)$ with center $u_{0}$ and radius $\rho$, and set :

[TABLE]

Denote also by $\bar{B}_{t,\alpha}(u_{0}\mathbf{1}_{]0,t[},\rho)$ its closure in $B_{p,q}^{1/p+\alpha}(]0,t[,E)$ (similar notation for $\mathscr{B}_{t,\alpha}(u_{0}\mathbf{1}_{]0,t[},\rho)$). Until the end of the paper, we often and abusively identify $u_{0}\mathbf{1}_{]0,t[}$ with $u_{0}$. We write for instance, $B_{t,\alpha}(u_{0},\rho)$ in place of $B_{t,\alpha}(u_{0}\mathbf{1}_{]0,t[},\rho)$. In the sequel, we implicitely use the following

Lemma 3.1**.**

Let $1\leq p,q\leq\infty$, $0<\alpha<1$, $u_{0}\in E$, $\rho>0$ and $0\leq t_{1}\leq t_{2}$. Then, $B_{t_{1},\alpha}(u_{0},\rho)=B_{t_{2},\alpha}(u_{0},\rho)|_{]0,t_{1}[}$ and $\mathscr{B}_{t_{1},\alpha}(u_{0},\rho)=\mathscr{B}_{t_{2},\alpha}(u_{0},\rho)|_{]0,t_{1}[}$.

Proof.

We only prove the first equality. Inclusion $B_{t_{2},\alpha}(u_{0},\rho)|_{]0,t_{1}[}\subset B_{t_{1},\alpha}(u_{0},\rho)$ is clear. For the opposite inclusion, let $u\in B_{t_{1},\alpha}(u_{0},\rho)$ and let $\epsilon=(\rho-\|u-u_{0}\|_{B_{p,q}^{1/p+\alpha}(]0,t_{1}[,E)})/2$. There exists $u_{*}\in B_{p,q}^{1/p+\alpha}(\mathbb{R},E)$ such that

[TABLE]

[TABLE]

Set $u_{**}=u_{*}+u_{0,*}$ with $u_{0,*}\in B_{p,q}^{1/p+\alpha}(\mathbb{R},\mathrm{E})$ and $u_{0,*}|_{]0,t_{2}[}=u_{0}$. By 3.2, $u_{**}(0)=u(0)=u_{0}$. Hence, by 3.3, $u_{**}|_{]0,t_{2}[}\in B_{t_{2},\alpha}(u_{0},\rho)$. With 3.2, this provides the lemma. ∎

Let now $R>0$, $T>0$, $1\leq p,q\leq\infty$, $0<\alpha<\eta<1$, and $u_{0}\in E$ be fixed. Let $\mathcal{H}_{T}:\mathscr{B}_{T,\alpha}(u_{0},R)\rightarrow B_{p,q}^{-1/p^{\prime}+\eta}(]0,T[,E)$. Consider the following properties: for any $(u,v,t)\in\mathscr{B}_{T,\alpha}(u_{0},R)^{2}\times]0,T[$, we have

• •

($L1$) $\phantom{OO}\|\mathcal{H}_{T}(u)-\mathcal{H}_{T}(v)\|_{B_{p,q}^{-1/p^{\prime}+\eta}(]0,T[,E)}\leq\mathfrak{C}_{T}\|u-v\|_{B_{p,q}^{1/p+\alpha}(]0,T[,E)}$

• •

($L2$) $\phantom{OO}\textrm{If }u|_{]0,t[}=v|_{]0,t[},\textrm{ then }\mathcal{H}_{T}(u)|_{]0,t[}=\mathcal{H}_{T}(v)|_{]0,t[}$

When condition $L2$ is satisfied, we define for any $t\in]0,T[$ an operator:

[TABLE]

by restriction. It means that, for any $u\in\mathscr{B}_{t,\alpha}(u_{0},R)$ we have:

[TABLE]

with $U\in\mathscr{B}_{T,\alpha}(u_{0},R)$ and $U|_{]0,\mathrm{t}[}=u$.

With these notations, the main result is the following

Theorem 3.1**.**

Let $T>0$, $R>0$, $1\leq p,q\leq\infty$, $0<\alpha<\eta<1$, and $u_{0}\in E$. Assume that $\mathcal{H}_{T}:\mathscr{B}_{T,\alpha}(u_{0},R)\rightarrow B_{p,q}^{-1/p^{\prime}+\eta}(]0,T[,E)$ satisfies conditions $(L_{1})$ and $(L_{2})$. Then, there exists $0<\rho<R$ and $0<\mathscr{T}<T$ such that, for any $0<t_{0}\leq\mathscr{T}$ the problem: find $u\in\mathscr{B}_{t_{0},\alpha}(u_{0},\rho)$ with:

[TABLE]

admits exactly one distributional solution. This solution belongs to $B_{p,q}^{1/p+\eta}(]0,t_{0}[,E)$.

The proof requires some lemmas which are detailed in the following section.

4 Uniform estimates.

The main goal of this section is to get uniform (in $t$) bounds in the required inequalities. In the sequel, for $0<t<T$, we denote by $P_{0,t}:B_{p,q}^{s}(]0,t[,E)\rightarrow B_{p,q}^{s}(\mathbb{R},E)$ the zero-extension operator.

Lemma 4.1**.**

Let $T>0$, $1\leq p,q\leq\infty$ and $-1/p^{\prime}<s<1/p$. The family $(P_{0,t})_{0<t\leq T}$ is equicontinuous.

Proof.

Let $u\in B_{p,q}^{s}(]0,t[,E)$ and let $\phi\in B_{p,q}^{s}(\mathbb{R},E)$ be any extension of $u$. Since $-1/p^{\prime}<s<1/p$, $\chi_{]0,t[}$ is a multiplier for $B_{p,q}^{s}(\mathbb{R},E)$. Hence, $\phi\chi_{]0,t[}$ is well defined in $B_{p,q}^{s}(\mathbb{R},E)$ and $P_{0,t}u=\phi\chi_{]0,t[}$. Therefore

[TABLE]

Taking the inf on all the extensions $\phi\in B_{p,q}^{s}(\mathbb{R},E)$ of $u$, we get

[TABLE]

∎

We deduce from lemma 4.1 the integral formulation of the problem

Lemma 4.2**.**

Let $T>0$, $1\leq p,q\leq\infty$, $1/p<s<1+1/p$. Let $\phi\in B_{p,q}^{s-1}(]0,T[,E)$ and $u_{0}\in E$. Then, problem: find $u\in B_{p,q}^{s}(]0,T[,E)$ with

[TABLE]

admits exactly one solution, given by

[TABLE]

Proof.

We only prove formula 4.3. For $\phi\in C^{\infty}([0,T],E)$, formula 4.3 reduces to the usual integral formula. In the case $\phi\in B_{p,q}^{s-1}(]0,T[,E)$, let $(\phi_{n})_{n\in\mathbb{N}}$ be a sequence of $\mathrm{C}^{\infty}([0,T],\mathrm{E})$ functions converging to $\phi$ in $B_{p,q}^{s-1}(]0,T[,E)$. Set $u(t)=u_{0}+<\phi|_{]0,t[},\mathbf{1}_{]0,t[}>$ and $u_{n}(t)=u_{0}+<\phi_{n}|_{]0,t[},\mathbf{1}_{]0,t[}>$. By the continuity of the bracket $<,>_{\mathbb{R}}$ and the equicontinuity of $(P_{0,t})_{0<t<T}$ (lemma 4.1), we get, for any $t\in]0,T[$ (see part 2, 9.)

[TABLE]

Therefore

[TABLE]

so that $u_{n}\rightarrow u$ in $L^{\infty}(]0,T[,E)$. Hence, from $u_{n}^{\prime}=\phi_{n}$ we get $u^{\prime}=\phi$. The rest of the proof is omitted. ∎

We need two additional fractional inequalities. The first one (cf. b) in theorem 4.1) replaces the full integration in use in the standard proof of Cauchy-Lipschitz theorem.

Theorem 4.1**.**

a) Let $0<t<T$, $1\leq m\leq\infty$ and $0\leq\epsilon<1/m$. Then

[TABLE]

b) Let $T>0$, $R>0$, $1\leq p,q\leq\infty$, $-1/p^{\prime}<s<\sigma<1/p$. For any $u\in B_{p,q}^{\sigma}(]0,T[,E)$ and any $t\in]0,T]$ we have: $\|u\|_{B_{p,q}^{s}(]0,t[,E)}\leq C_{T}\|u\|_{B_{p,q}^{\sigma}(]0,t[,E)}t^{\sigma-s}$.

Proof.

a)

[TABLE]

(see [12] page 206, or 4.13 below)

b) We first show that, for $0<t<T$, and for any $\theta\in B_{p,q}^{\sigma}(\mathbb{R},E)$, the following inequality holds true:

[TABLE]

Set $\epsilon=\sigma-s$. Taking in account $-\epsilon<0$ and $\sigma-1/p<0$, we get:

[TABLE]

[TABLE]

Since $\sigma+\frac{1}{p^{\prime}}-\epsilon=s+1/p^{\prime}>0$, we have, for the remainder:

[TABLE]

Notice that $\chi_{1/t}\in B_{m,q}^{1/m-\epsilon}(\mathbb{R})$ for $m=p$ and for $m=p^{\prime}$. Using 4.9, 4.10, 4.11 and a), inequality 4.8 a follows.

Now, for $u\in B_{p,q}^{\sigma}(]0,T[,E)$ and $0<t\leq T$, denoting by $\theta\in B_{p,q}^{\sigma}(\mathbb{R},E)$ any extension of $u|_{]0,t[}\in B_{p,q}^{\sigma}(]0,t[,E)$ and invoking 4.8:

[TABLE]

We take the inf on all the extensions $\theta$, and get b). ∎

The second inequality is a uniform fractional Poincaré’s inequality. We give the proof for a restricted range of values $1/p<s<1$. The general proof $1/p<s<1+1/p$ relies on tedious extension-retractation arguments and is omitted.

In the proof, for any open subset $\Omega$ of $\mathbb{R}$, we use the function $I_{\lambda,\Omega}:\Omega\rightarrow\mathbb{R}$ (or simply $I_{\lambda}$) defined by $I_{\lambda}(t)=\lambda t$. Recall the following inequality (see [12]), valid for any $\lambda\geq 1$, $1\leq p,q\leq\infty$, $s>0$ and $u\in B_{p,q}^{s}(\Omega,E)$

[TABLE]

By a duality argument, this inequality holds true for $s<0$ and $0<\lambda\leq 1$ (see [4] prop. 2.76). Therefore

Lemma 4.3**.**

Assume that $1/p<s<1+1/p$, and let $\Omega$ be a bounded interval of $\mathbb{R}$. There exists $C_{\Omega}>0$ such that for any $0<\lambda\leq 1$ and any $u\in B_{p,q}^{s}(I_{\lambda}(\Omega),E)$ with $u(0)=0$, we have

[TABLE]

Proof.

a) Assume that $1/p<s<1+1/p$. We first prove inequality 4.14 for $\lambda=1$. Theorem 3.3.5, p. 202 in [12] gives

[TABLE]

Note that

[TABLE]

for $\epsilon>0$ arbitrary small. With 4.15, it provides

[TABLE]

Next, arguing as in 4.4 and 4.5, we get

[TABLE]

Therefore, the case $\lambda=1$, follows from 4.16 and 4.17.

b) Assume that $1/p<s<1$. In the general case $0<\lambda\leq 1$, set $v=uoI_{\lambda}$. We have

[TABLE]

due to 4.13 since $s>0$ and $\lambda^{-1}\geq 1$. Next, the case $\lambda=1$ provides

[TABLE]

by inequality 4.13 since $0<\lambda\leq 1$ and $s-1<0$. Inequality 4.14 follows from 4.18 and 4.19. ∎

5 Proof of the theorem.

Before proceeding, we need a last uniform lemma.

Lemma 5.1**.**

Let $T>0$, $1\leq p,q<\infty$, $0<\alpha<\eta<1$ , $u_{0}\in E$ and $R>0$. Let also $\mathcal{H}_{T}:B_{T,\alpha}(u_{0},R)\rightarrow B_{p,q}^{-1/p^{\prime}+\eta}(]0,T[,E)$ satisfies properties $L1$ and $L2$. Then, there exists $\Lambda_{T}>0$ such that for any $0<t<T$ and $(u,v)\in\mathrm{B}_{t,\alpha}(u_{0},\mathrm{R}/\Lambda_{T})^{2}$ we have

[TABLE]

Proof.

a) We first define an equicontinuous family $(Q_{t})_{0<t<T}$ of extension operators. Let $\theta\in\mathscr{D}(\mathbb{R},\mathbb{R}_{+})$ with $\theta(s)=1$ for $|s|\leq 2T$. For any $0<t<T$ and $u\in B_{p,q}^{1/p+\alpha}(]0,t[,E)$ set

[TABLE]

$(\tau\in\mathbb{R})$. It follows from lemma 4.2 that $Q_{t}:B_{p,q}^{1/p+\alpha}(]0,t[,E)\rightarrow B_{p,q}^{1/p+\alpha}(\mathbb{R},E)$ is an extension operator. Moreover, due to the continuity of the bracket $<,>_{\mathbb{R}}$ and the equicontinuity of $(P_{0,t})_{0<t<T}$ (see lemma 4.1), we have

[TABLE]

(the last inequality holds with $\mathbb{R}$ in place of $]0,t[$, and follows on $]0,t[$ using by the definition of the norms). Hence, $(Q_{t})_{0<t<T}$ is equicontinuous. We denote by $\Lambda_{T}$ a bound of the norms of the $Q_{t}$.

b) Let now $u\in\mathrm{B}_{t,\alpha}(u_{0},\mathrm{R}/\Lambda_{T})$. Note that $u_{0}\mathbf{1}_{]0,T[}=Q_{t}(u_{0}\mathbf{1}_{]0,t[})|_{]0,T[}$. With a), it provides $\|Q_{t}(u)-u_{0}\|_{B_{p,q}^{\frac{1}{p}+\alpha}(]0,T[,E)}\leq\Lambda_{T}\|u-u_{0}\|_{B_{p,q}^{\frac{1}{p}+\alpha}(]0,t[,E)}$, hence

[TABLE]

Taking another $v\in\mathrm{B}_{t,\alpha}(u_{0},R/\Lambda_{T})$, we get

[TABLE]

∎

We now prove theorem 3.1.

Proof.

We use Picard fixed point theorem. Let $0<\rho<R/\Lambda_{T}$ and let $0<t_{0}<T$ to be precised. Define:

[TABLE]

where $u$ is given by equation 4.3 with $\mathcal{H}_{t_{0}}(\tilde{u})$ in place of $\phi$ and $s=1/p+\eta>1/p+\alpha$.

We prove that $\mathcal{S}_{t_{0}}(\bar{B}_{t_{0},\alpha}(u_{0},\rho))\subset\bar{B}_{t_{0},\alpha}(u_{0},\rho)$ for $t_{0}>0$ small enough.

Let $\tilde{u}\in\bar{B}_{t_{0},\alpha}(u_{0},\rho)$ and $u=\mathcal{S}_{t_{0}}(\tilde{u})$. Appealing to lemmas 4.2, 4.3 and theorem 4.1 for $0<\alpha<\eta$ we have:

[TABLE]

Due to lemma 5.1 and $\|\tilde{u}-u_{0}\|_{B_{p,q}^{1/p+\alpha}(]0,t_{0}[)}\leq\rho$, we get:

[TABLE]

for $t_{0}>0$ small enough. It proves the stability. The proof that $\mathcal{S}_{t_{0}}$ is a contraction is similar. ∎

6 Generalization.

Theorem 3.1 is not satisfactory for an operator $\mathcal{H}_{T}:U\subset B_{p,q}^{1/p+\alpha}(]0,T[,E)\rightarrow B_{p,q}^{-1/p^{\prime}+\eta}(]0,T[,E)$ defined on an arbitrary open set $U$. We have to identify

[TABLE]

The following proposition asserts that $I(U)=U(0)$ (set of initial values of elements of $U$) and provides a uniform estimates on the time $T_{u_{0}}$.

Proposition 6.1**.**

Let $T>0$, $1\leq p,q\leq\infty$, $0<\alpha<1$, and let $U$ be an open subset of $B_{p,q}^{1/p+\alpha}(]0,T[,E)$. Then $I(U)=U(0)$. Moreover, for any $u_{0}\in I(U)$ there exists $\gamma>0$, $R>0$ and $T_{0}>0$ such that for any $u_{1}\in E$ with $\|u_{1}-u_{0}\|_{E}\leq\gamma$, we have

[TABLE]

Proof.

The inclusion $I(U)\subset U(0)$ is clear. We prove the reverse inclusion -i.e that for any $u\mathbf{1}_{]0,T_{0}[}\in U$, $u(0)\in U|_{]0,T_{0}[}$ for some $0<T_{0}\leq T$- and 6.1 at the same time.

Let $u\in U$. For $R>0$ small enough, we have $\mathscr{B}_{\mathrm{T},\alpha}(u,R)\subset U$. Denote by $C_{\infty}$ a constant of continuity for the embeddings $B_{p,q}^{1/p+\alpha}(]0,T[,E)\hookrightarrow L^{\infty}(]0,T[,E)$ and $E\hookrightarrow B_{p,q}^{1/p+\alpha}(]0,T[,E)$, and set \epsilon=R/\big{[}4(2C_{\infty}+1)\big{]}. Pick up $\psi_{\epsilon}\in\mathscr{B}_{T,\alpha}(u,R)\cap C^{\infty}([0,T],E)$ with $\|u-\psi_{\epsilon}\|_{\mathrm{B}_{p,q}^{\frac{1}{p}+\alpha}(]0,T[,E)}\leq\epsilon$ and define $\phi_{\epsilon}:=\psi_{\epsilon}-\psi_{\epsilon}(0)+u(0)$. We have $\|u(0)-\psi_{\epsilon}(0)\|_{E}\leq\mathrm{C}_{\infty}\|u-\psi_{\epsilon}\|_{B_{p,q}^{1/p+\alpha}(]0,T[,E)}$. By definition of $\phi_{\epsilon}$ and $\psi_{\epsilon}$, this implies that $\|u-\phi_{\epsilon}\|_{B_{p,q}^{1/p+\alpha}(]0,T[,E)}\leq(C_{\infty}+1)\epsilon$.

Let now $u_{1}\in E$ with $\|u_{1}-u(0)\|_{E}\leq\epsilon$ , and let $\alpha<\delta<1$. Since $\phi_{\epsilon}(0)=u(0)$, appealing to theorem 4.1 b), we get, for any $0<t<T$

[TABLE]

Set T_{0}=inf\Big{\{}\Big{(}C_{T}^{-1}\|\phi^{\prime}_{\epsilon}\|_{B_{p,q}^{-1/p^{\prime}+\delta}(]0,T[)}^{-1}[R/2-(2C_{\infty}+1)\epsilon]\big{)}^{1/(\delta-\alpha)},T\Big{\}}. From inequality 6.2 and lemma 3.1 2) we get $u_{1}\in\mathscr{B}_{T_{0},\alpha}(u,R)=\mathscr{B}_{T,\alpha}(u,R)|_{]0,T_{0}[}\subset U|_{]0,T_{0}[}$, which proves the proposition. ∎

Due to corollary 3.1 , lemma 6.1 (and uniform estimates of the time of existence in the above proofs), we get corollary 6.1 below. We extend without comments the range of indexes, since the proof is easier for spaces $B_{p,q}^{s}$ of positive differential dimension $s-1/p\geq 0$. We also give a statement in the case of a continuous operator $\mathcal{H}_{T}$.

Corollary 6.1**.**

*Let $T>0$, $R>0$, $1\leq p,q\leq\infty$, $\eta>\alpha>0$, and let $U$ be an open subset of $B_{p,q}^{1/p+\alpha}(]0,T[,E))$. Let $u_{0}\in U(0)$.

a) Assume that $\mathcal{H}_{T}:U\rightarrow B_{p,q}^{-1/p^{\prime}+\eta}(]0,T[,E)$ satisfies condition $L_{1}$ and $L_{2}$. Then, there exists $0<\rho_{1}\leq\rho_{2}<R$ and $0<\mathscr{T}<T$ such that, for any $0<t_{0}\leq\mathscr{T}$ and any $u_{1}\in E$ with $\|u_{1}-u_{0}\|\leq\rho_{1}$, the problem: find $u\in\mathscr{B}_{t_{0},\alpha}(u_{0},\rho_{2})$ with:

[TABLE]

*admits exactly one solution. This solution belongs to $B_{p,q}^{1/p+\eta}(]0,t_{0}[,E)$.

b) Assume that $U=B_{p,q}^{1/p+\alpha}(]0,T[,E)$ in a). Then, the solution exists on the whole interval $]0,T[$.

c) Same assumptions as in a) except that $E$ is finite dimensional and $\mathcal{H}$ is not Lipschitzian but continuous. In the conclusions of a), uniqueness is lost.

Proof.

We only prove b). Appealing to standard arguments, it’s enough to get a priori bounds in $B_{p,q}^{1/p+\eta}$ for a local solution $u$ defined on an interval $[0,t_{0}[$. For $0<t<t_{0}$, arguing as in the proof of theorem 3.1, we get, for any $0<t<t_{0}$

[TABLE]

We do not prove that the constant $A_{\epsilon}$ is independent of $t$. This can be done by using the operators $Q_{t}$ (see proof of lemma 5.1). Hiding the term $C_{T}\epsilon\|u-u_{0}\|_{B_{p,q}^{1/p+\eta}(]0,t[)}$ in the left hand side of 6.4, and using $B_{p,q}^{1/p+\eta}(]0,t[)\hookrightarrow L^{\infty}(]0,t[)$ (equicontinuous family of embeddings) we get

[TABLE]

Set $z(t)=\|u\|_{L^{p}(]0,t[)}$. Inequality 6.5 implies $z^{\prime}(t)\leq C_{T}(z(t)+1)$. By Gronwall lemma, it follows that $z$ is bounded on $[0,t_{0}[$. Invoking one again 6.5, we get that $\|u\|_{B_{p,q}^{1/p+\eta}(]0,t[)}$ is bounded on $[0,t_{0}[$

∎

7 Examples.

Examples of operators $\mathcal{H}_{T}:B_{p,q}^{1/p+\alpha}(]0,T[,E)\rightarrow B_{p,q}^{-\frac{1}{p^{\prime}}+\eta}(]0,T[,E)$ endowed with properties $L1$ and $L2$ can be searched by means of Fourier series

[TABLE]

with $e_{j}(t)=exp(2ij\pi t/T)$ and where the $c_{j}:B_{p,q}^{1/p+\alpha}(]0,T[,E)\rightarrow\mathbb{C}$ are continuous functions. Nevertheless, property $L2$ is not easily characterized, and other decompositions must be looked for. See examples c) and d) below. We will need the following

Lemma 7.1**.**

Assume that $1\leq p,q\leq\infty$, $-1/p^{\prime}<\sigma\leq 1/p$, $1/p<s\leq 1/p+1$ with $\sigma+s>0$ Then, for any open interval $I$, and any $\psi\in B_{p,q}^{\sigma}(I,\mathscr{L}(F,E))$, $u\in B_{p,q}^{s}(I,F)$, we have

[TABLE]

Proof.

We only treat the case $I=\mathbb{R}$. For the remainder term, assume first that $p\geq 2$. Then

[TABLE]

Since $s+\sigma>0$. And for $1\leq p\leq 2$

[TABLE]

since $(s-1/p)+(1/p^{\prime}+\sigma)>0$. And for the paraproducts (see [4], p.103):

[TABLE]

[TABLE]

since $\sigma-1/p<0$ and $s-1/p>0$. ∎

In particular, the product is well defined and continuous in

[TABLE]

for $s>1/2$ and $\sigma>-1/2$.

We now proceed with the examples. In the sequel, we restrict our use of theorem 6.1 to the initial range of values $1\leq p,q\leq\infty$ and $0<\alpha<\beta<1$.

a) (Cauchy-Lipschitz, see [3], [9]) Let $\Omega$ be an open subset of $E$ and $f:\Omega\rightarrow E$ be a Lipschitz function. Define an operator

[TABLE]

by $\mathcal{H}_{T}(u)=f(u)$. Here, $U:=\{v\in B_{p,q}^{1/p+\alpha}(]0,T[,E)\textrm{ s.t }v([0,T])\subset\Omega\}$. Operator $\mathcal{H}_{T}$ satisfies properties $L1$ and $L2$. Hence, problem 3.5 is locally well posed in $B_{p,q}^{1/p+\alpha}$. This solution belongs to $W^{2,\infty}$ and is unique in this space.

b) Recall that for $0<\beta<1$, $B_{\infty,\infty}^{\beta}(]0,T[)$ is the space of bounded Holder functions with exponent $\beta$.

In this b), we consider an operator of the form $\mathcal{H}_{T}(u)=A(u)D^{\beta}u$, where $D^{\beta}$ is either the Riemann-Liouville either the Caputo fractional derivative, which we now define.

Let $T>0$, $0<\beta<1$, and let $u\in L^{\infty}(]0,T[,\mathbb{R})$. We set

[TABLE]

$0<t\leq T$. For $u\in W^{s,\infty}(]0,T[,\mathbb{R})$ ($1/2<s\leq 1$), define the Caputo derivative of order $\beta$ as

[TABLE]

Let $0<t<t+h<T$. For $u\in L^{\infty}(]0,T[,\mathbb{R})$, we have

[TABLE]

and similarly for $h<0$. It follows that

[TABLE]

continuously. Notice now that, for any $u\in W^{1,\infty}(]0,T[,\mathbb{R})$, we have $D^{\beta}_{c}u(t)=\int_{0}^{t}u^{\prime}(s)/(t-s)^{\beta}ds$. Hence, 7.5 provides

[TABLE]

continuously. Similarly, for $u\in W^{s,\infty}(]0,T[,\mathbb{R})$, $1/2<s<1$, using formulas 7.3 and 7.5, we obtain

[TABLE]

continuously. By standard embeddings and real interpolation formulas (see [12] p. 204) we get, for $\epsilon>0$ small enough

[TABLE]

continuously.

We now work with vector valued fuctions, for which the above notations and results can readily be extended. Until the end of this b), $n\in\mathbb{N}^{*}$ and A\in W^{1,\infty}\big{(}\mathbb{R}^{n},\mathbf{M}_{n\times n}(\mathbb{R})\big{)} are fixed. For $0<\beta_{j}<1$ $(1\leq j\leq n)$ and $u\in W^{s,\infty}(]0,T[,\mathbb{R}^{n})$, $1/2<s<1$, we write $D^{\beta}_{c}u=(D^{\beta_{j}}_{c}u_{j})_{1\leq j\leq n}$. In the sequel, $\beta^{*}=sup(\beta_{1},...,\beta_{n})$. Using 7.2, we can define, for $\epsilon>0$ small enough

[TABLE]

by $\mathcal{H}_{T}(u)=A(u)D^{\beta}_{c}u$. Operator $\mathcal{H}_{T}$ satifies properties $L1$ and $L2$. Hence, under condition $0<\beta_{j}<1$, problem 3.5 admits a unique solution in $B^{1-\epsilon}_{\infty,\infty}(]0,t_{0}[,\mathbb{R}^{n})$. This solution belongs to $B^{2-\beta^{*}}_{\infty,\infty}(]0,t_{0}[,\mathbb{R}^{n})$.

Similarly, the Riemann-Liouville derivative is given, for $u\in L^{\infty}(]0,T[,\mathbb{R})$, by

[TABLE]

Using 7.5 we get, for $1/2<s\leq 1$

[TABLE]

continuously. Coming back to the vector-valued case, let $0<\beta_{j}<1/2$ and let $\epsilon>0$ such that $1/2<1-\beta_{j}-\epsilon<1$ $(1\leq j\leq n)$. Using 7.7 and 7.2, we get that

[TABLE]

with $\mathcal{H}_{T}(u)=A(u)D_{r}^{\beta}u$ is well defined. Operator $\mathcal{H}_{T}$ satifies properties $L1$ and $L2$. Therefore, under condition $0<\beta_{j}<1/2$, problem 3.5 is locally well posed, with a solution $u\in B^{1-\beta^{*}}_{\infty,\infty}(]0,t_{0}[)$.

c) We assume here that $1<p<\infty$. In this c), we abusively write $\chi_{]s,T[}$ in place of $\chi_{]s,T[}|_{]0,T[}$.

Let \kappa\in C^{0}\big{(}[0,T],B_{p,q}^{-1/p^{\prime}+\eta}(]0,T[,\mathbb{R})\big{)}. Our goal is to give a meaning to the formula $\mathcal{H}_{T}(u)(t)=\int_{0}^{t}u(s)\kappa(s,t)ds$ for $u\in L^{\infty}(]0,T[,\mathbb{R})$. Let $\phi\in B_{p^{\prime},q^{\prime}}^{1/p^{\prime}-\eta}(]0,T[,\mathbb{R})$ and $s\in]0,T[$. We have

[TABLE]

using the continuity of the bracket and the fact that $\chi_{]s,T[}$ is a multiplier for $B_{p^{\prime},q^{\prime}}^{1/p^{\prime}-\eta}(]0,T[)$. Hence

[TABLE]

By theorem 11 in [10] and the fact that $B_{p^{\prime},q^{\prime},[0,T]}^{1/p^{\prime}-\eta}(\mathbb{R})=B_{p^{\prime},q^{\prime}}^{1/p^{\prime}-\eta}(]0,T[)$ within the range of indexes $-1/p<1/p^{\prime}-\eta<1/p^{\prime}$, we conclude that

[TABLE]

with $<\mathcal{H}_{T}(u),\phi>=\int_{0}^{T}u(s)<\kappa(s),\chi_{]s,T[}\phi>ds$ for any $\phi\in B_{p^{\prime},q^{\prime}}^{1/p^{\prime}-\eta}(]0,T[,\mathbb{R})$, is a well defined and continuous operator. Restricted to $B_{p,q}^{1/p+\alpha}(]0,T[,\mathbb{R})$, it satisfies properties $L1$ and $L2$. Due to theorem 6.1 b), we thus have the existence and uniqueness of a solution $u\in B_{p,q}^{1/p+\eta}(]0,T[,\mathbb{R})$ of problem $u^{\prime}(t)=\int_{0}^{t}u(s)\kappa(s,t)ds$ with $u(0)=u_{0}$.

This can be generalized to related operators, for instance $\int_{0}^{t}f(u(s))\kappa(s,t)ds$, in the scalar or Banach-valued case; or operators \int_{0}^{t}f\big{(}m(u)(s)\big{)}\kappa(s,t)ds with $m(u)(s)=sup_{0\leq\sigma<s}1/(s-\sigma)\int_{\sigma}^{s}|u(z)|dz$ etc…

d) Let $\mathcal{A}$ be a Banach algebra. Let $sup\{-1/p^{\prime},-1/2\}<\sigma<1/p$. Let $\psi_{j}\in B_{p,q}^{\sigma}(]0,T[,\mathcal{A})$ and let $f_{j}:\mathcal{A}\rightarrow\mathcal{A}$ be given Lipschitz functions $(j\in\mathbb{Z})$. Consider

[TABLE]

with

[TABLE]

(for results on composition operators, see [7]). Due to the hypothesis on $\sigma$, one can find $s$ with $1/p<s<\sigma+1<1/p+1$ and $s+\sigma>0$. For such $s$, lemma 7.1 applies and the product is continuous in $B_{p,q}^{s}(]0,T[,\mathcal{A})\times B_{p,q}^{\sigma}(]0,T[,\mathcal{A})\rightarrow B_{p,q}^{s}(]0,T[,\mathcal{A})$. Hence, $\mathcal{H}_{T}$ is well defined, continuous and satisfies properties L1 and L2 under the (sufficient) assumption that, for any $(u,v)\in B_{p,q}^{s}(]0,T[,\mathcal{A})^{2}$ with $u\neq v$, we have

[TABLE]

Definition 7.10 can be generalized for instance as

[TABLE]

with \kappa\in C^{0}\big{(}\mathbb{R}^{n},B^{\sigma}_{p,q}(]0,T[,\mathcal{A})\big{)}, \phi\in C^{0}\big{(}\mathbb{R}^{n}\times B^{s}_{p,q}(]0,T[,\mathcal{A}),B^{s}_{p,q}(]0,T[,\mathcal{A})\big{)}, $\mu$ a Radon measure on $\mathbb{R}^{\mathrm{n}}$ and with an extra condition analogous to 7.11.

$\bf{Acknowledgements}$. This work was supported by the LABEX MILYON (ANR-10-LABX-0070) of University de Lyon, within the program ”Investissement d’Avenir” (ANR-11-IDEX 0007) operated by the French National Research Agency (ANR).