Real Variable Methods in Fourier Analysis (eBook)

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1981 | 1. Auflage
391 Seiten
Elsevier Science (Verlag)
978-0-08-087157-8 (ISBN)

Infos

Real variable methods in Fourier analysis

Inhalt

Front Cover 1
Real Variable Methods in Fourier Analysis 4
Copyright Page 5
TABLE OF CONTENTS 12
DEDICATION 6
PREFACE 8
CHAPTER 1. POINTWISE CONVERGENCE OF A SEQUENCE OF OPERATORS 16
1.1. Finiteness a.e. and Continuity in measure of the maximal operator 23
1.2. Continuity in measure at o e X of the maximal operator and a.e. convergence 26
CHAPTER 2. FINITENESS A.E. AND THE TYPE OF THE MAXIMAL OPERATOR 28
2.1. A result of A. P. calderón on the partial sums of the fourier series of f e L2(T) 29
2.2. Commutativity of T* with mixing transformations. Positive operators. The theorem of Sawyer 34
2.3. Commutativity of T* with mixing transformations. The theorem of Stein 38
2.4. The theorem of Nikishin 44
CHAPTER 3. GENERAL TECHNIQUES FOR THE STUDY OF THE MAXIMAL OPERATOR 50
3.1. Reduction to a dense subspace 50
3.2. Covering and decomposition 54
3.3. Kolmogorov condition and the weak type of an operator 65
3.4. Interpolation 69
3.5. Extrapolation 75
3.6. Majorization 78
3.7. Linearization 81
3.8. Summation 83
CHAPTER 4. ESPECIAL TECHNIQUES FOR CONVOLUTION OPERATORS 88
4.1. The type (1,1) of maximal convolution operators 89
4.2. The type (p,p), p> 1, of maximal convolution operators
CHAPTER 5. ESPECIAL TECHNIQUES FOR THE TYPE (2,2) 106
5.1. Fourier transform 106
5.2. Cotlar's lemma 107
5.3. The method of rotation 111
CHAPTER 6. COVERINGS, THE HARDY-LITTLEWOOD MAXIMAL OPERATOR AND DIFFERENTIATION. SOME GENERAL THEOREMS 118
6.1. Some notation 119
6.2. Covering lemmas imply weak type properties of the maximal operator and differentiation 120
6.3. From the maximal operator to covering properties 129
6.4. Differentiation and the maximal operator 133
6.5. Differentiation properties imply covering properties 151
6.6. The halo problem 164
CHAPTER 7. THE BASIS OF INTERVALS 174
7.1. The interval basis B2 does not have the Vitali property. It does not differentiate L1 175
7.2. Differentiation properties of B2. Weak type inequality for a basis which is the Cartesian product of another two 175
7.3. The halo function of B2. Saks rarity theorem 180
7.4. A theorem of Besicovitch on the possible values of the upper and lower derivatives with respect to B2 186
7.5. A theorem of Marstrand and some generalizations 192
7.6. A problem of Zygmund solved by Moriyón 197
7.7. Covering properties of the basis of intervals . A theorem of Córdoba and R. Fefferman 199
7.8. Another problem of Zygmund. Solution by Córdoba 208
CHAPTER 8. THE BASIS OF RECTANGLES 214
8.1. The Perron tree 216
8.2. A lemma of Fefferman 222
8.3. The Kakeya problem 224
8.4. The Besicovitch set 225
8.5. The Nikodym set 230
8.6. Differentiation properties of some bases of rectangles 239
8.7. Some results concerning bases of rectangles in lacunary directions 248
CHAPTER 9. THE GEOMETRY OF LINEARLY MEASURABLE SETS 256
9.1. Linearly measurable sets 257
9.2. Density. Regular and irregular sets 260
9.3. Tangency properties 267
9.4. Projection properties 273
9.5. Sets of polar lines 283
9.6. Some applications 291
CHAPTER 10. APPROXIMATIONS OF THE IDENTITY 296
10.1. Radial kernels 297
10.2. Kernels non-increasing along rays 301
10.3. A theorem of F. Zo 307
10.4. Some necessary conditions on the kernel to define a good approximation of the identity 311
CHAPTER 11. SINGULAR INTEGRAL OPERATORS 320
11.1. The Hilbert transform 321
11.2. The Calderón-Zygmund operators 328
11.3. Singular integral operators with generalized homogeneity 342
CHAPTER 12. DIFFERENTIATION ALONG CURVES. A RESULT OF STEIN AND WAINGER 352
12.1. The strong type (2,2) for a homogeneous curve 353
12.2. The type (p,p) 1< p<
12.3. An application. Differentiation by rectangles determined by a field of directions 365
CHAPTER 13. MULTIPLIERS AND THE HARDY-LITTLEWOOD MAXIMAL OPERATOR 374
13.1. The characteristic function of the unit disk. A theorem of C. Fefferman 377
13.2. Polygons with in finitely many sides 383
13.3. The maximal operator with respect to a collection of rectangles. A theorem of A. Córdoba and R. Fefferman 386
REFERENCES 394
A LIST OF SUGGESTED PROBLEMS 404
INDEX 406

Produkt-Details

• Erscheint lt. Verlag: 1.1.1981
• Mitarbeit: Herausgeber (Serie): Miguel De Guzman
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Allgemeines / Lexika
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Naturwissenschaften
• Themenwelt: Technik
• ISBN-10: 0-08-087157-7 / 0080871577
• ISBN-13: 978-0-08-087157-8 / 9780080871578