In this subsection we will consider a variant of the square function appearing at the righthand side of y where we replace the frequencyprojections j bybetterbehavedones. There exist, however, various unified approaches to their study, the littlewood paley theory, which arises naturally from the consideration of the dirichlet problem, provides one of the most successful unifying perspectives on these function spaces see 18 for. A geometric theory of littlewood paley theory sergiu klainerman and igor rodnianski abstract. Research article estimates for parameter littlewoodpaley. 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. Research article littlewoodpaley operators on morrey spaces with variable exponent. Download fulltext pdf on littlewoodpaley functions and singular integrals article pdf available in hokkaido mathematical journal 293 february 2000 with 49 reads. For example, in, sakamoto and yabuta in considered the parameter function. The littlewood paley theory is a powerful tool to investigate the boundedness property of. In harmonic analysis the lebesgue spaces l p, the hardy spaces h p, various forms of lipschitz spaces and the space bmo are important.
Littlewoodpaley theory on spaces of homogeneous type and. We shall here single out several features of the theory o f the hilbert. Littlewoodpaley theory refers to decomposing a function f as fx x k2z p 2kf x where pd 2kf. Function spaces and potential theory download ebook pdf. Similarly, the sobolev spaces l k p are basic in the study of partial differential equations. This can be used to relate the lp norm of a function to besov norms. This work deals with an extension of the classical littlewood paley theory in the context of symmetric diffusion semigroups. Littlewoodpaley theory and the study of func tion spaces. Sharp maximal function estimates for parameterized littlewood paley operators and area integrals lin, y. A basic example which lies at the source of the theory of singular integrals is given by the hilbert transform. Research article littlewoodpaley operators on morrey. From the original definitions of these spaces, it may not appear that they are closely. Boundedness of littlewoodpaley operators and their.
This work deals with an extension of the classical littlewoodpaley theory in the context of symmetric diffusion semigroups. The littlewood paley theory provides one of the most successful unifying perspectives on these and other function spaces. Littlewoodpaley theory on spaces of homogeneous type and the classical function spaces memoirs of the american mathematical society y. 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. Download limit exceeded you have exceeded your daily download allowance. This means that, to obtain the littlewoodpaley characterization of w. Applications of littlewoodpaley theory for morrey spaces to. We establish littlewoodpaley characterizations of the sobolev spaces w. Newest littlewoodpaleytheory questions mathematics. We will be providing unlimited waivers of publication charges for accepted articles related to covid19. The main purpose of this paper is to prove that these two equivalent relations 1. Our intention is to introduce the concept of the littlewood paley decomposition for periodic functions and derive related tools so that the periodic case and the whole space case can be treated in the same way.
In particular, we introduce matrixweighted homogeneous besov spaces b. There are, however, various unified approaches to their study. Littlewoodpaley theory and the study of function spaces, 1991. Many fields in analysis require the study of specific function spaces. Topics in harmonic analysis related to the littlewoodpaley. The study of transversality is based on the virtual fundamental chain techniques the theory of kuranishi structures and their multisections and chain level intersection theories. Next, we dene and study the littlewoodpaley gfunction associated with the riemannliouville operator for which we prove the l pboundedness for p. A detailed analysis comparing the orientations of the moduli spaces and their fiber products is carried out. Singular integrals and differentiability properties of.
Sis dense in hs for all s 2r, since sis dense in l2. The theory originally proceeded along three main lines, each interesting in its own right. Littlewood paley theory was developed to study function spaces in harmonic analysis and partial differential equations. Recently, it has contributed to the development of the \\varphi\transform and wavelet decompositions. Multiparameter hardy spaces and discrete littlewoodpaley theory 5 for every x. Sawyer in this work, han and sawyer extend littlewoodpaley theory, besov spaces, and triebellizorkin spaces to the general setting of a space of homogeneous type. Topics in harmonic analysis related to the littlewood. Research article littlewoodpaley operators on morrey spaces. We develop some aspects of littlewoodpaley function space theory in the matrix weight setting. Littlewood paley decompositions do not only play an important role in the theory of the classical besov and triebellizorkin spaces but also in the study of scales of function spaces associated with laplacebeltrami operators on manifolds, laplacetype operators on. Littlewoodpaley operators on morrey spaces with variable. Some recent works on multiparameter hardy space theory. Sawyer 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.
Littlewoodpaley theory and the study of function spaces by frazier, michael, 1956publication date 1991. From the original definitions of these spaces, it may not appear that they are closely related. Pdf on littlewoodpaley functions and singular integrals. Classical and multilinear harmonic analysis by camil muscalu. Jul 25, 2014 the extent to which littlewoodpaley theory characterizes function spaces is remarkable. As an application of this fact we show the boundedness property of and singular integral operators. Singular integral operators, littlewoodpaley theory, and wavelet approximation math 758s, spring 2000 mwf 9. Multiparameter hardy spaces and discrete littlewood paley theory 5 for every x. Paleylittlewood decomposition for sectorial operators and. First, we study the gauss and poisson semigroups connected with the riemannliouville operator. Fourier analysis and function spaces download ebook pdf. Fractional sobolev spaces on the circle with a littlewood. We develop a geometric invariant littlewood paley theory for arbitrary tensors of a compact 2 dimensional manifold.
Parametrized multilinear littlewood paley operators on hardy spaces he, sha and xue, qingying, taiwanese journal of mathematics, 2019. The goal of the present paper is to show that these function spaces fall under the scope of the littlewood paley theory. Chapter 2 function spaces many di erential equations of physics are relations involving linear di erential operators. We show that all the important features of the classical lp theory survive with estimates which. It is also well known that one can define fourier transform directly on the circle and get a fourier series as a result, that are somewhat easier to work with. R is the space of functions p k ka with each a a rectangle atom and p k jkj the present volume develops the theory of integration in banach spaces, martingales and umd spaces, and culminates in a treatment of the hilbert transform, littlewood paley theory and the vectorvalued mihlin multiplier theorem. Littlewoodpaley functions and sobolev spaces sciencedirect. Request pdf littlewoodpaley spaces we introduce the littlewoodpaley spaces in which the lebesgue spaces, the hardy spaces, the orlicz spaces, the lorentz. R is the space of functions p k ka with each a a rectangle atom and p k jkj space of smooth tensor. Download chapter pdf this orthogonality is easily understood on l 2, but at this point it is not clear how it manifests itself on other spaces.
Littlewood paley theory and the study of function spaces by frazier, michael, 1956publication date 1991. Weiss, littlewoodpaley theory and the study of function spaces, cbms. Littlewood paley theory on spaces of homogeneous type and the classical function spaces memoirs of the american mathematical society y. Littlewoodpaley theory for matrixweighted function spaces. The aim of this paper is to establish the vectorvalued inequalities for littlewoodpaley operators, including the lusin area integrals, the littlewoodpaley gfunctions and g. The new feature is that the elements of the vector spaces are functions, and the spaces are in. Under some primitive assumptions on the geometry of. Citeseerx scientific documents that cite the following paper. Littlewoodpaley theory and the study of function spaces. It is well known that using local charts one can define sobolev spaces on manifolds, and get the littlewoodpaley characterisation as well. The littlewood paley theory of onedimensional fourier series, and its applications, represents one of the most farreaching advances of that subject. In harmonic analysis, a field within mathematics, littlewoodpaley theory is a theoretical framework used to extend certain results about l 2 functions to l p functions for 1 pdf littlewoodpaley spaces we introduce the littlewoodpaley spaces in which the lebesgue spaces, the hardy spaces, the orlicz spaces, the lorentz. Calderonzygmund operator theory and function spaces. The littlewood paley decomposition and besov space techniques have become an indispensable tool for pdes on rd.
Littlewood paley theory and the study of function spaces cbms regional conference series in mathematics by bjorn jawerth, and guido weiss michael frazier 19910604 on. We establish littlewood paley characterizations of the sobolev spaces w. For questions about the littlewoodpaley theory, a theoretical framework used to extend certain results about l2 functions to lp functions for 1 of function spaces table of contents 2016 journal of function spaces publishes research on all aspects of function spaces, functional analysis, and their employment across other mathematical disciplines. With further research about littlewood paley operators, some authors turn their attentions to study the parameter littlewoodpaley operators. We provide a simple proof so that we are able to extend and improve many results published in recent papers. By applying the vectorvalued inequalities for the littlewoodpaley operators and their commutators on lebesgue spaces with variable exponent, the boundedness of the littlewoodpaley operators, including the lusin area integrals, the littlewoodpaley gfunctions and g. In this general setting there are applications to a variety of problems, such as those arising in the study of the expansions coming from. It is well known that using local charts one can define sobolev spaces on manifolds, and get the littlewood paley characterisation as well. The extent to which littlewoodpaley theory characterizes function spaces is remarkable. These operators, like matrices, are linear maps acting on vector spaces. Littlewood paley theory and the study of function spaces by frazier, michael, 1956. Intoduction to pde 1 littlewoodpaley decomposition we start with a smooth, nonincreasing, radial nonnegative function.
Littlewoodpaley theory on spaces of homogeneous type and the. Cbms regional conference series in mathematics, vol. May 28, 2011 in this final set of notes we will study the littlewood paley decomposition and the littlewood paley inequalities. For questions about the littlewoodpaley theory, a theoretical framework used to extend certain results about l2 functions to lp functions for 1 to ensure their research is fasttracked and made available on a preprint server as soon as possible.
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