On necessary and sufficient conditions for $L^p$-estimates of Riesz transforms associated to elliptic operators on $\RR^n$ and related estimates

TL;DR

This paper establishes necessary and sufficient conditions for $L^p$ estimates of Riesz transforms related to elliptic operators, revealing distinct behaviors for $p<2$ and $p>2$ and unifying many existing results.

Contribution

It introduces four critical numbers that determine the $L^p$ boundedness ranges for elliptic operator-associated objects, providing a unified framework for $p<2$ and $p>2$ cases.

Findings

Identifies four critical numbers governing $L^p$ estimates.

Reveals the $p<2$ case is fundamentally different from $p>2$.

Provides criteria for $L^p$ boundedness in new exponent ranges.

Abstract

This article focuses on $L^p$ estimates for objects associated to elliptic operators in divergence form: its semigroup, the gradient of the semigroup, functional calculus, square functions and Riesz transforms. We introduce four critical numbers associated to the semigroup and its gradient that completely rule the ranges of exponents for the $L^p$ estimates. It appears that the case $p<2$ already treated earlier is radically different from the case $p>2$ which is new. We thus recover in a unified and coherent way many $L^p$ estimates and give further applications. The key tools from harmonic analysis are two criteria for $L^p$ boundedness, one for $p<2$ and the other for $p>2$ but in ranges different from the usual intervals $(1,2)$ and $(2,\infty)$.