The multiplication operator in Sobolev spaces with respect to measures (Q5933474)

Let \(1 \leq p < \infty\) and \((\mu_0,\ldots,\mu_k)\) be a vector of Borel measures on the real line. The Sobolev norm of a function \(f \in C^k({\mathbb R})\) is defined by NEWLINE\[NEWLINE \|f\|_p = \left( \sum_{j=0}^k \int |f^{(j)}(x)|^p d\mu_j(x)\right) ^{1/p}. NEWLINE\]NEWLINE The author only considers measures for which all polynomials have a bounded Sobolev norm. The multiplication operator \(M\), for which \((Mf)(x) = xf(x)\), is considered in the space \(P^{k,p}\) which is the completion of polynomials with the Sobolev norm on \(W^{k,p}\). Boundedness of the multiplication operator \(M\) plays an important role when one wants to investigate orthogonal polynomials with respect to the inner product induced by \(\|\cdot \|_2\): it implies for instance that the zeros of these Sobolev orthogonal polynomials are bounded. The present paper gives a characterization (Theorem 4.1) of those vector measures \((\mu_0,\ldots,\mu_k)\) for which the multiplication operator is bounded in \(P^{k,p}\), in terms of the notion \textit{extended sequentially dominated measures}, i.e., \(d\mu_j = f_j d\mu_{j-1}\), with \(f_j\) bounded functions and \(1 \leq j \leq k\). Furthermore, the author also gives a sufficient condition for boundedness of the multiplication operator (Theorem 4.3) and necessary conditions (Theorems 4.4 and 4.5). This is a very technical paper with many definitions and lemmas, and relies heavily on two (yet) unpublished manuscripts [\textit{J. M. Rodríguez}, ``Generalized weighted Sobolev spaces and applications to Sobolev orthogonal polynomials'', I, II].