Hilbert scales and Sobolev spaces defined by associated Legendre functions

TL;DR

This paper explores Hilbert scales generated by associated Legendre functions, linking them to weighted Sobolev spaces and analyzing their spectral properties for various parameters.

Contribution

It provides new characterizations of these Hilbert spaces as weighted Sobolev spaces and establishes identities among spaces with different regularity levels.

Findings

Characterization of Hilbert scales via associated Legendre functions

Identification of these spaces as weighted Sobolev spaces

Derivation of identities among spaces with varying regularity

Abstract

In this paper we study the Hilbert scales defined by the associated Legendre functions for arbitrary integer values of the parameter. This problem is equivalent to study the left-definite spectral theory associated to the modified Legendre equation. We give several characterizations of the spaces as weighted Sobolev spaces and prove identities among the spaces corresponding to lower regularity index.