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.
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
Many phenomena in engineering and mathematical physics can be modeled by means of boundary value problems for a certain elliptic differential operator in a given domain. When the differential operator under discussion is of second order a variety of tools are available for dealing with such problems, including boundary integral methods, variational methods, harmonic measure techniques, and methods based on classical harmonic analysis. When the differential operator is of higher-order (as is the case, e.g., with anisotropic plate bending when one deals with a fourth order operator) only a few options could be successfully implemented. In the 1970s Alberto Calderón, one of the founders of the modern theory of Singular Integral Operators, advocated the use of layer potentials for the treatment of higher-order elliptic boundary value problems. The present monograph represents the first systematic treatment based on this approach.This research monograph lays, for the first time, the mathematical foundation aimed at solving boundary value problems for higher-order elliptic operators in non-smooth domains using the layer potential method and addresses a comprehensive range of topics, dealing with elliptic boundary value problems in non-smooth domains including layer potentials, jump relations, non-tangential maximal function estimates, multi-traces and extensions, boundary value problems with data in Whitney-Lebesque spaces, Whitney-Besov spaces, Whitney-Sobolev- based Lebesgue spaces, Whitney-Triebel-Lizorkin spaces,Whitney-Sobolev-based Hardy spaces, Whitney-BMO and Whitney-VMO spaces.
This monograph is the first one to systematically present a series of local and global estimates and inequalities for differential forms, in particular the ones that satisfy the A-harmonic equations. The presentation focuses on the Hardy-Littlewood, Poincare, Cacciooli, imbedded and reverse Holder inequalities. Integral estimates for operators, such as homotopy operator, the Laplace-Beltrami operator, and the gradient operator are discussed next. Additionally, some related topics such as BMO inequalities, Lipschitz classes, Orlicz spaces and inequalities in Carnot groups are discussed in the concluding chapter. An abundance of bibliographical references and historical material supplement the text throughout.This rigorous presentation requires a familiarity with topics such as differential forms, topology and Sobolev space theory. It will serve as an invaluable reference for researchers, instructors and graduate students in analysis and partial differential equations and could be used as additional material for specific courses in these fields.
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
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
'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.
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.
On the integrability of Banach space valued Walsh polynomials.- Le principe de sous-suites dans les espaces de Banach.- Domains of attraction in Banach spaces.- Charges, poids et mesures de Levy dans les espaces vectoriels localement convexes.- Random fourier series on locally compact abelian groups.- Une solution simple au probleme de Skorokhod.- Demonstration elementaire d'un resultat d'Azema et Jeulin.- Sauts additifs et sauts multiplicatifs des semi-martingales.- Le support exact du temps local d'une martingale continue.- Martingales locales a accroissements independants.- Decomposition de martingales locales et rarefaction des sauts.- Un critere previsible pour l'uniforme integrabilite des semimartingales exponentielles.- Sur la convergence des martingales indexees par ? ¿?.- Arret de certaines suites multiples de variables aleatoires independantes.- Une remarque sur le calcul stochastique dependant d'un parametre.- Quasimartingales et formes lineaires associees.- Sur la p-variation d'une surmartingale continue.- Sur la p-variation des surmartingales.- Une remarque sur l'expose precedent.- Representations multiplicatives de sousmartingales.- Caracterisation d'une classe de semimartingales.- Sur les integrales stochastiques de L.C. Young.- Une topologie sur l'espace des semimartingales.- Equations differentielles lipschitziennes etude de la stabilite.- Fonction maximale et variation quadratique des martingales en presence d'un poids.- Weighted norm inequalities for martingales.- Inegalites de normes avec poids.- In¿lit¿e Hardy, semimartingales, et faux-amis.- Sur l'expression de la dualit¿ntre H1 et BMO.- Inegalites de convexite pour les processus croissants et les sousmartingales.- Theoreme de separation dans le probleme d'arret optimal.- Martingales et changements de temps.- Quelques epilogues.- En cherchant une d¿nition naturelle des int¿ales stochastiques optionnelles.- Les filtrations de certaines martingales du mouvement brownien dans ?n.- Demonstration simple d'un resultat sur le temps local.- Temps local et balayage des semi-martingales.- Sur le balayage des semi-martingales continues.- Semimartingales et valeur absolue.- Sur une formule de la theorie du balayage.- Construction de quasimartingales s'annulant sur un ensemble donne.- Conditional excursion theory.- Problemes a frontiere libre et arbres de mesures.- Un th¿¿ de J.W. Pitman.- Mesures de probabilite sur les entiers et ensembles Progressions.- On the uniqueness of optimal controls.- Processus de diffusion gouverne par la forme de dirichlet de l'operateur de Schr¿dinger.- Operateur de Schr¿dinger a resolvante compacte.- Grossissement d'une filtration et applications.- Encore une remarque sur la ? formule de balayage ?.- Presentation de l'?inegalite de doob ? de metivier-pellaumail.- Solution explicite de l'equation .- Caracterisation des semimartingales, d'apres dellacherie.- Un exemple de J. Pitman.- Le probleme de skorokhod : complements a l'expose precedent.- A propos de la formule d'Azema-Yor.- Martingales de valeur absolue donnee, d'apres Protter et Sharpe.- On the left end points of Brownian excursions.