Localisation et multiplicateurs des espaces de Sobolev homogènes. (Localization and multipliers of homogeneous Sobolev spaces) (Q1102499)

Let \(\psi \geq 0\) be a radial \(C^{\infty}\) function in \({\mathbb{R}}^ n\) with supp \(\psi =\{x\in {\mathbb{R}}^ n;\leq | x| \leq 2\}\), such that \(\sum_{j\in {\mathbb{Z}}}\psi_ j(x)=1\), \(x\in R^ n_*=R^ n\setminus \{0\}\), \(\psi_ j(x)=\psi (2^{-j}x)\). The operators \(T_{\psi}:{\mathcal D}'({\mathbb{R}}^ n_*)\to {\mathcal D}'({\mathbb{R}}^ n_*)^{{\mathbb{Z}}}\) and \(U_{\psi}:{\mathcal D}'({\mathbb{R}}^ n_*)^{{\mathbb{Z}}}\to {\mathcal D}'({\mathbb{R}}^ n_*)\) are defined by \(T_{\psi}(u)=(\psi_ ju)_{j\in {\mathbb{Z}}}\), \(U_{\psi}((u_ j)_{j\in {\mathbb{Z}}})=\sum_{j\in {\mathbb{Z}}}\psi_ ju_ j.\) Let E be a Banach space of distributions in \({\mathbb{R}}^ n_*\) such that the injection \(E\to {\mathcal D}'({\mathbb{R}}^ n_*)\) is continuous. One also suppose that E is a \({\mathcal D}({\mathbb{R}}^ n_*)\)-module and that E is homogeneous, i.e. invariant under dilatations \(x\mapsto \lambda^{-1}x\), \(\lambda >0\). E is \(\ell^ p\)-localizable \((1<p<\infty)\) if \(T_{\psi}\) is bounded from E into \(\ell^ p(E)\) and \(U_{\psi}\) is bounded from \(\ell^ p(E)\) into E. Let \({\mathcal S}'\) be the space of tempered distributions in \({\mathbb{R}}^ n\) and \({\mathcal S}_ 0'\) the quotient space \({\mathcal S}'/\{polynomials\}\). The operators \(\Delta_ j\), \(j\in {\mathbb{Z}}\), are defined in \({\mathcal S}'\) by \((\Delta_ ju){\hat{\;}}=\psi_ j\hat u\). \(\Delta_ j\) is also defined in \({\mathcal S}_ 0'\). For \(s\in {\mathbb{R}}\), \(1<p<\infty\), the homogeneous Sobolev space \(\dot H^ s_ p\) is the subspace of \({\mathcal S}_ 0'\) defined by \((\sum_{j\in {\mathbb{Z}}}4^{sj}| \Delta_ ju|^ 2)^{1/2}\in L^ p\). For \(s- n/p \not\in {\mathbb{N}}\) and s-n/p'\(\not\in {\mathbb{N}}\), \(p'=p/(p-1)\), \(H^ s_ p\) can be realized as a subspace \(Z^ s_ p\) of \({\mathcal D}'({\mathbb{R}}^ n_*).\) One of the main results in this paper is that \(Z^ s_ p\) is \(\ell^ p\)-localizable. Another important result characterizes the multipliers of \(Z^ s_ p\).