On necessary and sufficient conditions for Lp-estimates of Riesz transforms associated to elliptic operators on Rn and related estimates / Pascal Auscher.

Format:

Book

Author/Creator:

Auscher, Pascal, author.

Series:

Memoirs of the American Mathematical Society ; Volume 186, Number 871.

Memoirs of the American Mathematical Society, 0065-9266 ; Volume 186, Number 871

Language:

English

Subjects (All):

Singular integrals.

Littlewood-Paley theory.

Calderón-Zygmund operator.

Elliptic operators.

Semigroups.

Physical Description:

1 online resource (102 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 2007.

Language Note:

English

Summary:

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 $(2, \infty)$.

Contents:

""Contents""; ""Acknowledgements""; ""Introduction""; ""Notation""; ""Chapter 1. Beyond Caldero'n-Zygmund operators""; ""Chapter 2. Basic L[sup(2)] theory for elliptic operators""; ""2.1. Definition""; ""2.2. Holomorphic functional calculus on L[sup(2)]""; ""2.3. L[sup(2)] off-diagonal estimates""; ""2.4. Square root""; ""2.5. The conservation property""; ""Chapter 3. L[sup(P)] theory for the semigroup""; ""3.1. Hypercontr activity and uniform boundedness""; ""3.2. W[sup(1,p)] elliptic estimates and hypercontr activity""; ""3.3. Gradient estimates""; ""3.4. Summary""

""3.5. Sharpness issues""""3.6. Analytic extension""; ""Chapter 4. L[sup(p)] theory for square roots""; ""4.1. Riesz transforms on L[sup(p)]""; ""4.2. Reverse inequalities""; ""4.3. Invertibility""; ""4.4. Applications""; ""4.5. Riesz transforms and Hodge decomposition""; ""Chapter 5. Riesz transforms and functional calculi""; ""5.1. Blunck & Kunstmann's theorem""; ""5.2. Hardy-Littlewood-Sobolev estimates""; ""5.3. The Hardy-Littlewood-Sobolev-Kato diagram""; ""5.4. More on the Kato diagram""; ""Chapter 6. Square function estimates""

""6.1. Necessary and sufficient conditions for boundedness of vertical square functions""""6.2. On inequalities of Stein and Fefferman for non-tangential square functions""; ""Chapter 7. Miscellani""; ""7.1. Local theory""; ""7.2. Higher order operators and systems""; ""Appendix A. Calderon-Zygmund decomposition for Sobolev functions""; ""Appendix. Bibliography""

Notes:

"Volume 186, Number 871 (first of five numbers)."

Includes bibliographical references.

Description based on print version record.

ISBN:

1-4704-0475-3