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# On Necessary and Sufficient Conditions for Lp-estimates of Riesz Transforms Associated to Elliptic Operators on Rn and Related Estimates

## Pascal Auscher

Part of the Memoirs of the American Mathematical Society Series

## Synopsis

On Necessary and Sufficient Conditions for Lp-estimates of Riesz Transforms Associated to Elliptic Operators on Rn and Related Estimates by Pascal Auscher

This memoir focuses on $Lp$ estimates for objects associated to elliptic operators in divergence form: its semigroup, the gradient of the semigroup, functional calculus, square functions and Riesz transforms. The author introduces four critical numbers associated to the semigroup and its gradient that completely rule the ranges of exponents for the $Lp$ estimates. It appears that the case $p2$ which is new. The author thus recovers in a unified and coherent way many $Lp$ estimates and gives further applications. The key tools from harmonic analysis are two criteria for $Lp$ boundedness, one for $p2$ but in ranges different from the usual intervals $(1,2)$ and $(2,\infty)$.

## Book Info

#### Publication date

15th February 2007

#### Author

Pascal Auscher

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#### Publisher

American Mathematical Society

Paperback
75 pages

#### Categories

Functional analysis & transforms
Differential calculus & equations

#### ISBN

9780821839416

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