G i gaudry this book is intended to be a detailed and carefully written account of various versions of the littlewoodpaley theorem and of some of its applications, together with indications of its general. We should emphasize that we have replaced the usual definition of homogeneous spaces which are quotient distribution spaces modulo polynomials by something better adapted to the study. It extends some of the benefits of orthogonality to situations where orthogonality doesnt really make sense. John edensor littlewood was a fellow for sixtynine years, lecturer in the college and professor in the university. The littlewoodpaley decomposition is of interest in multiple areas of. The book begins with some simple examples which provide an overview of the classical littlewoodpaley theory. The classical fourier series expansion is not convenient for this purpose, especially when we estimate the l p norm of a series for p6 2.
Marco cannone, in handbook of mathematical fluid dynamics, 2005. Moreover, it is shown that the t1 theorem for ndimensional calderonzygmund operators, without doubling assumptions, can be proved using the littlewoodpaley type decomposition that is obtained for functions in l 2. Im referring to the book of bahouri, chemin, danchin and during a. Littlewoodpaley decomposition of operator densities and. We develop a geometric invariant littlewood paley theory for arbitrary tensors of a compact 2 dimensional manifold.
We show that all the important features of the classical lp theory survive with estimates which. Littlewoodpaley decomposition of operator densities and application to a new proof of. These wavelets are compactly supported and oscillating functions. Topics in harmonic analysis related to the littlewood. The class will cover part of chapter 7 of book 2 and chapters 14 of book 3. Not only is it possible to build littlewoodpaley theory on a torus, but this is even often used. Maximal operators, littlewoodpaley theory, and wavelet. Fourier analysis and nonlinear partial differential equations 343 by hajer bahouri, raphael danchin and jeanyves chemin 2011, hardcover at the best online prices at ebay. Fourier coefficients and holder continuity mathoverflow. A class of bilinear multipliers given by littlewoodpaley. A proof of the doobmeyer decomposition theorem pdf file a generalization of the littlewood paley inequality and some other results related to stochastic partial differential equations, ulam quarterly, vol 2, no. Littlewoodpaley theory on spaces of homogeneous type and the.
In particular, those techniques based on the littlewoodpaley decomposition have proved to be very efficient for the study of evolution equations. Littlewoodpaley theory and the study of function spaces michael frazier bjorn jawerth guido weiss. Littlewoodpaley decomposition of operator densities and application to a new proof of the liebthirring inequality julien sabin to cite this version. The aim of the present monograph is to introduce methods from fourier analysis, and in particular techniques based on the littlewood paley decomposition, for the solution of nonlinear partial differential equations. Littlewoodpaley decomposition is a particular way of decomposing the phase space which takes a single function and writes it as a superposition of a countably infinite family of functions of varying frequencies. Littlewoodpaley and multiplier theory, bulletin of the american. Lecture notes on introduction to harmonic analysis. However, thanks to littlewoodpaley theory, which has much better lp stability properties than the fourier transform, one can get good control in terms of littlewoodpaley components of the function, as bazin points out below. This book is intended to be a detailed and carefully written account of various versions of the littlewoodpaley theorem and of some of its applications, together with indications of its general significance in fourier multiplier theory. Hardy littlewoodpaley inequalities and fourier multipliers on su2 by rauan akylzhanov london, erlan nurlustanov astana and michael ruzhansky london abstract. Littlewoodpaley theory, distorted fourier transform, zero energy. Littlewood paley theory on spaces of homogeneous type and the classical function spaces memoirs of the american mathematical society uk ed. Conference board of the mathematical sciences regional conference series in mathematics, 01607642.
Littlewoodpaley theory and the study of function spaces by michael frazier, bjorn jawerth, and guido weiss. Littlewoodpaley decompositions and besov spaces on lie. Fourier analysis and nonlinear partial differential. The result for 1 book aims at presenting selfcontained, state of the art models of those techniques with applications to different classes of partial differential equations. In harmonic analysis, a field within mathematics, littlewoodpaley theory is a theoretical. Littlewoodpaley decomposition from wolfram mathworld. Littlewoodpaley theory uses a decomposition of a function f into a sum of functions f. The wavelet decomposition allows us to obtain an even better localization for these functions, say. The littlewoodpaley decomposition allows us to describe an arbitrary tempered distribution into the sum of regular functions that are well localized in the frequency variable.
Wellknown decompositions in several contexts are as special examples. In the fields of functional and harmonic analysis, the littlewoodpaley decomposition is a particular way of decomposing the phase plane which takes a single function and writes it as a superposition of a countably infinite family of functions of varying frequencies. We have striven to make the presentation selfcontained and. Im referring to the book of bahouri, chemin, danchin and during a proof page 88. These consist of very basic tools in analysis which allow us to decompose a function, on the frequency side, to pieces that have almost disjoint frequency supports.
In this case, the besov spaces defined in terms of the littlewoodpaley decomposition coincide. In this work, han and sawyer extend littlewood paley theory, besov spaces, and triebellizorkin spaces to the general setting of a space of homogeneous type. Buy littlewoodpaley theory and the study of function spaces cbms regional conference series in mathematics on free shipping on qualified orders. Tempered distribution an overview sciencedirect topics. A unified littlewoodpaley decomposition of abstract besov spaces. The fourier transform and tempered distributions, interpolation of operators, the maximal function and calderonzygmund decomposition, singular integrals, riesz transforms and spherical harmonics, the littlewoodpaley gfunction and multipliers, sobolev spaces. In this chapter, littlewoodpaley theory is developed using the theory of vectorvalued singular integral operators.
Section 3 presents simple case studies in applications to di. We prove noncommutative versions of hardylittlewood and paley inequalities relating a function and its fourier coe cients on the group su2. Fourier analysis and nonlinear partial differential equations. John edensor littlewood frs 9 june 1885 6 september 1977 was an english mathematician. A geometric theory of littlewood paley theory sergiu klainerman and igor rodnianski abstract. Littlewood paley decomposition is a particular way of decomposing the phase space which takes a single function and writes it as a superposition of a countably infinite family of functions of varying frequencies. We should emphasize that we have replaced the usual definition. He worked on topics relating to analysis, number theory, and differential equations, and had a lengthy collaboration with g. In terms of its onedimensional fourier series, they define the dyactic decomposition of a function and, by employing the poisson integral, a certain nonlinear operator they called the gfunction. A proof of the doobmeyer decomposition theorem pdf file a generalization of the littlewoodpaley inequality and some other results related to stochastic partial differential equations, ulam quarterly, vol 2, no. Littlewoodpaley decomposition of operator densities and application to a new proof of the liebthirring inequality. Littlewoodpaley decomposition mathematics stack exchange.
A remark on littlewoodpaley theory for the distorted. Global wellposedness for the kellersegel system of. We introduce a littlewood paley decomposition related to any sublaplacian on a lie group g of polynomial volume growth. Weiss, g littlewoodpaley theory and the study of function spaces, vol. Fourier transform on l1, tempered distribution, fourier transform on l2, interpolation of operators, hardylittlewood maximal function, singular integrals, littlewoodpaley theory, fractional integration, singular multipliers, bessel functions, restriction to the sphere and uniform sobolev inequality.
Maximal operators, littlewoodpaley theory, and wavelet approximation lectures for fall 1999. Im working on a thesis concerning the littlewoodpaley decomposition of distributions and the use of paraproducts. Potential theory and a result of muckenhouptwheeden 91 10. Littlewoodpaley theory, lpmultipliers and function spaces. The topics will fall under the following three headlines. In this general setting there are applications to a variety of problems, such as those arising in the study of the expansions coming from second order elliptic operators. In this work, han and sawyer extend littlewoodpaley theory, besov spaces, and triebellizorkin spaces to the general setting of a space of homogeneous type.
In particular, those techniques based on the littlewoodpaley decomposition have proved to be very efficient for. Section 2 introduces the corresponding colombeautheoretic notion and discusses basic properties and illustrative examples. Various applications including multiplier theorem and strichartz estimates for the wave operator are then discussed. In particular, those techniques based on the littlewood paley decomposition have proved to be very efficient for the study of evolution equations. Rigorously, denoting 2k, c skp 2kfklp kjdj sp 2kfklp c skp 2kfklpfor all 1 p 1 a more technical estimate relating the lpnorm. Let ej v2i 1 be the usual littlewoodpaley decomposition of unity and. Littlewoodpaley theory on spaces of homogeneous type and the classical function spaces by. Hardylittlewoodpaley inequalities and fourier multipliers on su2. The paper used in this book is acidfree and falls within the guidelines. After a talk with my advisor i believe that i have a clear view of the matter. However, formatting rules can vary widely between applications and fields of interest or study.
Weighted littlewoodpaley theory and exponentialsquare. Littlewoods three principles 2 littlewoods third principle egorovs theorem is different from the other two, in that it holds in any measure space. The littlewoodpaley decomposition we start our analysis with forming a smooth littlewoodpaley decomposition as follows. Littlewoodpaley theory refers to decomposing a function f as fx x k2z p 2kf x where pd 2kf. There are several ways to construct such a decomposition. By means of the fourier frequency localization and bony paraproduct decomposition, we show a lo. Littlewoodpaley theory is an essential tool of fourier analysis, with applications and connections to pdes, signal processing, and probability. For this purpose, they establish a suitable analogue of the calderon reproducing formula and use it to extend classical results on atomic decomposition, interpolation, and t1 and tb theorems. Littlewoodpaley decompositions and besov spaces on lie group. In this final set of notes we will study the littlewoodpaley decomposition and the littlewoodpaley inequalities. Littlewoodpaley decomposition, introduction to pseudodifferential operators, symbols and schwartz kernels chap section 5, chap 7 section 12. These pieces, the littlewoodpaley pieces of the function, are almost orthogonal to.
Destination page number search scope search text search scope search text. In chapter 2 we give a detailed presentation on littlewood paley decomposition and define homogeneous and nonhomogeneous besov spaces. Besov spaces and littlewoodpaley decomposition 23 the littlewoodpaley. His skill in mathematics was remarkable and even in extreme old age he was busy in solving many extremely difficult problems. The littlewoodpaley decomposition is of interest in multiple areas of mathematics and forms the basis for the socalled littlewoodpaley theory.
A critical parabolic sobolev embedding via littlewood paley decomposition h. Littlewoodpaley theory and the t1 theorem with non. Littlewoodpaley decompositions and besov spaces related to. His father, a graduate of peterhouse, cambridge, took his family to south africa in 1892 when he became headmaster of a school at wynberg, near cape town. Littlewoodpaley theory on spaces of homogeneous type and. The \\varphi\transform, wavelet, and smooth atomic expansions are presented as natural extensions of the classical theory. Hardylittlewoodpaley inequalities and fourier multipliers. Suppose that f is a schwartz function whose fourier transform is supported away from. In this book, we have been studying the huge development of the rich theory of function spaces. A higher dimensional version of the classical littlewood paley identity is obtained on the ball. Pdf unified littlewoodpaley decomposition of abstract.
Let be a smooth real radial function supported on the closed ball of the frequency plane, which is identically equal to on. For this purpose, they establish a suitable analogue of the calder on reproducing formula and use it to extend classical results on atomic decomposition, interpolation, and t1 and tb. Intoduction to pde 1 littlewoodpaley decomposition we start with a smooth, nonincreasing, radial nonnegative function. This leads to a characterization of bmoa in terms of carleson measures as in the setting of the disc. This is a result historically due to littlewood and paley, which explains the name. On some weighted norm inequalities for littlewoodpaley. Littlewoodpaley decomposition, square function, manifolds with ends. Wavelet transform using appropriate spline wavelets in the littlewood paley decomposition scheme. The wavelet decomposition allows us to obtain an even better localization for these functions, say in both space and frequency. Littlewoods work with paley represents one of the most farreaching advances in fourier analysis. Global wellposedness for the kellersegel system of equations in critical spaces hantaek bae july 22, 2010 key words. A critical parabolic sobolev embedding via littlewoodpaley.