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1.
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
2.
Mieczysław Mastyło 《Potential Analysis》2011,35(4):301-328
We define a general variant of the modulus of smoothness in metric spaces and show that under mild condition it is equivalent
to the K-functional of a couple of Besov type spaces which in special cases coincide with spaces defined by Korevaar and Schoen. We
prove various symmetrization inequalities which involve the modulus, the K-functional and the isoperimetric estimators. We also characterize the Hajłasz-type Sobolev spaces defined not necessarily
on doubling measure spaces by means of generalized Poincaré inequalities. This require to study of some variants of the Fefferman–Stein
sharp functions as well as the Hardy–Littlewood maximal operators. 相似文献
3.
Facundo Mémoli 《Foundations of Computational Mathematics》2011,11(4):417-487
This paper discusses certain modifications of the ideas concerning the Gromov–Hausdorff distance which have the goal of modeling
and tackling the practical problems of object matching and comparison. Objects are viewed as metric measure spaces, and based
on ideas from mass transportation, a Gromov–Wasserstein type of distance between objects is defined. This reformulation yields
a distance between objects which is more amenable to practical computations but retains all the desirable theoretical underpinnings.
The theoretical properties of this new notion of distance are studied, and it is established that it provides a strict metric
on the collection of isomorphism classes of metric measure spaces. Furthermore, the topology generated by this metric is studied,
and sufficient conditions for the pre-compactness of families of metric measure spaces are identified. A second goal of this
paper is to establish links to several other practical methods proposed in the literature for comparing/matching shapes in
precise terms. This is done by proving explicit lower bounds for the proposed distance that involve many of the invariants
previously reported by researchers. These lower bounds can be computed in polynomial time. The numerical implementations of
the ideas are discussed and computational examples are presented. 相似文献
4.
Some function spaces on spaces of homogeneous type 总被引:2,自引:0,他引:2
Manfred Kronz 《manuscripta mathematica》2001,106(2):219-248
We introduce Campanato, Morrey, BMO and Sobolev-type spaces for mappings from a space of homogeneous type into a complete
metric space which possess properties comparable to their classical analogues. In particular we show integral characterizations,
the validity of the John–Nirenberg theorem, Poincarè and Sobolev inequalities, Sobolev's embedding theorem and estimates
on the pointwise behavior of Sobolev-type mappings.
Received: 4 December 2000 / Revised version: 5 July 2001 相似文献
5.
Generalizations of the Trudinger–Moser inequality to Sobolev–Lorentz spaces with weights are considered. The weights in these
spaces allow for the addition of certain lower order terms in the exponential integral. We prove an explicit relation between
the weights and the lower order terms; furthermore, we show that the resulting inequalities are sharp, and that there are
related phenomena of concentration–compactness.
相似文献
6.
Francisco J. Aragón Artacho Boris S. Mordukhovich 《Journal of Global Optimization》2011,50(1):145-167
This paper mainly concerns the study of a large class of variational systems governed by parametric generalized equations,
which encompass variational and hemivariational inequalities, complementarity problems, first-order optimality conditions,
and other optimization-related models important for optimization theory and applications. An efficient approach to these issues
has been developed in our preceding work (Aragón Artacho and Mordukhovich in Nonlinear Anal 72:1149–1170, 2010) establishing qualitative and quantitative relationships between conventional metric regularity/subregularity and Lipschitzian/calmness
properties in the framework of parametric generalized equations in arbitrary Banach spaces. This paper provides, on one hand,
significant extensions of the major results in op.cit. to partial metric regularity and to the new hemiregularity property.
On the other hand, we establish enhanced relationships between certain strong counterparts of metric regularity/hemiregularity
and single-valued Lipschitzian localizations. The results obtained are new in both finite-dimensional and infinite-dimensional
settings. 相似文献
7.
By taking an interest in a natural extension to the small parameters of the trace inequality for Morrey spaces, Orlicz–Morrey
spaces are introduced and some inequalities for generalized fractional integral operators on Orlicz–Morrey spaces are established.
The local boundedness property of the Orlicz maximal operators is investigated and some Morrey-norm equivalences are also
verified. The result obtained here sharpens the one in our earlier papers. 相似文献
8.
9.
In this paper, we consider an extension of well-posedness for a minimization problem to a class of variational–hemivariational
inequalities with perturbations. We establish some metric characterizations for the well-posed variational–hemivariational
inequality and give some conditions under which the variational–hemivariational inequality is strongly well-posed in the generalized
sense. Under some mild conditions, we also prove the equivalence between the well-posedness of variational–hemivariational
inequality and the well-posedness of corresponding inclusion problem. 相似文献
10.
We study the class of those locally radial stratified spaces which are (C)–regular for standard control functions. These spaces generalise Whitney stratified spaces, but are shown to have the same
topological and metric properties. The weakly Whitney stratified spaces that we have introduced recently form an intermediate
class between Whitney stratified spaces and these locally radial (C)–regular spaces.
Received: 14 January 2000 / Revised version: 27 November 2002
Published online: 20 March 2003
Mathematics Subject Classification (2000): 58A35, 58C35, 26B20 相似文献
11.
Karl-Theodor Sturm 《Acta Mathematica》2006,196(1):133-177
We introduce a curvature-dimension condition CD (K, N) for metric measure spaces. It is more restrictive than the curvature bound
(introduced in [Sturm K-T (2006) On the geometry of metric measure spaces. I. Acta Math 196:65–131]) which is recovered as
the borderline case CD(K, ∞). The additional real parameter N plays the role of a generalized upper bound for the dimension. For Riemannian manifolds, CD(K, N) is equivalent to
and dim(M) ⩽ N.
The curvature-dimension condition CD(K, N) is stable under convergence. For any triple of real numbers K, N, L the family of normalized metric measure spaces (M, d, m) with CD(K, N) and diameter ⩽ L is compact.
Condition CD(K, N) implies sharp version of the Brunn–Minkowski inequality, of the Bishop–Gromov volume comparison theorem and of the Bonnet–Myers
theorem. Moreover, it implies the doubling property and local, scale-invariant Poincaré inequalities on balls. In particular,
it allows to construct canonical Dirichlet forms with Gaussian upper and lower bounds for the corresponding heat kernels. 相似文献
12.
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
13.
Kasso A. Okoudjou Laurent Saloff-Coste Alexander Teplyaev 《Transactions of the American Mathematical Society》2008,360(7):3857-3873
We develop a new approach to formulate and prove the weak uncertainty inequality, which was recently introduced by Okoudjou and Strichartz. We assume either an appropriate measure growth condition with respect to the effective resistance metric, or, in the absence of such a metric, we assume the Poincaré inequality and reverse volume doubling property. We also consider the weak uncertainty inequality in the context of Nash-type inequalities. Our results can be applied to a wide variety of metric measure spaces, including graphs, fractals and manifolds.
14.
In this paper, we consider some generalization of maximally movable spaces of Finsler type. Among them, there are locally
conic spaces (Riemannian metrics of their tangent spaces are realized on circular cones) and generalized Lagrange spaces with
Tamm metrics (their tangent Riemannian spaces admit all rotations). On the tangent bundle of a Riemannian manifold, we study
a special class of almost product metrics, generated Tamm metric. This class contains Sasaki metric and Cheeger–Gromol metric.
We determine the position of this class in the Naveira classification of Riemannian almost product metrics. 相似文献
15.
T. V. Mikhailova 《Journal of Mathematical Sciences》1996,80(5):2052-2083
The paper deals with the uniqueness and stability of generalized solutions to the mixed boundary-value problem for the elasticity
theory system in an unbounded domain, coinciding with a cone in a neighborhood of infinity. It is assumed that the boundary
of the domain consists of two parts: the Dirichlet condition is prescribed on one of them, denoted by Γ1, and the Neumann condition is prescribed on the other. The paper contains sufficient conditions (in terms of metric properties
of Γ1) for the validity of the Korn and Hardy inequalities, which imply the uniqueness and stability of the solution to the considered
problem in appropriate function spaces. Bibliography: 8 titles.
Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 18, pp. 106–156, 1995. 相似文献
16.
We investigate the metric generalized inverses of linear operators in Banach spaces and their homogeneous selections, which
was the research suggestion given by Nashed and Votruba (Bull. Am. Math. Soc. 80:831–835, 1974). We construct a kind of the bounded homoneneous selections for the set-valued metric generalized inverse. Criteria for the
metric generalized inverses of linear operators and their homogeneous selections are given in terms of Moore–Penrose conditions.
The research was supported in part by the National Science Foundation Grant (10671049) and the Science Foundation Grant of
Heilongjiang Province. 相似文献
17.
Various Poincaré–Sobolev type inequalities are studied for a reaction–diffusion model of particle systems on Polish spaces.
The systems we consider consist of finite particles which are killed or produced at certain rates, while particles in the
system move on the Polish space interacting with one another (i.e. diffusion). Thus, the corresponding Dirichlet form, which
we call reaction–diffusion Dirichlet form, consists of two parts: the diffusion part induced by certain Markov processes on
the product spaces En (n≥1) which determine the motion of particles, and the reaction part induced by a Q-process on ℤ+ and a sequence of reference probability measures, where the Q-process determines the variation of the number of particles and the reference measures describe the locations of newly produced
particles. We prove that the validity of Poincaré and weak Poincaré inequalities are essentially due to the pure reaction
part, i.e. either of these inequalities holds if and only if it holds for the pure reaction Dirichlet form, or equivalently,
for the corresponding Q-process. But under a mild condition, stronger inequalities rely on both parts: the reaction–diffusion Dirichlet form satisfies
a super Poincaré inequality (e.g., the log-Sobolev inequality) if and only if so do both the corresponding Q-process and the diffusion part. Explicit estimates of constants in the inequalities are derived. Finally, some specific examples
are presented to illustrate the main results.
Mathematics Subject Classifications (2000) 4FD0F, 60H10.
Feng-Yu Wang: Supported in part by the DFG through the Forschergruppe “Spectral Analysis, Asymptotic Distributions and Stochastic
Dynamics”, the BiBoS Research Centre, NNSFC(10121101), and RFDP(20040027009). 相似文献
18.
Filippo Gazzola Hans-Christoph Grunau Guido Sweers 《Annali di Matematica Pura ed Applicata》2010,189(3):475-486
We prove that the best constant for the critical embedding of higher order Sobolev spaces does not depend on all the traces.
The proof uses a comparison principle due to Talenti (Ann Scuola Norm Sup Pisa Cl Sci 3(4): 697–718, 1976) and an extension
argument which enables us to extend radial functions from the ball to the whole space with no increase of the Dirichlet norm.
Similar arguments may also be used to prove the very same result for Hardy-Rellich inequalities. 相似文献
19.
We discuss the Grüss inequalities on spaces of continuous functions defined on a compact metric space. Using the least concave
majorant of the modulus of continuity, we obtain the Grüss inequality for the functional L(f) = H(f; x), where H:C[a, b] → C[a, b] is a positive linear operator and x ∈ [a, b] is fixed. We apply this inequality in the case of known operators, e.g., the Bernstein operator, the Hermite–Fejér interpolation
operator, and convolution-type operators. Moreover, we deduce Grüss-type inequalities using the Cauchy mean-value theorem,
thus generalizing results of Chebyshev and Ostrowski. The Grüss inequality on a compact metric space for more than two functions
is given, and an analogous Ostrowski-type inequality is obtained. The latter, in turn, leads to one further version of the
Grüss inequality. In the appendix, we prove a new result concerning the absolute first-order moments of the classic Hermite–Fejér
operator. 相似文献
20.
Snorre H. Christiansen 《Numerische Mathematik》2007,107(1):87-106
We elaborate on the interpretation of some mixed finite element spaces in terms of differential forms. In particular we define
regularization operators which, combined with the standard interpolators, enable us to prove discrete Poincaré–Friedrichs
inequalities and discrete Rellich compactness for finite element spaces of differential forms of arbitrary degree on compact
manifolds of arbitrary dimension. 相似文献