دانلود کتاب Singular Integrals and Differentiability Properties of Functions.

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Title page......Page 1
Date-line......Page 2
Dedication......Page 3
Preface......Page 5
Notation......Page 7
Contents......Page 11
I. Some Fundamental Notions of Real-Variable Theory......Page 15
1. The maximal function......Page 16
2. Behavior near general points of measurable sets......Page 24
3. Decomposition in cubes of open sets in $\\mathbb{R}^n$......Page 28
4. An interpolation theorem for $L^p$......Page 32
5. Further results......Page 34
II. Singular Integrals......Page 38
1. Review of certain aspects of harmonic analysis in $\\mathbb{R}^n$......Page 39
2. Singular integrals: the heart of the matter......Page 40
3. Singular integrals: some extensions and variants of the preceding......Page 46
4. Singular integral operators which commute with dilations......Page 50
5. Vector-valued analogues......Page 57
6. Further results......Page 60
1. The Riesz transforms......Page 66
2. Poisson integrals and approximations to the identity......Page 72
3. Higher Riesz transforms and spherical harmonics......Page 80
4. Further results......Page 89
IV. The Littlewood-Paley Theory and Multipliers......Page 93
1. The Littlewood-Paley $g$-function......Page 94
2. The function $g_\\lambda^\\ast$......Page 98
3. Multipliers (first version)......Page 106
4. Application of the partial sums operators......Page 111
5. The dyadic decomposition......Page 115
6. The Marcinkiewicz multiplier theorem......Page 120
7. Further results......Page 124
V. Differentiability Properties in Terms of Function Spaces......Page 128
1. Riesz potentials......Page 129
2. The Sobolov spaces, $L_k^p(\\mathbb{R}^n)$......Page 133
3. Bessel potentials......Page 142
4. The spaces $\\Lambda_\\alpha$ of Lipschitz continuous functions......Page 153
5. The spaces $\\Lambda_\\alpha^{p,q}$......Page 162
6. Further results......Page 171
VI. Extensions and Restrictions......Page 178
1. Decomposition of open sets into cubes......Page 179
2. Extension theorems of Whitney type......Page 182
3. Extension theorem for a domain with minimally smooth boundary......Page 192
4. Further results......Page 204
1. Non-tangential convergence and Fatou\'s theorem......Page 208
2. The area integral......Page 217
3. Application of the theory of $H^p$ spaces......Page 229
4. Further results......Page 247
VIII. Differentiation of Functions......Page 252
1. Several notions of pointwise differentiability......Page 253
2. The splitting of functions......Page 258
3. A characterization of differentiability......Page 262
4. Desymmetrization principle......Page 269
5. Another characterization of differentiability......Page 274
6. Further results......Page 278
A. Some Inequalities......Page 283
B. The Marcinkiewicz Interpolation Theorem......Page 284
C. Some Elementary Properties of Harmonic Functions......Page 286
D. Inequalities for Rademacher Functions......Page 288
Bibliography......Page 291
Index......Page 301