Fock-Sobolev spaces of fractional order

TL;DR

This paper introduces and analyzes weighted Fock-Sobolev spaces of fractional order, showing their equivalence with weighted Fock spaces and providing explicit kernel formulas, boundedness results, duality, interpolation, and Carleson measure characterizations.

Contribution

It establishes the equivalence of two definitions of weighted Fock-Sobolev spaces and connects them to weighted Fock spaces, simplifying their study.

Findings

Weighted Fock-Sobolev spaces are equivalent to weighted Fock spaces with shifted parameters.

Explicit formulas for reproducing kernels are derived.

Boundedness of integral operators and dual space characterizations are established.

Abstract

For the full range of index $0<p\leq\infty$, real weight $\alpha$ and real Sobolev order $s$, two types of weighted Fock-Sobolev spaces over $\mathbb C^n$, $F^p_{\alpha, s}$ and $\widetilde F^p_{\alpha,s}$, are introduced through fractional differentiation and through fractional integration, respectively. We show that they are the same with equivalent norms and, furthermore, that they are identified with the weighted Fock space $F^p_{\alpha-sp,0}$ for the full range of parameters. So, the study on the weighted Fock-Sobolev spaces is reduced to that on the weighted Fock spaces. We describe explicitly the reproducing kernels for the weighted Fock spaces and then establish the boundedness of integral operators induced by the reproducing kernels. We also identify dual spaces, obtain complex interpolation result and characterize Carleson measures.