On Necessary and Sufficient Conditions for $L^p$-Estimates of Riesz Transforms Associated to Elliptic Operators on $\mathbb {R}^n$ and Related EstimatesThis 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 $(2,\infty)$. |
Contents
Chapter 1 Beyond CalderonZygmund operators | 1 |
Chapter 2 Basic Lsup2 theory for elliptic operators | 9 |
Chapter 3 LsupP theory for the semigroup | 15 |
Chapter 4 Lsupp theory for square roots | 25 |
Chapter 5 Riesz transforms and functional calculi | 41 |
Chapter 6 Square function estimates | 51 |
Chapter 7 Miscellani | 65 |
Appendix A CalderonZygmund decomposition for Sobolev functions | 69 |
| 73 | |
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On Necessary and Sufficient Conditions for $L^p$-Estimates of Riesz ... Pascal Auscher No preview available - 2007 |
Common terms and phrases
AMS-LATEX argument Assume assumption ball Blunck & Kunstmann bounded holomorphic functional bounded on LP bounded operator Calderón-Zygmund decomposition Calderón-Zygmund theory Chapter coefficients complex interpolation Corollary 3.6 critical numbers cube Q defined denotes Department of Mathematics depending diagonal estimates dimension duality elliptic operators ellipticity constants extends family Vt follows GL(f Hence Hölder Hölder inequality holomorphic functional calculus hypercontractivity implicit constants Int 7(L integral number interval of exponents isomorphism Kato's conjecture kernel Lebesgue LEMMA Let f e LP bounded LP boundedness LP estimates LP norms LP off-diagonal estimates LP spaces LP theory maximal interval Minkowski inequality obtain off-diagonal estimates imply operator norm p_(L p+(L pl(L Proof of Step PROOF OF THEOREM properties Proposition 3.2 proved q_(L q+(L r_(L radius resp result reverse inequalities Riesz transform LP satisfies LP semigroup Sobolev space square root strong type Theorem 1.1 Vf|p weak type write www.ams yields