Sobolev interpolation inequalities with optimal Hardy-Rellich inequalities and critical exponents

TL;DR

This paper develops sharp higher order Sobolev interpolation inequalities for radial and non-radial functions, extending previous first-order results using advanced Fourier analysis and comparison principles.

Contribution

It introduces a new family of sharp Sobolev interpolation inequalities involving optimal Hardy-Rellich forms for higher order derivatives, expanding prior first-order work.

Findings

Established sharp higher order Sobolev inequalities for radial and non-radial functions.

Extended first-order results to higher order derivatives using Fourier analysis.

Developed a new critical Hardy-Sobolev inequality involving Lorentz norms.

Abstract

We establish a new family of the critical higher order Sobolev interpolation inequalities for radial functions as well as for non-radial functions. These Sobolev interpolation inequalities are sharp in the sense that they use the optimal quadratic forms of the sharp Hardy-Rellich inequalities and cover the Sobolev critical exponents. Our results extend those studied by Dietze and Nam in [15] for the first order derivative case to higher order setting. The well-known P\'{o}lya-Szeg\"{o} symmetrization principle and the nonlinear ground state representation play an important role in the work of [15]. To overcome the absence of the P\'{o}lya-Szeg\"{o} principle and the nonlinear ground state representation in the higher order case, our proofs rely on the Fourier analysis and a higher order verion of the Talenti comparison principle. We also study a new version of the critical Hardy-Sobolev interpolation inequality involving the critical quadratic form of the Hardy inequality and Lorentz norms. Our critical Hardy-Sobolev interpolation inequality complements the result of Dietze and Nam in [15].