Ebook: Bounded and Compact Integral Operators

The monograph presents some of the authors' recent and original results concerning boundedness and compactness problems in Banach function spaces both for classical operators and integral transforms defined, generally speaking, on nonhomogeneous spaces. Itfocuses onintegral operators naturally arising in boundary value problems for PDE, the spectral theory of differential operators, continuum and quantum mechanics, stochastic processes etc. The book may be considered as a systematic and detailed analysis of a large class of specific integral operators from the boundedness and compactness point of view. A characteristic feature of the monograph is that most of the statements proved here have the form of criteria. These criteria enable us, for example, togive var­ ious explicit examples of pairs of weighted Banach function spaces governing boundedness/compactness of a wide class of integral operators. The book has two main parts. The first part, consisting of Chapters 1-5, covers theinvestigation ofclassical operators: Hardy-type transforms, fractional integrals, potentials and maximal functions. Our main goal is to give a complete description of those Banach function spaces in which the above-mentioned operators act boundedly (com­ pactly). When a given operator is not bounded (compact), for example in some Lebesgue space, we look for weighted spaces where boundedness (compact­ ness) holds. We develop the ideas and the techniques for the derivation of appropriate conditions, in terms of weights, which are equivalent to bounded­ ness (compactness).

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The monograph presents some of the authors' recent and original results concerning boundedness and compactness problems in Banach function spaces both for classical operators and integral transforms defined, generally speaking, on nonhomogeneous spaces. It focuses on integral operators naturally arising in boundary value problems for PDE, the spectral theory of differential operators, continuum and quantum mechanics, stochastic processes, etc. The book may be considered as a systematic and detailed analysis of a large class of specific integral operators from the boundedness and compactness point of view. A characteristic feature of the monograph is that most of the statements proved here have the form of criteria. We provide a list of problems which were open at the time of completion of the book.
Audience: The book is aimed at a rather wide audience, ranging from researchers in functional and harmonic analysis to experts in applied mathematics and prospective students.

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The monograph presents some of the authors' recent and original results concerning boundedness and compactness problems in Banach function spaces both for classical operators and integral transforms defined, generally speaking, on nonhomogeneous spaces. It focuses on integral operators naturally arising in boundary value problems for PDE, the spectral theory of differential operators, continuum and quantum mechanics, stochastic processes, etc. The book may be considered as a systematic and detailed analysis of a large class of specific integral operators from the boundedness and compactness point of view. A characteristic feature of the monograph is that most of the statements proved here have the form of criteria. We provide a list of problems which were open at the time of completion of the book.
Audience: The book is aimed at a rather wide audience, ranging from researchers in functional and harmonic analysis to experts in applied mathematics and prospective students.
Content:
Front Matter....Pages i-xvi
Hardy-Type Operators....Pages 1-76
Fractional Integrals on the Line....Pages 77-250
One-Sided Maximal Functions....Pages 251-316
Ball Fractional Integrals....Pages 317-342
Potentials on R N ....Pages 343-366
Fractional Integrals on Measure Spaces....Pages 367-446
Singular Numbers....Pages 447-499
Singular Integrals....Pages 501-592
Multipliers of Fourier Transforms....Pages 593-615
Problems....Pages 617-621
Back Matter....Pages 622-643

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The monograph presents some of the authors' recent and original results concerning boundedness and compactness problems in Banach function spaces both for classical operators and integral transforms defined, generally speaking, on nonhomogeneous spaces. It focuses on integral operators naturally arising in boundary value problems for PDE, the spectral theory of differential operators, continuum and quantum mechanics, stochastic processes, etc. The book may be considered as a systematic and detailed analysis of a large class of specific integral operators from the boundedness and compactness point of view. A characteristic feature of the monograph is that most of the statements proved here have the form of criteria. We provide a list of problems which were open at the time of completion of the book.
Audience: The book is aimed at a rather wide audience, ranging from researchers in functional and harmonic analysis to experts in applied mathematics and prospective students.
Content:
Front Matter....Pages i-xvi
Hardy-Type Operators....Pages 1-76
Fractional Integrals on the Line....Pages 77-250
One-Sided Maximal Functions....Pages 251-316
Ball Fractional Integrals....Pages 317-342
Potentials on R N ....Pages 343-366
Fractional Integrals on Measure Spaces....Pages 367-446
Singular Numbers....Pages 447-499
Singular Integrals....Pages 501-592
Multipliers of Fourier Transforms....Pages 593-615
Problems....Pages 617-621
Back Matter....Pages 622-643
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