A simple characterization of weighted Sobolev spaces with bounded multiplication operator
The definition of weighted Sobolev spaces introduced by \textit{A. Kufner} and \textit{B. Opic} [Commentat. Math. Univ. Carol. 25, 537--554 (1984; Zbl 0557.46025)] is extended to a wider class of vectorial measures, with the property of making them a Banach space. In this context, the author obtains a characterization for the boundedness of the multiplication operator \(\mathcal{M}f(x)=xf(x)\), a condition which in turn implies that the zeros of Sobolev orthogonal polynomials are bounded by \(\|\mathcal{M}\|\). Following the steps of Kufner--Opic, it is introduced a class \(B_ p(A)\) of weight functions in the set \(A\) whose \(1/(1-p)\)-th powers are locally integrable. The main result of the paper adds the assumption that the weights are comparable to a piecewise monotone function. Several examples are given at the end of the article.
scientific article; zbMATH DE number 5306487
scientific article