Compact embeddings for fractional super and sub harmonic functions with radial symmetry

TL;DR

This paper establishes the compactness of Sobolev space embeddings for radially symmetric fractional super and sub harmonic functions, and proves the existence of maximizers for related interpolation inequalities.

Contribution

It introduces new compactness results for embeddings involving radially symmetric fractional harmonic functions and demonstrates the existence of maximizers for associated inequalities.

Findings

Proves compactness of Sobolev embeddings for radially symmetric fractional harmonic functions.

Establishes existence of maximizers for interpolation inequalities in this context.

Provides decay estimates for radially symmetric functions in the specified function space.

Abstract

We prove compactness of the embeddings in Sobolev spaces for fractional super and sub harmonic functions with radial symmetry. The main tool is a pointwise decay for radially symmetric functions belonging to a function space defined by finite homogeneous Sobolev norm together with finite $L^2$ norm of the Riesz potentials. As a byproduct we prove also existence of maximizers for the interpolation inequalities in Sobolev spaces for radially symmetric fractional super and sub harmonic functions.