Improved Lerey inequality and Trudinger-Moser type inequality involving the Leray potential

TL;DR

This paper enhances Leray's inequality with reminder terms, introduces new Hilbert spaces based on these inequalities, and establishes Trudinger-Moser type inequalities involving the Leray potential in the unit ball of R2.

Contribution

It provides improved versions of Leray's inequality, constructs associated Hilbert spaces, and derives new Trudinger-Moser inequalities involving the Leray potential.

Findings

Enhanced Leray inequalities with reminder terms

New Hilbert spaces with specific embedding properties

Trudinger-Moser inequalities involving Leray potential

Abstract

We obtain three types of results in this paper. Firstly we improve Leray's inequality by providing several types of reminder terms, secondly we introduce several Hilbert spaces based on these improved Leray inequalities and discuss their embedding properties, thirdly we obtain some Trudinger-Moser type inequalities in the unit ball of R2 associated with the norms of these Hilbert spaces where the Leray potential is used. Our approach is based on analysis of radially symmetric functions.