Characterization of Higher-Order Sobolev Spaces on the Sphere via Generalized Averaging of Function

TL;DR

This paper introduces a refined characterization of higher-order Sobolev spaces on the sphere using a generalized averaging approach, broadening the class of weight functions and localizing the averaging domain.

Contribution

It extends previous methods by refining the square function, allowing for more general weight functions and localized averaging on the sphere.

Findings

A new square function characterization for Sobolev spaces on the sphere.

Broader class of weight functions suitable for averaging.

Localization of averaging domain to a coordinate patch.

Abstract

We present a new characterization of higher-order Sobolev spaces on the sphere. Building on the approach of Barcel\'o et al. (2020), we refine the square function they introduced for this purpose. In particular, we provide a detailed analysis of the weight function and the averaging range in its definition. Our results demonstrate that averaging can be performed using a weight function from a broad class, and that the averaging domain can be restricted to a local coordinate patch on the sphere.