On \(L^ 1\)-criteria for quasi-radial Fourier multipliers with applications to some anisotropic function spaces
The purpose of this paper is to extend the quasi-convexity theorem and its variants [cf. \textit{W. Trebels}, Rev. Roum. Math. Pures Appl. 20, 1173-1185 (1975; Zbl 0328.42003)] from the radial to the quasi-radial case. By S we denote the Schwartzian space and by S' its dual, the space of tempered distributions. Define on S the Fourier transformation F[f], the inverse Fourier transformation \(F^{-1}\) and \([L^ p]{\hat{\;}}=\{g\in S':\quad F^{-1}[g]\in L^ p\}.\) Consider the dilation matrix \(A_ t=diag(t^{\alpha_ 1},...,t^{\alpha_ n})\), \(t>0\), \(\alpha \in(0,\infty)^ n\), \(| \alpha | =\psi\), and associate with \(A_ t\) a positive \(A_ t\)-homogeneous distance function \(\rho\) (\(\xi)\), i.e. a continuous function \(\rho\) on \({\mathbb{R}}^ n\) with \(\rho(\xi)>0\), \(\xi \neq 0\), \(\rho(A_ t\xi)=t\rho(\xi)\), \(t>0\), \(\xi \in {\mathbb{R}}^ n\). The paper shows the following: Theorem. let \(1\leq p<2\), 1\(\leq q\leq 2\) and \(1/p+1/q<2\); let g be measurable with distributional derivatives \(D^{\sigma}g\in L^ q(Q_ k)\) for all \(\sigma \in \{0,1\}^ n\), \(k\in {\mathbb{Z}}\), where \(Q_ k=\{\xi \in {\mathbb{R}}^ n:\quad 2^{k-1}\leq \rho(\xi)\leq 2^{k+1}\},\quad \rho \in C^{\infty}({\mathbb{R}}^ n_ 0),\) satisfying \(\sum_{k\in {\mathbb{Z}}}2^{k(\alpha \sigma -\nu(1/p+1/q- 1))}\| D^{\sigma}g\|_{L^ p(Q_ k)}\leq B<\infty\) for all \(\sigma \in \{0,1\}^ n\). Then \(g\in [L^ p]{\hat{\;}}\) and \(\| F^{-1}[g]\|_ p\leq cB\). They also discuss an inequality \(\| F^{-1}[m(\rho(\xi))]\|_ p\leq C\| m\|_{BV_{N+1}^{\nu /p'}},\) where \(\| m\|_{BV^{\omega}_{k+1}}=\int^{\infty}_{0}t^{k+\omega}| dm^{(k)}(t)|<\infty.\) These results in combination with already known ones allow a simple derivation of standard embedding properties for anisotropic Bessel, Sobolev and Besov spaces on \({\mathbb{R}}^ n\).
scientific article; zbMATH DE number 3865922