This book provides a comprehensive introduction to the systematic theory of tensor products and tensor norms within the framework of operator spaces. The use of tensor products has significantly advanced functional analysis and other areas of mathematics and physics, and the field of operator spaces is no exception. Building on the theory of tensor products in Banach spaces, this work adapts the definitions and results to the operator space context. This approach goes beyond a mere translation of existing results. It introduces new insights, techniques, and hypotheses to address the many challenges of the non-commutative setting, revealing several notable differences to the classical theory. This text is expected to be a valuable resource for researchers and advanced students in functional analysis, operator theory, and related fields, offering new perspectives for both the mathematics and physics communities. By presenting several open problems, it also serves as a potential source for further research, particularly for those working in operator spaces or operator algebras.
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In many branches of mathematical analysis and mathematical physics, the Hardy operator and Hardy inequality are fundamentally important and have been intensively studied ever since the pioneer researches. This volume presents new properties of higher-dimensional Hardy operators obtained by the authors and their collaborators over the last decade. Its prime focus is on higher-dimensional Hardy operators that are based on the spherical average form. The key motivation for this monograph is based on the fact that the Hardy operator is generally smaller than the Hardy–Littlewood maximal operator, which leads to, on the one hand, the operator norm of the Hardy operator itself being smaller than the latter. On the other hand, the former characterizing the weight function class or function spaces is greater than the latter. Sample Chapter(s) Preface Chapter 2: Hardy operators on other function spaces Contents: Sharp Bounds for Hardy Operators on Lebesgue Spaces Hardy Operators on Other Function Spaces Weighted Inequalities for Hardy Type Operators Commutators of Hardy Operators Hardy Operators on Heisenberg Groups and p-Adic Fields Sharp Constants for Hausdorff q-Inequalities Readership: This book is suitable for undergraduate and graduate students as well as researchers in theoretical and applied mathematics. It will be a useful teaching material at seminars as well as an invaluable reference source in all science libraries.
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