On necessary and sufficient conditions for \(L^p\)-estimates of Riesz transforms associated to elliptic operators on \(\mathbb{R}^n\) and related estimates.

*(English)*Zbl 1221.42022
Mem. Am. Math. Soc. 871, 75 p. (2007).

A good idea of the book is given by the author’s abstract and by the titles of chapters and sections which are reproduced below.

Author’s abstract: This memoir 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 \(p< 2\) already treated earlier is radically different from 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 \(p< 2\) and the other for \(p> 2\) but in ranges different from the usual intervals \((1, 2)\) and \((2,\infty)\).

Chapters and sections titles: Chapter 1. Beyond Calderón-Zygmund operators.

Chapter 2. Basic \(L^2\) theory for elliptic operators. Section 2.1 Definition. Section 2.2 Holomorphic functional calculus on \(L^2\). Section 2.3 \(L^2\) off-diagonal estimates. Section 2.4 Square root. Section 2.5 The conservation property.

Chapter 3. \(L^p\) theory for the semigroup. Section 3.1 Hypercontractivity and uniform boundedness. Section 3.2 \(W^{1,p}\) elliptic estimates and hypercontractivity. Section 3.3 Gradient estimates. Section 3.4 Summary. Section 3.5 Sharpness issues. Section 3.6 Analytic extension.

Chapter 4. \(L^p\) theory for square roots. Section 4.1 Riesz transforms on \(L^p\). Section 4.2 Reverse inequalities. Section 4.3 Invertibility. Section 4.4 Applications. Section 4.5 Riesz transforms and Hodge decomposition.

Chapter 5. Riesz transforms and functional calculi. Section 5.1, Blunek & Kunstmann’s theorem. Section 5.2 Hardy-Littlewood-Sobolev estimates. Section 5.3 The Hardy-Littlewood-Sobolev-Kato diagram. Section 5.4 More on the Kato diagram.

Chapter 6. Square function estimates. Section 6.1 Necessary and sufficient conditions for boundedness of vertical square functions. Section 6.2 On inequalities of Stein and Fefferman for non-tangential square functions.

Chapter 7. Miscellani. Section 7.1 Local theory. Section 7.2 Higher-order operators and systems.

Appendix A. Calderón-Zygmund decomposition for Sobolev functions.

Author’s abstract: This memoir 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 \(p< 2\) already treated earlier is radically different from 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 \(p< 2\) and the other for \(p> 2\) but in ranges different from the usual intervals \((1, 2)\) and \((2,\infty)\).

Chapters and sections titles: Chapter 1. Beyond Calderón-Zygmund operators.

Chapter 2. Basic \(L^2\) theory for elliptic operators. Section 2.1 Definition. Section 2.2 Holomorphic functional calculus on \(L^2\). Section 2.3 \(L^2\) off-diagonal estimates. Section 2.4 Square root. Section 2.5 The conservation property.

Chapter 3. \(L^p\) theory for the semigroup. Section 3.1 Hypercontractivity and uniform boundedness. Section 3.2 \(W^{1,p}\) elliptic estimates and hypercontractivity. Section 3.3 Gradient estimates. Section 3.4 Summary. Section 3.5 Sharpness issues. Section 3.6 Analytic extension.

Chapter 4. \(L^p\) theory for square roots. Section 4.1 Riesz transforms on \(L^p\). Section 4.2 Reverse inequalities. Section 4.3 Invertibility. Section 4.4 Applications. Section 4.5 Riesz transforms and Hodge decomposition.

Chapter 5. Riesz transforms and functional calculi. Section 5.1, Blunek & Kunstmann’s theorem. Section 5.2 Hardy-Littlewood-Sobolev estimates. Section 5.3 The Hardy-Littlewood-Sobolev-Kato diagram. Section 5.4 More on the Kato diagram.

Chapter 6. Square function estimates. Section 6.1 Necessary and sufficient conditions for boundedness of vertical square functions. Section 6.2 On inequalities of Stein and Fefferman for non-tangential square functions.

Chapter 7. Miscellani. Section 7.1 Local theory. Section 7.2 Higher-order operators and systems.

Appendix A. Calderón-Zygmund decomposition for Sobolev functions.

Reviewer: Michael Shapiro (Mexico City)

##### MSC:

42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |

42B25 | Maximal functions, Littlewood-Paley theory |

47F05 | General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX) |

47B44 | Linear accretive operators, dissipative operators, etc. |

35J15 | Second-order elliptic equations |

35J30 | Higher-order elliptic equations |