Traces of multipliers in the space of Bessel potentials (Q909200)

The space of the multipliers on a Banach function space S is denoted by MS. Let \(H_ p^{\ell}({\mathbb{R}}^ n)\), \(\ell \geq 0\), \(1<p<\infty\), be the space of Bessel potentials on \({\mathbb{R}}^ n\), defined as the completion of \(C^{\infty}_ 0({\mathbb{R}}^ n)\) by the norm \(\| (1- \Delta)^{\ell /2}u;{\mathbb{R}}^ n\|_{L_ p}\). According to E. M. Stein, let us define the extension \(T\gamma\) (\(\xi\),y) on \({\mathbb{R}}^{n+m}\) of a function \(\gamma\) (x) on \({\mathbb{R}}^ n\) by \(T\gamma (x,y)=\int_{{\mathbb{R}}^ n}\xi (t)\gamma (x+| y| t)dt,\) where the kernel \(\zeta \in C^{\infty}({\mathbb{R}}^ n)\cap L({\mathbb{R}}^ n)\) satisfies (*) \(\int_{{\mathbb{R}}^ n}\zeta (x)dx=1\), \(\int_{{\mathbb{R}}^ n}x^{\alpha}\zeta (x)dx=0\), \(0<| \alpha | \leq [r]\) for some \(r>0\) and (**) \(K\equiv \int_{{\mathbb{R}}^ n}(1+| x|)\sum^{[r]}_{j=0}\sup_{\partial B_{| x|}}| \nabla_{j,x}\zeta | (1+| x|)^ jdx<\infty\), \(B_{| x|}=\{t\in {\mathbb{R}}^ n:\) \(| t| <| x| \}.\) In this paper it is proved that (i) if \(\Gamma \in MH^{\ell}_ p({\mathbb{R}}^{n+m})\) and \(\gamma (x)=\Gamma (x,0),\) it follows that \(\gamma \in MW_ p^{\ell - m/p}({\mathbb{R}}^ n)\) with \(\| \gamma;{\mathbb{R}}^ n\|_{MW_ p^{\ell -m/p}}\leq c\| \Gamma;{\mathbb{R}}^{n+m}\|_{MH^{\ell}_ p},\), where \(W_ p^{\ell -m/p}({\mathbb{R}}^ n)\), \(p\ell >m\), is the trace of \(H^{\ell}_ p({\mathbb{R}}^{n+m})\) on \({\mathbb{R}}^ n,\) (ii) if \(\zeta\) satisfies (*) and (**) with \(r=1\) and if \(\gamma \in MW_ p^{\ell -m/p}({\mathbb{R}}^ n)\) (resp. \(\gamma \in L_{\infty}({\mathbb{R}}^ n)\) and \(p\ell <m)\), then it follows that \(T\gamma \in MH^{\ell}_ p({\mathbb{R}}^{n+m})\) with \(\| T\gamma;{\mathbb{R}}^{n+m}\|_{MH^{\ell}_ p}\leq cK\| \gamma;{\mathbb{R}}^ n\|_{MW_ p^{\ell -m/p}}\) (resp. \(\leq cK\| \gamma;{\mathbb{R}}^ n\|_{L^{\infty}})\) (Theorem 1, 2). The proofs are considerably hard partly because some of the notations adopted there are not familiar to the present reviewer.