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

Nguyen Anh Dao, Anh Xuan Do, Nguyen Lam, Guozhen Lu

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 Pólya-Szegö symmetrization principle and the nonlinear ground state representation play an important role in the work of [15]. To overcome the absence of the Pólya-Szegö 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].

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

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 Pólya-Szegö symmetrization principle and the nonlinear ground state representation play an important role in the work of [15]. To overcome the absence of the Pólya-Szegö 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].

Paper Structure

This paper contains 4 sections, 19 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Proofs of Theorem \ref{['T3.1']}, Theorem \ref{['T3.2']} and Theorem \ref{['T3']}
  3. Proof of Theorem \ref{['T1']}
  4. Proof of Theorem \ref{['T2']}

Key Result

Theorem A

Let $m$, $k$ be integers with $0\leq k<m$ and $s\geq0$ such that $k+s>0$. Let $u\in\mathcal{S}^{\prime}(\mathbb{R}^{N})$ be such that $\nabla^{m}u\in L^{p}(\mathbb{R}^{N})$, $1\leq p<\infty$ and $u\in\dot{B}^{-s}(\mathbb{R}^{N})$. Then, we have $\nabla^{k}u\in L^{r}(\mathbb{R}^{N})$, $r=p\left( \df where we denote the Besov space $\dot{B}^{-s}=\dot{B}_{\infty,\infty}^{-s}$.

Theorems & Definitions (28)

  • Theorem A
  • Proposition 1.1
  • Theorem B
  • Theorem C
  • Theorem 1.1
  • Proposition A: Higher order Caffarelli-Kohn-Nirenberg inequality
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.1
  • ...and 18 more