Potential and Sobolev Spaces Related to Symmetrized Jacobi Expansions

TL;DR

This paper explores potential and Sobolev spaces linked to symmetrized Jacobi expansions, establishing isomorphisms, characterizations, and embedding theorems for these function spaces.

Contribution

It introduces a symmetrization approach to Jacobi expansions and characterizes potential spaces as Sobolev spaces, including fractional square function and embedding results.

Findings

Potential spaces of integer orders are isomorphic to Sobolev spaces.

Fractional square function characterization of potential spaces.

Structural and embedding theorems for these spaces.

Abstract

We apply a symmetrization procedure to the setting of Jacobi expansions and study potential spaces in the resulting situation. We prove that the potential spaces of integer orders are isomorphic to suitably defined Sobolev spaces. Among further results, we obtain a fractional square function characterization, structural theorems and Sobolev type embedding theorems for these potential spaces.