Differentiation of integrals in Rn

M.D. Guzman9780387073996, 038707399X

Table of contents :
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 $L^p$, $1leq p