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Weighted Norm Inequalities for Integral Transforms with Product Kernels
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Authors: Vakhtang Kokilashvili (A. Razmadze Mathematical Institute and International. Black Sea Univ., GA) Alexander Meskhi (A. Razmadze Mathematical Inst., Georgia ) Lars-Erik Persson (Lulea Univ. of Technology, Sweden) 
Book Description:
The book may be considered as a systematic and detailed analysis of a wide class of integral transforms with product kernels from the two-weighted boundedness point of view. The considered product kernels cover that case when factors of kernels have essential (less than one) singularities. The book intends to make a breakthrough in two directions: to cover multidimensional potentials, Hilbert transforms, strong maximal functions and at the same time, to present solutions of two-weighted problems for them which are much more complicated than one-weighted ones.
In the given monograph two-weighted boundedness criteria for multiple Hardy transforms is reflected in the case when the two-dimensional weight on the right-hand side of an appropriate inequality is a product of two weight functions of single variable. In this case we present simpler and transparent criteria than those of E.Sawyer for general weights. Moreover, we prove some new multidimensional Hardy-
type inequalities with general kernels. Weighted integral in-equalities for monotonic functions of several variables are also discussed. The main subjects of this book can be useful for applications both within various areas of the mathematical sciences (e.g. Fourier and Harmonic analysis, Fractional Calculus, BVP of PDE in Mathematical Physics, Stochastic Processes, Error Estimates in Numerical Analysis, etc.) as well as directly in some applied sciences.

Table of Contents:
Preface

Acknowledgement

Basic Notation

1. HARDY AND POLYA-KNOPP INEQUALITIES pp.1-70
1.1 A Two-dimensional Hardy-type Inequality
1.2 The Two-dimensional Plya-Knopp Type Inequality
1.3 The Multidimensional Case: 1 < p q <
1.4 The Multidimensional Case: 1 < q < p <
1.5 Multi-dimensional Plya-Knopp Type Inequalities
1.6 Double Riemann-Liouville Transform Without Singularity
1.7 Further Results
1.8 Notes and Comments on Chapter 1

2. WEIGHTED BOUNDEDNESS CRITERIA FOR INTEGRAL TRANSFORMS WITH PRODUCT KERNELS pp.71-128
2.1 Integrals with General Product Kernels
2.2 Truncated Potentials and Ball Fractional Integrals
2.3 The Case of m-Multiple Kernels
2.4 Multiple One-sided Potentials. Trace Inequality
2.5 Multidimensional Hardy-Type Inequalities with General Kernels
Via Convexity
2.6 Weighted Integral Inequalities for Monotonic Functions, the Case
p q
2.7 Weighted Integral Inequalities for Monotone Functions, the Case
0 < q < p <
2.8 Further Results and Applications
2.9 Notes and Comments on Chapter 2

3. ONE-SIDED FRACTIONAL MULTIPLE OPERATORS pp.129-176
3.1 One-dimensional Operators
3.2 One-sided Strong Fractional Maximal Functions
3.3 Mixed type Operators
3.4 One-sided Potentials with Product Kernels
3.5 One-weight Inequalities
3.6 Weighted Strichartz Estimates for Semilinear Wave Equations
3.7 Notes and Comments on Chapter 3

4. STRONG FRACTIONAL MAXIMAL FUNCTIONS AND MULTIPLE RIESZ POTENTIALS pp.177-214
4.1 Single Kernel Operators
4.2 Two-weight Problem for Strong Fractional Maximal Functions
4.3 Mixed Multiple Operators
4.4 Solution of the Trace Problem
4.5 Riesz Potentials with Product Kernels
4.6 Some Remarks
4.7 Notes and Comments on Chapter 4

5. STRONG MAXIMAL FUNCTIONS AND HILBERT TRANSFORMS WITH PRODUCT KERNELS pp.215-286
5.1 Single Maximal Functions
5.2 Strong Maximal Functions
5.3 Two-weight Estimates for Hilbert Transforms with Single Kernel
5.4 Hilbert Transforms with Product Kernels
5.5 Examples
5.6 Applications to the Fourier Multipliers
5.7 Notes and Comments on Chapter 5

6. TWO-WEIGHT ESTIMATES FOR FOURIER OPERATORS AND BERNSTEIN INEQUALITIES pp.287-318
6.1 Two-weight Inequalities for Cesaro and Abel-Poisson Means of
Fourier Series
6.2 On the Means of Fourier Integrals
6.3 Bernstein inequalities in the Two-weighted Setting
6.4 Notes and Comments on Chapter 6

Appendix pp.319-322

Open Problems pp.323-326

Bibliography pp.327-340

Index pp.341-342

   Series:
      Mathematics Research Developments
   Binding: Hardcover
   Pub. Date: 2009
   Pages: 342.pp
   ISBN: 978-1-60741-591-6
   Status: AV
  
Status Code Description
AN Announcing
FM Formatting
PP Page Proofs
FP Final Production
EP Editorial Production
PR At Prepress
AP At Press
AV Available
  
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Weighted Norm Inequalities for Integral Transforms with Product Kernels