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.
Preface; Hardy & Polya-Knopp Inequalities; Weighted Boundedness Criteria for Integral Transforms with Product Kernels; One-Sided Fractional Multiple Operators; Strong Fractional Maximal Functions & Multiple Riesz Potentials; Strong Maximal Functions & Hilbert Transforms with Product Kernels; Two-Weight Estimates for Fourier Operators & Bernstein Inequalities; Index.