Applications of $p$-harmonic transplantation for functional inequalities involving a Finsler norm

TL;DR

This paper utilizes $p$-harmonic transplantation to establish various functional inequalities involving a general Finsler norm, providing precise extremal information and applicable to symmetric functions.

Contribution

It introduces the use of Ioku's transformation, a specific $p$-harmonic transplantation, to derive key inequalities with detailed extremal characterizations.

Findings

Established Sobolev, Poincaré, and logarithmic Sobolev inequalities involving Finsler norms.

Provided accurate extremal functions for these inequalities.

Demonstrated the applicability of $p$-harmonic transplantation to symmetric functions.

Abstract

In this paper, we prove several inequalities such as Sobolev, Poincar\'e, logarithmic Sobolev, which involve a general norm with accurate information of extremals, and are valid for some symmetric functions. We use Ioku's transformation, which is a special case of $p$-harmonic transplantation, between symmetric functions.