Calderon-Zygmund capacities and operators on nonhomogeneous spaces / Alexander Volberg.

Format:

Book

Author/Creator:

Volberg, Alexander, 1956-

Series:

Regional conference series in mathematics ; no. 100.

Regional conference series in mathematics

Language:

English

Subjects (All):

Calderón-Zygmund operator.

Genre:

Conference papers and proceedings.

Physical Description:

iv, 167 pages ; 25 cm.

Place of Publication:

Providence, RI : American Mathematical Society, 2003.

Contents:

Chapter 2. Preliminaries on Capacities 7

Chapter 3. Localization of Newton and Riesz Potentials 11

3.1. Localization lemmas 11

3.2. A building block for the construction of special measures 13

3.3. Localization on special cubes 13

3.4. Modification of distribution S. Construction of auxiliary measures 14

3.5. Ahlfors balls 15

3.6. The principal estimate for auxiliary measures 16

Chapter 4. From Distribution to Measure. Carleson Property 21

Chapter 5. Potential Neighborhood that has Properties (3.13)-(3.14) 25

5.1. Capacities with Calderon-Zygmund (CZ) kernels 26

5.2. Variational capacity and extremal measures 33

5.3. L[superscript p] theory of nonhomogeneous CZ operators. Measure of order m 42

5.4. Riesz and Cauchy kernels: [gamma]+ [characters not reproducible] [gamma subscript op] 45

5.5. Cauchy kernel and analytic capacity 47

Chapter 6. The Tree of the Proof 51

Chapter 7. The First Reduction to Nonhomogeneous Tb Theorem 55

Chapter 8. The Second Reduction 61

8.1. Suppressed kernels 61

8.2. From real-valued kernel to vector valued kernel 67

8.3. From one lattice to two lattices 68

8.4. Core suppression 69

Chapter 9. The Third Reduction 71

Chapter 10. The Fourth Reduction 73

10.1. [mu], b, D, [eta] decomposition 73

10.2. Good functions and bad functions 74

10.3. Estimates of nonhomogeneous Calderon-Zygmund operators on good functions 76

10.4. The reduction of Theorem 9.1 to estimates of nonhomogeneous Calderon-Zygmund operator, namely to Theorem 10.6 78

Chapter 11. The Proof of Nonhomogeneous Cotlar's Lemma. Arbitrary Measure 83

Chapter 12. Starting the Proof of Nonhomogeneous Nonaccretive Tb Theorem 93

12.1. Terminal and transit cubes 94

12.2. Projections [Lambda] and [Delta subscript Q] 96

Chapter 13. Next Step in Theorem 10.6. Good and Bad Functions 101

13.1. Good functions and bad functions again 101

13.2. Reduction to estimates on good functions 102

13.3. Splitting [left angle bracket]T[open phi subscript good], [psi subscript good right angle bracket] to three sums 103

13.4. Three types of estimates of [function of] k(x,y)f(x)g(y)d[mu](x)d[mu](y) 104

13.5. Estimate of long range interaction sum [sigma subscript 2] 106

13.6. Short range interaction sum [sigma subscript 3]. Nonhomogeneous paraproducts 109

Chapter 14. Estimate of the Diagonal Sum. Remainder in Theorem 3.3 121

14.1. Estimate of [Sigma subscript term] 123

14.2. Estimate of [Sigma subscript tr] 126

Chapter 15. Two Weight Estimate for the Hilbert Transform. Preliminaries 127

Chapter 16. Necessity in the Main Theorem 133

Chapter 17. Two Weight Hilbert Transform. Towards the Main Theorem 135

17.1. Bad and good parts of f and g 136

17.2. Estimates on good functions 137

Chapter 18. Long Range Interaction 139

Chapter 19. The Rest of the Long Range Interaction 143

Chapter 20. The Short Range Interaction 145

20.1. The estimate of neighbor-terms 145

20.2. The estimate of stopping terms 145

20.3. The choice of stopping intervals 147

Chapter 21. Difficult Terms and Several Paraproducts 153

21.1. First paraproduct 154

21.2. Two more paraproducts 156

21.3. Second paraproduct: miraculous improvement of the Carleson property 159

Chapter 22. Two-Weight Hilbert Transform and Maximal Operator 161

22.1. Doubling 161

22.2. No doubling 163.

Notes:

"CBMS Regional Research Conference, Nonhomogeneous Harmonic Analysis, Weights, and Applications to Problems in Complex Analysis and Operator Theory, University of North Carolina, May 13-17, 2002"--t.p. verso.

Includes bibliographical references.

ISBN:

0821832522

OCLC:

53276503