Optimality of Cwikel-Solomyak estimates in Orlicz spaces

TL;DR

This paper investigates the optimal bounds of Cwikel-Solomyak estimates within Orlicz spaces, enhancing understanding of operator norm bounds and Sobolev embeddings.

Contribution

It establishes the optimality of Solomyak's results in Orlicz spaces using refined Sobolev embedding techniques.

Findings

Confirmed the optimality of Cwikel-Solomyak estimates in Orlicz spaces

Extended the understanding of Sobolev embeddings in the context of operator estimates

Provided a framework for analyzing operator bounds in generalized function spaces

Abstract

We discuss the optimality of Cwikel-Solomyak estimates for the uniform operator norm and establish optimality of M.Z Solomyak's results \cite{Solomyak1995} within the class of Orlicz spaces. Our methods are based on finding the optimal version of the Sobolev embedding theorem.