Leray-Trudinger Type Exponential Integrability in Log-Weighted Sobolev Spaces

TL;DR

This paper extends the analysis of exponential integrability in logarithmically weighted Sobolev spaces from radial to non-radial functions, establishing optimal inequalities and highlighting fundamental differences from previous radial-only results.

Contribution

It introduces a new framework connecting logarithmic weights and Leray energy, proving optimal exponential integrability for general functions and sharp inequalities for radial functions.

Findings

Established optimal exponential integrability for non-radial functions.

Proved sharp inequalities for radial functions in weighted Sobolev spaces.

Demonstrated fundamental differences from previous radial-only inequalities.

Abstract

In this article, we conduct a comprehensive study of weighted Sobolev spaces with logarithmic weights, orginially introduced by Calanchi and Ruf to analyze the sharp exponential integrability of radial functions belonging to these spaces. By exploring the connection between these logarithmically weighted energies and the Leray energy, we expand the framework to incorporate non-radial functions. More precisely, we establish optimal exponential integrability for general functions in the spirit of optimal Leray-Trudinger inequalities established by Di Blasio, Pisante and Psaradakis. Furthermore, we prove sharp versions of these inequalities when restricted to radial functions. Notably, the inequalities presented here are fundamentally different in nature from those of Calanchi and Ruf, for which the non-radial extension fails to hold.