共查询到20条相似文献,搜索用时 31 毫秒
1.
We discuss the potential theory related to the variational capacity and the Sobolev capacity on metric measure spaces. We prove our results in the axiomatic framework of [17].46E35, 31C15, 31C45 相似文献
2.
In this paper, we introduce some new function spaces of Sobolev type on metric measure spaces. These new function spaces are defined by variants of Poincaré inequalities associated with generalized approximations of the identity, and they generalize the classical Sobolev spaces on Euclidean spaces. We then obtain two characterizations of these new Sobolev spaces including the characterization in terms of a variant of local sharp maximal functions associated with generalized approximations of the identity. For the well-known Hajłasz–Sobolev spaces on metric measure spaces, we also establish some new characterizations related to generalized approximations of the identity. Finally, we clarify the relations between the Sobolev-type spaces introduced in this paper and the Hajłasz–Sobolev spaces on metric measure spaces. 相似文献
3.
Recently, in the article [LW], the authors use the notion of polynomials in metric spaces of homogeneous type (in the sense of Coifman-Weiss) to prove a relationship between high order Poincaré inequalities and
representation formulas involving fractional integrals of high order, assuming only that is a doubling measure and that geodesics exist. Motivated by this and by recent work in [H], [FHK], [KS] and [FLW] about
first order Sobolev spaces in metric spaces, we define Sobolev spaces of high order in such metric spaces . We prove that several definitions are equivalent if functions of polynomial type exist. In the case of stratified groups,
where polynomials do exist, we show that our spaces are equivalent to the Sobolev spaces defined by Folland and Stein in [FS].
Our results also give some alternate definitions of Sobolev spaces in the classical Euclidean case.
Received: 10 February 1999 / Published online: 1 February 2002 相似文献
4.
In this article we study metric measure spaces with variable dimension. We consider Lebesgue spaces on these sets, and embeddings
of the Riesz potential in these spaces. We also investigate Hajłasz-type Sobolev spaces, and prove Sobolev and Trudinger inequalities
with optimal exponents. All of these questions lead naturally to function spaces with variable exponents.
Supported the Research Council of Norway, Project 160192/V30. 相似文献
5.
Heli Tuominen 《Annali di Matematica Pura ed Applicata》2009,188(1):35-59
We study the regularity of Orlicz–Sobolev functions on metric measure spaces equipped with a doubling measure. We show that each Orlicz–Sobolev function is quasicontinuous and has Lebesgue points outside a set of capacity zero and that the discrete maximal operator is bounded in the Orlicz–Sobolev space. We also show that if the Hardy–Littlewood maximal operator is bounded in the Orlicz space $L^{\Psi}(X)We study the regularity of Orlicz–Sobolev functions on metric measure spaces equipped with a doubling measure. We show that
each Orlicz–Sobolev function is quasicontinuous and has Lebesgue points outside a set of capacity zero and that the discrete
maximal operator is bounded in the Orlicz–Sobolev space. We also show that if the Hardy–Littlewood maximal operator is bounded
in the Orlicz space , then each Orlicz–Sobolev function can be approximated by a H?lder continuous function both in the Lusin sense and in norm.
The research is supported by the Centre of Excellence Geometric Analysis and Mathematical Physics of the Academy of Finland. 相似文献
6.
N. N. Romanovskiĭ 《Siberian Mathematical Journal》2014,55(3):511-529
We use a new method to prove the Sobolev embedding theorem for functions on a metric space and study other questions of the theory of Sobolev spaces on a metric space. We prove the existence and uniqueness of solution to a variational problem. 相似文献
7.
Recently Korevaar and Schoen developed a Sobolev theory for maps from smooth (at least ) manifolds into general metric spaces by proving that the weak limit of appropriate average difference quotients is well
behaved. Here we extend this theory to functions defined over Lipschitz manifold. As an application we then prove an existence
theorem for harmonic maps from Lipschitz manifolds to NPC metric spaces.
Received December 6, 1996 / Accepted March 4, 1997 相似文献
8.
Juha Kinnunen 《Journal of Mathematical Analysis and Applications》2008,344(2):1093-1104
We discuss Maz'ya type isocapacitary characterizations of Sobolev inequalities on metric measure spaces. 相似文献
9.
杨大春 《中国科学A辑(英文版)》2003,46(5)
This paper introduces the fractional Sobolev spaces on spaces of homogeneous type,includingmetric spaces and fractals. These Sobolev spaces include the well-known Hajfasz-Sobolev spaces as specialmodels.The author establishes varions chaaracterizations of(sharp)maximal functions for these spaces.Asapplications,the author identifies the fractional Sobolev spaces with some Lipscitz-type spaces.Moreover;some embedding theorems are also given. 相似文献
10.
We establish a modified segment inequality on metric spaces that satisfy a generalized volume doubling property. This leads
to Sobolev and Poincaré inequalities for such spaces. We also give several examples of spaces that satisfy the generalized
doubling condition. 相似文献
11.
We define and study Musielak-Orlicz-Sobolev spaces with zero boundary values on any metric space endowed with a Borel regular measure. We extend many classical results, including completeness, lattice properties and removable sets, to Musielak-Orlicz-Sobolev spaces on metric measure spaces. We give sufficient conditions which guarantee that a Sobolev function can be approximated by Lipschitz continuous functions vanishing outside an open set. These conditions are based on Hardy type inequalities. 相似文献
12.
We consider quasi-isometric mappings of domains in multidimensional Euclidean spaces. We establish that a mapping depends continuously in the sense of the topology of Sobolev classes on its metric tensor to within isometry of the space. In the space of metric tensors we take the topology determined by means of almost everywhere convergence. We show that if the metric tensor of a mapping is continuous then the length of the image of a rectifiable curve is determined by the same formula as in the case of mappings with continuous derivatives. (Continuity of the metric tensor of a mapping does not imply continuity of its derivatives.) 相似文献
13.
This paper introduces the fractional Sobolev spaces on spaces of homogeneous type, including metric spaces and fractals. These
Sobolev spaces include the well-known Hajłasz-Sobolev spaces as special models. The author establishes various characterizations
of (sharp) maximal functions for these spaces. As applications, the author identifies the fractional Sobolev spaces with some
Lipscitz-type spaces. Moreover, some embedding theorems are also given. 相似文献
14.
Siberian Mathematical Journal - We study the properties of the so-called grand Sobolev spaces on a metric measure space. The introduction of the spaces is motivated by the available... 相似文献
15.
Juha Heinonen Pekka Koskela Nageswari Shanmugalingam Jeremy T. Tyson 《Journal d'Analyse Mathématique》2001,85(1):87-139
We give a definition for the class of Sobolev functions from a metric measure space into a Banach space. We give various characterizations
of Sobolev classes and study the absolute continuity in measure of Sobolev mappings in the “borderline case”. We show under
rather weak assumptions on the source space that quasisymmetric homeomorphisms belong to a Sobolev space of borderline degree;
in particular, they are absolutely continuous. This leads to an analytic characterization of quasiconformal mappings between
Ahlfors regular Loewner spaces akin to the classical Euclidean situation. As a consequence, we deduce that quasisymmetric
maps respect the Cheeger differentials of Lipschitz functions on metric measure spaces with borderline Poincaré inequality.
J. H. supported by NSF grant DMS9970427. P. K. supported by the Academy of Finland, project 39788. N. S. supported in part
by Enterprise Ireland. J. T. T. supported by an NSF Postdoctoral Research Fellowship. 相似文献
16.
Sobolev Spaces with Zero Boundary Values on Metric Spaces 总被引:6,自引:0,他引:6
We generalize the definition of the first order Sobolev spaces with zero boundary values to an arbitrary metric space endowed with a Borel regular measure. We show that many classical results extend to the metric setting. These include completeness, lattice properties and removable sets. 相似文献
17.
Robert S. Strichartz 《Journal of Functional Analysis》2003,198(1):43-83
We construct function spaces, analogs of Hölder-Zygmund, Besov and Sobolev spaces, on a class of post-critically finite self-similar fractals in general, and the Sierpinski gasket in particular, based on the Laplacian and effective resistance metric of Kigami. This theory is unrelated to the usual embeddings of these fractals in Euclidean space, and so our spaces are distinct from the function spaces of Jonsson and Wallin, although there are some coincidences for small orders of smoothness. We show that the Laplacian acts as one would expect an elliptic pseudodifferential operator of order d+1 on a space of dimension d to act, where d is determined by the growth rate of the measure of metric balls. We establish some Sobolev embedding theorems and some results on complex interpolation on these spaces. 相似文献
18.
We extend Cheeger’s theorem on differentiability of Lipschitz functions in metric measure spaces to the class of functions
satisfying Stepanov’s condition. As a consequence, we obtain the analogue of Calderon’s differentiability theorem of Sobolev
functions in metric measure spaces satisfying a Poincaré inequality.
Communicated by Steven Krantz 相似文献
19.
In this paper, we consider the natural generalization of Cheeger type Sobolev spaces to maps into a metric space. We solve Dirichlet problem for CAT(0)-space targets, and obtain some results about the relation between Cheeger type Sobolev spaces for maps into a Banach space and those for maps into a subset of that Banach space. We also prove the minimality of upper pointwise Lipschitz constant functions for locally Lipschitz maps into an Alexandrov space of curvature bounded above. 相似文献
20.
《Expositiones Mathematicae》2020,38(4):480-495
The aim of this note is to explain in which sense an axiomatic Sobolev space over a general metric measure space (à la Gol’dshtein–Troyanov) induces – under suitable locality assumptions – a first-order differential structure. 相似文献