Embeddings into Orlicz spaces via the modified Riesz potential

TL;DR

This paper investigates how the modified Riesz potential can be used to embed functions from L^1_p spaces into Orlicz spaces, with the embeddings depending on domain geometry and being optimal for p=1.

Contribution

It establishes point-wise estimates for the modified Riesz potential that lead to domain-dependent embeddings into Orlicz spaces, including optimal results for p=1.

Findings

Point-wise estimates imply embeddings into Orlicz spaces.

Embeddings depend on the geometry of the domain.

The Orlicz function is optimal when p=1.

Abstract

We study point-wise estimates for the modified Riesz potential. We show that the point-wise estimates imply embeddings into Orlicz spaces from the L^1_p-space where the functions are defined in non-smooth domains. The Orlicz functions depend on the geometry of the domain. We show that the Orlicz function is optimal when p=1.