Sobolev spaces on an arbitrary metric space

We define Sobolev space W ^1,p for 1<p on an arbitrary metric space with finite diameter and equipped with finite, positive Borel measure. In the Euclidean case it coincides with standard Sobolev space. Several classical imbedding theorems are special cases of general results which hold in the metric case. We apply our results to weighted Sobolev space with Muckenhoupt weight.This work is supported by KBN grant no. 2 1057 91 01     

Generalized Cauchy functional equation and characterizations of inner product spaces

Summary Using a generalized Cauchy functional equation we show that some well-known characterizations of inner product spaces, such as those of Jordan—von Neumann, Johnson, and Rassias, can be proved without use of the triangle inequality.     

Maximal elements in topological and generalized metric spaces

In this note, some problems concerning existence of maximal elements in a topological as well as in a generalized metric space, equipped with an ordering, are studied. The results presented here may be considered as a partial refinement of those established in [2] for uniform structures.     

A splitting property of the upper bounded-Hausdorff convergence

We provide a characterization, based on a splitting property, for the upper bounded-Hausdorff convergence of a net of closed sets. Furthermore, we use this property to describe the local structure of the upper bounded-Hausdorff topology.The author is indebted to S. Levi and R. Lucchetti for continuous suggestions and encouragements.     

Two-weight Weak-type Norm Inequalities for the Commutators of Fractional Integrals

In this paper, we obtain a sufficient condition for two-weight weak type norm inequality for the commutators of fractional integral , where and .     

New characterizations of Sobolev metric spaces

We provide new characterizations of Sobolev ad BV spaces in doubling and Poincaré metric spaces in the spirit of the Bourgain–Brezis–Mironescu and Nguyen limit formulas holding in domains of ${\mathbb{R}}^N$.     

Probability distribution of metric measure spaces

In this paper we are going to generalize Gromov's mm-Reconstruction theorem (cf. [Metric Structures for Riemannian and Non-Riemannian Spaces, Birkhäuser, Basel, 1999] ) to a probability measures on the spaces of mm-spaces. And for this purpose, we give alternative proof of mm-Reconstruction theorem.     

The Continuity of Commutators on Triebel-Lizorkin Spaces

The purpose of this paper is to study the continuity in the context of Triebel-Lizorkin spaces for some commutators related to certain convolution operators. The operators include Littlewood-Paley operator, Marcinkiewicz integral and Bochner-Riesz operator.     

Some remarkable lines of triangles in real normed spaces and characterizations of inner product structures

Summary In this work we consider the heights and the bisectrices of a triangle in a real normed space. Using well-known formulas which can be generalized to real normed spaces we obtain a collection of new characterizations of inner product spaces.     

Densely two-valued metric projections in uniformly convex Banach spaces

For every uniformly convex Banach spaceX with dimX2 there is a residual setU in the Hausdorff metric spaceB(X) of bounded and closed sets inX such that the metric projection generated by a set fromU is two-valued and upper semicontinuous on a dense and everywhere continual subset ofX. For any two closed and separated subsetsM ₁ andM ₂ ofX the points on the equidistant hypersurface which have best approximations both inM ₁ andM ₂ form a dense G set in the induced topology.The author is partially supported by the National Fund for Scientific Research at the Bulgarian Ministry of Science and Education under contract MM 408/94.     

Heat kernel estimates for jump processes of mixed types on metric measure spaces

In this paper, we investigate symmetric jump-type processes on a class of metric measure spaces with jumping intensities comparable to radially symmetric functions on the spaces. The class of metric measure spaces includes the Alfors d-regular sets, which is a class of fractal sets that contains geometrically self-similar sets. A typical example of our jump-type processes is the symmetric jump process with jumping intensity where ν is a probability measure on , c(α, x, y) is a jointly measurable function that is symmetric in (x, y) and is bounded between two positive constants, and c ₀(x, y) is a jointly measurable function that is symmetric in (x, y) and is bounded between γ₁ and γ₂, where either γ₂ ≥ γ₁ > 0 or γ₁ = γ₂ = 0. This example contains mixed symmetric stable processes on as well as mixed relativistic symmetric stable processes on . We establish parabolic Harnack principle and derive sharp two-sided heat kernel estimate for such jump-type processes. Dedicated to Professor Masatoshi Fukushima on the occasion of his 70th birthday. The research of Zhen-Qing Chen is supported in part by NSF Grants DMS-0303310 and DMS-06000206. The research of Takashi Kumagai is supported in part by the Grant-in-Aid for Scientific Research (B) 18340027.     

Convergence in measure with respect to a matrix-valued measure and some matrix completion problems

For a matrix-valued measure M we introduce a notion of convergence in measure M, which generalizes the notion of convergence in measure with respect to a scalar measure and takes into account the matrix structure of M. Let S be a subset of the set of matrices of given size. It is easy to see that the set of S-valued measurable functions is closed under convergence in measure with respect to a matrix-valued measure if and only if S is a ρ-closed set, i.e. if and only if SP is closed for any orthoprojector P. We discuss the behaviour of ρ-closed sets under operations of linear algebra and the ρ-closedness of particular classes of matrices.     

Littlewood–Paley characterizations of higher‐order Sobolev spaces via averages on balls

In this article, the authors characterize higher‐order Sobolev spaces , with , and , or with , and , via the Lusin area function and the Littlewood–Paley ‐function in terms of ball averages, where denotes the maximal integer not greater than . Moreover, the authors also complement the above results in the endpoint cases of p via establishing some weak type estimates. These improve and develop the corresponding known results for Sobolev spaces with smoothness order .     

The doubling property of conformal measures of infinite iterated function systems

We provide sufficient conditions for the conformal measures induced by regular conformal infinite iterated function systems to satisfy the doubling property. We apply these conditions to iterated function systems derived from the continued fraction algorithm—continued fractions with restricted entries. For these systems our conditions are expressed in terms of the asymptotic density properties of the allowed entries. As examples, we give some relatively large classes of sets of continued fractions with restricted entries for which the corresponding conformal measures have the doubling property. Similarly, we give some other classes for which the conformal measure does not have the doubling property.