Differentiation of Integrals in R to the Nth Power

Title page ......Page 1
Date-line ......Page 2
Dedication ......Page 3
Preface ......Page 4
CONTENTS ......Page 8
CHAPTER I SOME COVERING THEOREMS ......Page 12
1. Covering theorems of the Besicovitch type ......Page 13
2. Covering theorems of the Whitney type ......Page 20
3. Covering theorems of the Vitali type ......Page 30
CHAPTER II THE HARDY-LITTLEWOOD MAXIMAL OPERATOR ......Page 46
1. Weak type (1,1) of the maximal operator ......Page 47
2. Differentiation bases and the maximal operator associated to them ......Page 53
3. The maximal operator associated to a product of differentiation bases ......Page 55
4. The rotation method in the study of the maximal operator ......Page 62
5. A converse inequality for the maximal operator ......Page 67
6. The space $L(1+\\log^+L)$. Integrability properties of the maximal operator ......Page 71
CHAPTER III THE MAXIMAL OPERATOR AND THE DIFFERENTIATION PROPERTIES OF A BASIS ......Page 76
1. Density bases. Theorems of Busemann-Feller ......Page 77
2. Individual differentiation properties ......Page 88
3. Differentiation properties for classes of functions ......Page 92
1. The interval basis $\\mathcal{B}_2$ does not satisfy the Vitali property ......Page 103
2. Saks\' rarity theorem. A problem of Zygmund ......Page 107
3. A theorem of Besicovitch on the possible values of the upper and lower derivatives ......Page 111
1. The Perron tree. The Kakeya problem ......Page 120
2. The basis $\\mathcal{B}_3$ is not a density basis ......Page 126
3. The Nikodym set. Some open problems ......Page 131
1. An example of Hayes. A density basis $\\mathcal{B}$ in $\\mathbb{R}^1$ and a function $g$ in each $L^p$, $1\\leq  p<\\infty, such that $\\mathcal{B}$ does not differentiate $\\int g$ ......Page 145
2. Bases of convex sets ......Page 148
3. Bases of unbounded sets and star-shaped sets ......Page 152
4. A problem ......Page 158
1. The theorem of de Possel ......Page 159
2. An individual covering theorem ......Page 164
3. A covering theorem for a class of functions ......Page 169
4. A problem related to the interval basis ......Page 176
5. An example of Hayes. A basis $\\mathcal{B}$ differentiating $L^q$ but no $L^{q_1}$ with $q_1 < q$ ......Page 177
CHAPTER VIII ON THE HALO PROBLEM ......Page 188
1. Some properties of the halo function ......Page 190
2. A result of Hayes ......Page 191
3. An application of the extrapolation method of Yano ......Page 194
4. Some remarks on the halo problem ......Page 198
APPENDIX I On the Vitali covering properties of a differentiation basis. by Antonio Cordoba ......Page 201
APPENDIX II A geometric proof of the strong maximal theorem. by Antonio Cordoba and Robert Fefferman ......Page 207
APPENDIX III Equivalence between the regularity property and the differentiation of $L^1$ for a homothecy invariant basis. by Roberto Moriyon ......Page 217
APPENDIX IV On the derivation properties of a class of bases. by Roberto Moriyon ......Page 222
BIBLIOGRAPHY ......Page 226
A LIST OF SUGGESTED PROBLEMS ......Page 234
INDEX ......Page 235