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The book has grown out of readings of current papers during the past ten years. It is not a systematic treatise on the theory of partial differential equations. It contains short accounts (sections) of the elements of many sides of the theory not usually combined in a single volume. The main sides which are dealt with are existence, uniqueness, regularity of solutions to linear and nonlinear, stationary or evolution equations, qualitative properties of solutions (resonances, attractors, inertial manifolds), dynamical systems. Related Harnack, Moser-Trudinger, Sobolev-Poincaré, Strichartz inequalities, and function spaces such as BMO, BV, Morrey, Orlicz, Sobolev spaces, Kato classes of functions, are investigated. Sections include historical comments, definitions, main results without detailed proofs, examples and applications extracted from current works. This book has been written for young researchers in mathematics and applied sciences (dynamics, chemistry, biology). With sections presented in an alphabetical order, an author and subject indexes, it can be a useful tool for advanced students entering in the field. A solid undergraduate background in mathematics is required.

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Theory of Function Spaces II deals with the theory of function spaces of type Bspq and Fspq as it stands at the present. These two scales of spaces cover many well-known function spaces such as Hölder-Zygmund spaces, (fractional) Sobolev spaces, Besov spaces, inhomogeneous Hardy spaces, spaces of BMO-type and local approximation spaces which are closely connected with Morrey-Campanato spaces.Theory of Function Spaces II is self-contained, although it may be considered an update of the author's earlier book of the same title.The book's 7 chapters start with a historical survey of the subject, and then analyze the theory of function spaces in Rn and in domains, applications to (exotic) pseudo-differential operators, and function spaces on Riemannian manifolds.------ ReviewsThe first chapter deserves special attention. This chapter is both an outstanding historical survey of function spaces treated in the book and a remarkable survey of rather different techniques developed in the last 50 years. It is shown that all these apparently different methods are only different ways of characterizing the same classes of functions. The book can be best recommended to researchers and advanced students working on functional analysis.- Zentralblatt MATH

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Un nouveal espace fonctionnel adapte a l'etude des operateurs definis par des integrales singulieres.- Application of Carleson measures to partial differential equations and Fourier multiplier problems.- On the maximal function for the Mehler kernel.- Pointwise behavpour of solutions to Schr¿dinger equations.- An application of Lp estimates to scattering theory.- Elementary characterizations of the Morrey-Campanato spaces.- On nonisotropic Lipschitz spaces.- Lipschitz spaces on compact rank one symmetric spaces.- On the Sobolev spaces Wk,1(Rn).- Interval averages of H1-functions and BMO norm of inner functions.- An H1 function with non-summable Fourier expansion.- Integral characterization of a space generated by blocks.- Another characterization of Hp, 0

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Theory of Function Spaces II deals with the theory of function spaces of type Bspq and Fspq as it stands at the present. These two scales of spaces cover many well-known function spaces such as Hölder-Zygmund spaces, (fractional) Sobolev spaces, Besov spaces, inhomogeneous Hardy spaces, spaces of BMO-type and local approximation spaces which are closely connected with Morrey-Campanato spaces. Theory of Function Spaces II is self-contained, although it may be considered an update of the author&#8217;s earlier book of the same title. The book&#8217;s 7 chapters start with a historical survey of the subject, and then analyze the theory of function spaces in Rn and in domains, applications to (exotic) pseudo-differential operators, and function spaces on Riemannian manifolds. ------ Reviews The first chapter deserves special attention. This chapter is both an outstanding historical survey of function spaces treated in the book and a remarkable survey of rather different techniques developed in the last 50 years. It is shown that all these apparently different methods are only different ways of characterizing the same classes of functions. The book can be best recommended to researchers and advanced students working on functional analysis. - Zentralblatt MATH

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'Still waters run deep.' This proverb expresses exactly how a mathematician Akihito Uchiyama and his works were. He was not celebrated except in the field of harmonic analysis, and indeed he never wanted that. He suddenly passed away in summer of 1997 at the age of 48. However, nowadays his contributions to the fields of harmonic analysis and real analysis are permeating through various fields of analysis deep and wide. One could write several papers explaining his contributions and how they have been absorbed into these fields, developed, and used in further breakthroughs. Peter W. Jones (Professor of Yale University) says in his special contribution to this book that Uchiyama's decomposition of BMO functions is considered to be the Mount Everest of Hardy space theory. This book is based on the draft, which the author Akihito Uchiyama had completed by 1990. It deals with the theory of real Hardy spaces on the n-dimensional Euclidean space. Here the author explains scrupulously some of important results on Hardy spaces by real-variable methods, in particular, the atomic decomposition of elements in Hardy spaces and his constructive proof of the Fefferman-Stein decomposition of BMO functions into the sum of a bounded?function and Riesz transforms of bounded functions.

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The theory of interpolation spaces ¿ its origin, prospects for the future.- An interpolation theorem for modular spaces.- Some aspects of the minimal, M¿biusinvariant space of analytic functions on the unit disc.- A non-linear complex interpolation result.- A remark about Calder¿n's upper s method of interpolation.- The coincidence of real and complex interpolation methods for couples of weighted Banach lattices.- The K functional for (H1, BMO).- A relation between two interpolation methods.- Harmonic interpolation.- Higher order commutators of singular integral operators.- On interpolation between H1 and H?.- Interpolation theory and duality.- The K-functional for symmetric spaces.- Applications of interpolation with a function parameter to Lorentz, Sobolev and besov spaces.- On the smoothness of fourier transforms.- Rearrangements of BMO functions and interpolation.- Descriptions of some interpolation spaces in off-diagonal cases.- N.B. - Some of these problems were prepared already for the 1982 conference.

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This exposition of research on the martingale and analytic inequalities associated with Hardy spaces and functions of bounded mean oscillation (BMO) introduces the subject by concentrating on the connection between the probabilistic and analytic approaches. Short surveys of classical results on the maximal, square and Littlewood-Paley functions and the theory of Brownian motion introduce a detailed discussion of the Burkholder-Gundy-Silverstein characterization of HP in terms of maximal functions. The book examines the basis of the abstract martingale definitions of HP and BMO, makes generally available for the first time work of Gundy et al. on characterizations of BMO, and includes a probabilistic proof of the Fefferman-Stein Theorem on the duality of H11 and BMO.

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The purpose of this book is to present a modern account of mapping theory with emphasis on quasiconformal mapping and its generalizations. The modulus method was initiated by Arne Beurling and Lars Ahlfors to study conformal mappings, and later this method was extended and enhanced by several others. The techniques are geometric and they have turned out to be an indispensable tool in the study of quasiconformal and quasiregular mappings as well as their generalizations. The book is based on recent research papers and extends the modulus method beyond the classical applications of the modulus techniques presented in many monographs. TOC:Introduction and notation.- Moduli and capacity.- Moduli and domains.- Q-Homeomorphisms.- Q-Homeomorphisms with Q in BMO.- More General Q-Homeomorphisms.- Ring Q-Homeomorphisms.- Mappings With Finite Length Distortions (FLD).- Lower Q-Homeomorphisms.- Mappings With Finite Area Distortions (FAD).- On Ring Solutions of the Beltrami Equation.- Homeomorphisms with Finite Means Dilatations.- On the Mapping Theory in Metric Spaces.- Moduli Theory.- BMO Functions by John-Nirenberg.- References.- Index.

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Various applications of equimeasurable function rearrangements to the 'best constant'-type problems are considered in this volume. Several classical theorems are presented along with some very recent results. In particular, the text includes a product-space extension of the Rising Sun lemma, a product-space version of the John-Nirenberg inequality for bounded mean oscillation (BMO) functions with sharp exponent, a refinement of the Gurov-Reshetnyak lemma, sharp embedding theorems for Muckenhoupt, Gurov-Reshetnyak, reverse Hölder, and Gehring classes, etc. This volume is interesting for graduate students and mathematicians involved with these topics. TOC:Preface.- 1.Preliminaries and auxilliary results.- 2. Properties of oscillations and the definition of the BMO-class.- 3.Estimates of rearrangements and the John-Nirenberg theorem.- 4.The BMO-estimates of the Hardy-type transforms.- 5.The Gurov-Reshetnyak class of functions.- Appendix: A.The boundedness of the Hardy-Littlewood maximal operator from BMO into BLO.- B.The weighted analogs of the Riesz lemma and the Gurov-Reshetnyak theorem.- C.Classes of functions satisfying the reverse Hölder inequality.- References.- Index.

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