Weighted norm inequalities, off-diagonal estimates and elliptic operators

TL;DR

This paper reviews a generalized Calderón-Zygmund framework for non-integral singular operators, emphasizing off-diagonal estimates and their applications to weighted inequalities and elliptic operators.

Contribution

It introduces a broad theory for operators lacking kernel bounds but satisfying off-diagonal estimates, extending weighted and $L^p$ estimates to new classes of elliptic operators.

Findings

Off-diagonal estimates are crucial for weighted inequalities.

Semigroup methods apply to elliptic operators with new $L^p$ ranges.

The theory encompasses operators without traditional kernel bounds.

Abstract

We give an overview of the generalized Calder\'on-Zygmund theory for "non-integral" singular operators, that is, operators without kernels bounds but appropriate off-diagonal estimates. This theory is powerful enough to obtain weighted estimates for such operators and their commutators with $\BMO$ functions. $L^p-L^q$ off-diagonal estimates when $p\le q$ play an important role and we present them. They are particularly well suited to the semigroups generated by second order elliptic operators and the range of exponents $(p,q)$ rules the $L^p$ theory for many operators constructed from the semigroup and its gradient. Such applications are summarized.