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

By P. Auscher. 2005

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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 $p 2$ 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 $p 2$ but in ranges different from the usual intervals $(1,2)$ and $(2,\infty)$.

Subfields: Applied Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/7bfe48eae4de8282abc960f444fd4e22ad428e4f

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In Applied Mathematics

In Numerical Analysis

Citations in the corpus

New Orlicz–Hardy spaces associated with divergence form elliptic operators (2009, background): “Moreover, we show that Hω,L(Rn) ⊂ Hω(Rn) for all pω ∈ ( nn+1 , 1] in Remark 7.4 below.”