Free interpolation in some spaces of smooth functions

Yu. V. Netrusov

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 21, Tome 206 (1993), pp. 107-118

Résumé

This paper is devoted to the study of the traces of functions from the classes $B^l_{p,\theta}$ or $F^l_{p,\theta}$ on a set $A\subset\mathbb R^n$. In the proofs the results of [1] are essentially used. We consider the folowing questions: 1) Under what conditions on a compact set $K$, $K\subset\mathbb R^n$, do the traces on $K$ of functions from $B^l_{p,\theta}\cap\mathbb C(\mathbb R^n)$ (or $F^l_{p,\theta}\cap\mathbb C(\mathbb R^n)$ fill in the space $C(K)$? 2) Under what conditions on a Borel set $A$, does the space of traces on $A$ of functions from $F^l_{p,\theta}$, $0$, coincide with some quasi-Banach lattice? 3) What is the description of the space of traces in this case? See Theorem 2.1 for an answer to 1) and Theorem 2.2 for answer to 2) and 3). In the last part pf the paper we prove counterparts of Theorems 2.1 and 2.2 for spaces of analytic functions. Bibliography: 14 titles.