In harmonic analysis, a branch of mathematics, the space of functions of bounded mean oscillation (BMO), introduced by John & Nirenberg (1961), plays the same role in the theory of Hardy spaces that the space L of bounded functions plays in the theory of Lp-spaces. A general reference for functions of bounded mean oscillation is (Stein 1993, chapter IV).The space VMO of functions of vanishing mean oscillation is the closure in BMO of the continuous functions that vanish at infinity. It can also be defined as the space of functions whose "mean oscillations" on balls Q are not only bounded, but also tend to zero uniformly as the radius of the ball Q tends to 0 or infinity. The space VMO is a sort of Hardy space analogue of the space of continuous functions vanishing at infinity, and in particular the Hardy space H1 is the dual of VMO. (Stein 1993, p. 180)
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.
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
Noman enjoys creating new things and experimenting as a way of taking incremental innovation head-on. He believes in the idea of learning by doing, trying, failing, and learning from experiences of failure. This book took the same approach, it was not a straight journey and there was a lot of learning on the way. Support from Professor Rhys Rowland Jones made this journey complete at the end of the day. This book investigates the relationship between Innovation and organizational culture in meeting the dynamic requirements of business membership organizations' (BMOs) members. The literature review compliments the findings, hence, Norman is able to develop an effective BMO framework of this relationship as presented in this book. Finally, Norman used the grounded theory approach due to his inclination towards exploring, learning, as well as creating a new organizational model out of his work on innovation and organizational culture. In a global village, where change is the new normal, none of us can expect BMOs to continue to enjoy a status quo while their members operate in a dynamic business environment. BMOs must learn how to adapt or they risk losing relevance.
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.
It has been 15 years since the first edition of Stochastic Integration and Differential Equations , A New Approach appeared, and in those years many other texts on the same subject have been published, often with connections to applications, especially mathematical finance. Yet in spite of the apparent simplicity of approach, none of these books has used the functional analytic method of presenting semimartingales and stochastic integration. Thus a 2nd edition seems worthwhile and timely, though it is no longer appropriate to call it "a new approach".The new edition has several significant changes, most prominently the addition of exercises for solution. These are intended to supplement the text, but lemmas needed in a proof are never relegated to the exercises. Many of the exercises have been tested by graduate students at Purdue and Cornell Universities. Chapter 3 has been completely redone, with a new, more intuitive and simultaneously elementary proof of the fundamental Doob-Meyer decomposition theorem, the more general version of the Girsanov theorem due to Lenglart, the Kazamaki-Novikov criteria for exponential local martingales to be martingales, and a modern treatment of compensators. Chapter 4 treats sigma martingales (important in finance theory) and gives a more comprehensive treatment of martingale representation, including both the Jacod-Yor theory and Emery's examples of martingales that actually have martingale representation (thus going beyond the standard cases of Brownian motion and the compensated Poisson process). New topics added include an introduction to the theory of the expansion of filtrations, a treatment of the Fefferman martingale inequality, and that the dual space of the martingale space H^1 can be identified with BMO martingales. Solutions to selected exercises are available at the web site of the author, with current URL http://www.orie.cornell.edu/~protter/books.html.
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