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

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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 $(2, \infty)$.Algebra to ALEXANDER KLESHCHEV, Department of Mathematics, University of Oregon, Eugene, OR 97403-1222; email: amsOnoether.uoregon.edu ... Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109- 1109; email: FRSSumich.edu Complex analysis and harmonic analysis ... University of Southern California, Los Angeles, CA 90089- 1113; email: guralnicSmath.usc.edu.

 Title : On Necessary and Sufficient Conditions for Lp-estimates of Riesz Transforms Associated to Elliptic Operators on Rn and Related Estimates Author : Pascal Auscher Publisher : American Mathematical Soc. - 2007

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