Weighted Calderon-Zygmund and Rellich inequalities in L^p

TL;DR

This paper establishes necessary and sufficient conditions for weighted Rellich and Calderon-Zygmund inequalities in L^p spaces, including in whole and half-spaces with boundary conditions, and computes optimal constants in certain cases.

Contribution

It provides a comprehensive characterization of when these inequalities hold for general operators in L^p spaces, extending previous results.

Findings

Derived necessary and sufficient conditions for inequalities

Computed best constants in specific cases

Extended results to operators with singular coefficients

Abstract

We find necessary and sufficient conditions for the validity of weighted Rellich and Calderon-Zygmund inequalities in L^p, 1 \leq p \leq \infty, in the whole space and in the half-space with Dirichlet boundary conditions. General operators like L=\Delta+c\frac{x}{|x|^2}\cdot\nabla-\frac{b}{|x|^2} are considered. We compute best constants in some situations.