Hedberg, function spaces and potential theory, fundamental principles of mathematical sciences 314. Multiple solutions for a nonlinear and nonhomogeneous problem in. An approximation of hedbergs type in sobolev spaces with. Introduction we introduce the variable exponent morrey spaces lp.
Two approximation problems in function spaces 55 the capacitary potential satisfies 1. Later, potential theory was strongly influenced by functional analysis. Show full abstract on a regular perturbation expansion in terms of small parameters to the thirdorder and greens function integralequationmethod. Function spaces, especially those spaces that have become known as. Xiao, nonlinear potential analysis on morrey spaces and their capacities, indiana univ. Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle coronavirus. Potential theory, harmonic functions, stochastic process. Pdf developments and perspectives in nonlinear potential theory. Rn, well known in the case where p and are constant, see. Hedberg function spaces and potential theory the subject of this book is the interplay between function space theory and potential theory.
As an application, we give a new proof of a theorem of w. Applications of potential theory to weighted sobolev spaces include quasi continuity of sobolev functions, poincare inequalities, and spectral synthesis theorems. Maximal function, fractional maximal operator, riesz potential, morrey space, variable exponent, hardylittlewoodsobolev type estimate, bmo space. As an application, we extend the known relations between sriesz capacities, 0 adams lars i. Thinness in nonlinear potential theory for nonisotropic sobolev spaces 315 1. Numerous and frequentlyupdated resource results are available from this search. Lars inge hedberg the subject of this book is the interplay between function space theory and potential theory. A crucial step in classical potential theory is the identification of the potential energy of a charge. Recently, it has contributed to the development of the \\varphi\transform and wavelet decompositions. Littlewoodpaley theory was developed to study function spaces in harmonic analysis and partial differential equations. Carleson embeddings for sobolev spaces via heat equation. Bilinear forms on nonhomogeneous sobolev spaces in.
Nonlinear potential theory and weighted sobolev spaces. Littlewoodpaley theory and the study of function spaces. Hedberg function spaces and potential theory series. Carleson measures for weighted hardysobolev spaces. An extension of hedberg s convolution inequality and applications. Lqextensions of lpspaces by fractional diffusion equations. Boundedness of the maximal operator in the local morrey. Through a geometric capacitary analysis based on space dualities, this paper addresses several fundamental aspects of functional analysis and potential theory for the morrey spaces in harmonic analysis over the euclidean spaces. In particular, they play a decisive role in the modem theory of partial differential equations pde. Thinness in nonlinear potential theory for nonisotropic sobolev. Abstractwe study the mapping properties of fractional maximal operators in sobolev and campanato spaces in metric measure spaces. However, formatting rules can vary widely between applications and fields of interest or study. We obtain sharp constants for sobolev inequalities for higher order fractional derivatives.
Our aim in this paper is to deal with sobolev embeddings for riesz potentials of functions in morrey spaces of variable exponent. Regularity for the planar optimal pcompliance problem. On mazyas work in potential theory and the theory of. Adams, sobolev spaces academic press, new york, 1975. Also departing is michel jabbour who left kentucky for the ecole polytechnique in paris after spending. Moreover, the proof does not make use of the subadditivity of the capacity. The potential theory has been studied very much, especially after the 1 researches of gauss in 1840, where he studied important problems and methods which gave yet remained partly as basic ideas of modern researchs in this.
Fractional maximal functions in metric measure spaces. Hedberg, function spaces and potential theory, springerverlag, berlin, 1996. Sobolevs inequality for riesz potentials of functions in. Adams and extended to many functions spaces by adams, xiao and liu. In this section we list briefly some definitions and well known facts about sobolev spaces with variable exponent and bessel potential spaces with variable exponent. Let be an open subset of, by the symbol, we denote the family of all measurable functions. An extension of hedbergs convolution inequality and applications. The spaces appear naturally as interpolation spaces that are obtained from sobolev spaces by means of the complex interpolation method cf. For extensive treatises and notes on the history, see the monographs adams hedberg 1, heinonenkilpel. Function spaces and potential theory, springerverlag, berlin 1996. Estimates for bessel and riesz potentials springerlink. Acrobat reader function spaces and potential theory carefully and thoughtfully written and prepared with, in my opinion, just the right amount of detail included. More re cently, ideas from potential theory have enriched the theory of those more general function spaces that appear naturally in the study of nonlinear partial differential equations.
Click download or read online button to get function spaces and potential theory book now. Moreover, kellogg lemmas are established for various concepts of thinness. A fractional dirichlettoneumann operator on bounded. Nonlinear potentials, local solutions to elliptic equations. Ams proceedings of the american mathematical society. Carleson measures for weighted hardysobolev spaces nagoya. This boundedness principle is due to mazja and havin 31, theorem 3.
Harmonic functions matthew baker georgia institute of technology arizona winter school on padic geometry. They include the recent monographs by adams and hedberg ah on nonlinear potential theory and function spaces, and by heinonen, kilpelainen, and martio. Laceysharp estimates of the sobolev norm of u times the gradient v. Starting in the 1990s a lot of attention has been given to analysis on metric spaces. Kop function spaces and potential theory av david r adams, lars i hedberg pa. A note on dyadic hausdorff capacities sciencedirect. Adams, 9783540570608, available at book depository with free delivery worldwide. In this note we strengthen to morreylorentz spaces the sobolevtrace principle discovered by d. Hedberg, function spaces and potential theory, a series of. The library of congress has cataloged the original printing as follows. Connections between operator theory, function theory, and complex analysis summer graduate workshop, msri june 20july 1, 2011 books ah96 d. Singular operators with antisymmetric kernels, related. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel. In the chapter devoted to potential theory, several weighted capacities are investigated.
Developments and perspectives in nonlinear potential theory. Global bounded solutions of the higherdimensional kellersegel system under smallness conditions in optimal spaces. Function spaces and potential theory david r adams, lars i. Beckner concerning conformally invariant higherorder differential operators on the sphere. Adams, a note on the choquet integrals with respect to hausdor. Trace inequality, riesz potential, metric space, sobolev function, the poincare inequality. Adams, traces of potentials arising from translation invariant operators, ann. A banach space of integrable functions or distributions on the dimensional euclidean space, which generalizes the ordinary sobolev space of functions whose derivatives belong to classes, and their duals.
Nonlinear potential theory associated with pharmonic functions has been studied since the 1960s. We extend two versions of calderonzygmund capacities in to metric spaces and establish their equivalence under certain conditions. Bessel potential space encyclopedia of mathematics. Pdf nonlinear potential theory aims at replicating the classical linear. Throughout the paper we shall use the letter a to denote various positive con. Introduction it is well known that the hausdorff capacity and its dyadic version play an important role in potential theory, harmonic analysis and the modern theory of partial differential equations. They are included in the more general scale of lizorkintriebel spaces. This site is like a library, use search box in the widget to get ebook that you want. If denotes the laplace operator, the bessel potential space,, can be defined as the space of functions or distributions such that belongs to. Adams inequality on metric measure spaces project euclid. Sobolevs inequality for riesz potentials of functions in musielakorliczmorrey spaces over nondoubling metric measure spaces. Hedberg, function spaces and potential theory springerverlag, berlin, 1996. Estimates of the sobolev norm of a product of two functions.
We show that for 0 function spaces, especially those spaces that have become known as sobolev spaces, and their natural extensions, are now a central concept in analysis. Cordesnirenbergs imbedding and restricting with application. In potential theory, this type of inequalities arise from investigation. Hedberg, function spaces and potential theory, springer 1999. Morrey spaces in harmonic analysis morrey spaces in harmonic analysis adams, david.
The current state of the nonlinear potential theory is found in the new book by d. Yellow sale 2014 english titles sale prices valid until july 31, 2014 1 d. Function spaces and potential theory download ebook pdf. Potential theory, which grew out of the theory of the electrostatic or gravita tional potential, the laplace equation, the dirichlet problem. Jump to content jump to main navigation jump to main navigation. Best constants for sobolev inequalities for higher order. This paper not only presents a criterion for the cordesnirenberg space cnp, imbedding between the associate morrey space hs. Function spaces, especially those spaces that have become known as sobolev spaces, and their natural extensions, are now a central concept in analysis. Fine regularity of solutions of elliptic partial differential equations. An extension of hedbergs convolution inequality and. As an application, we extend the known relations between sriesz capacities, 0 oct 03, 2004 read on euler equations in higher order sobolev spaces, potential analysis on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Hedberg, function spaces and potential theory springer.
1234 285 332 847 1582 124 75 139 1104 701 92 798 860 929 374 1248 1335 1052 778 1435 1012 1560 696 787 30 759 923 339 1206 958 169 1268 794