Weighted inequalities of Hardy type

Cover......Page 1
Title......Page 2
ISBN 9789813140646......Page 3
Contents......Page 5
Preface......Page 8
Conventions and Notation......Page 13
Introduction......Page 17
1. Hardy\'s Inequality and Related Topics......Page 31
1.1. Weighted Lebesgue Spaces and the Hardy Operator......Page 31
1.2. Hardy\'s Inequality with Derivatives......Page 42
1.3. Some Notations and Modifications......Page 44
1.4. Hardy\'s Inequality for Some Special Classes of Functions......Page 47
1.5. The Role of the Interval......Page 50
1.6. Compactness of the Hardy Operator......Page 55
1.7. Some Limiting Inequalities - Preliminary Results......Page 62
1.8. Limiting Inequalities - General Results......Page 66
1.9. Miscellanea......Page 72
1.10. Comments and Remarks......Page 80
2. Some Weighted Norm Inequalities......Page 87
2.1. Preliminaries......Page 87
2.2. A Special Operator......Page 89
2.3. General Hardy-type Operators. The Fundamental Lemma......Page 98
2.4. General Hardy-type Operators. The Case p < q......Page 109
2.5. General Hardy-type Operators. The Case p > q......Page 118
2.6. Some Modifications and Extensions......Page 131
2.7. Comments and Remarks......Page 133
3. The Hardy-Steklov Operator......Page 137
3.1. Introduction......Page 137
3.2. Some Auxiliary Results......Page 145
3.3. The Case p < q......Page 147
3.4. The Case p > q......Page 153
3.5. Some Applications......Page 164
3.6. Some Generalizations and Extensions......Page 172
3.7. Comments and Remarks......Page 179
4. Higher Order Hardy Inequalities......Page 181
4.1. Preliminaries......Page 181
4.2. Some Special Cases I......Page 183
4.3. The General Case......Page 190
4.4. Some Special Cases II......Page 206
4.5. Reducing the Conditions......Page 211
4.6. Overdetermined Classes (k = 1)......Page 214
4.7. Overdetermined Classes (k > 1)......Page 222
4.8. Overdetermined Classes (Another Approach)......Page 236
4.9. Again the Interval (0,oo)......Page 252
4.10. Comments and Remarks......Page 257
5. Fractional Order Hardy Inequalities......Page 263
5.1. Introduction......Page 263
5.2. An Elementary Approach. The Unweighted Case......Page 266
5.3. The General Weighted Case......Page 271
5.4. Hardy-type Inequalities and Interpolation Theory......Page 291
5.5. Further Results......Page 300
5.6. Comments and Remarks......Page 310
6. Integral Operators on the Cone of Monotone Functions......Page 315
6.1. Introduction......Page 315
6.2. The Duality Principle of Sawyer......Page 318
6.3. Applications of the Duality Principle......Page 327
6.4. More General Integral Operators......Page 335
6.5. Comments and Remarks......Page 345
7. New and Complementary Results......Page 349
7.1. On the Prehistory and History of the Hardy Inequality......Page 349
7.2. A Convexity Approach to Prove Hardy Inequalities......Page 352
7.3. Scales of Conditions to Characterize Hardy-type Inequalities......Page 368
7.4. Hardy\'s Inequalities for all Parameters......Page 385
7.5. More on Hardy-type Inequalities for Hardy Operators with Kernel......Page 391
7.6. Hardy-type Inequalities in Other Function Spaces......Page 423
7.7. More on Multidimensional Hardy-type Inequalities......Page 445
References......Page 453
Index......Page 473