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

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.