Nonmultipliers of the Sobolev spaces \(W^{k,1}({\mathbb{R}}^ n)\)
Let \(W^{k,1}({\mathbb{R}}^ n)\) be the usual Sobolev space of order k on \(L^ 1({\mathbb{R}}^ n)\) and let \(\overset\circ W^{k,1}({\mathbb{R}}^ n)\) denote the corresponding homogeneous Sobolev space. A multiplier is defined as a tempered distribution S such that convolution with S defines a continuous linear operator on \(W^{k,1}({\mathbb{R}}^ n).\) Theorem 1 asserts that if m is a continuous function in \({\mathbb{R}}^ n\) satisfying some limit condition, then, if m is a Fourier multiplier of \(W^{k,1}({\mathbb{R}}^ n)\), the corresponding limit is a constant function. This implies that some natural singular integral operators are not multiplies. This result follows from Theorem 2, which says that if \(\Omega\) is a homogeneous function of degree 0, which is continuous in \(R^ n-\{0\}\), then \(\Omega\) is a Fourier multiplier of \(W^{k,1}\) if and only if it is a constant. The main tool for the proof is Lemma 4, asserting that if m is a bounded function vanishing in a neighborhood of the origin, it is a Fourier multiplier of \(W^{k,1}({\mathbb{R}}^ n)\) if and only if it is a Fourier multipler of \(\overset\circ W^{k,1}({\mathbb{R}}^ n)\).
scientific article; zbMATH DE number 4019754