A capacitary weak type inequality for Sobolev functions and its applications

In this paper a capacitary weak type inequality for Sobolev functions is established and is applied to reprove some well-known results concerning Lebesgue points, Taylor expansions in the -sense, and the Lusin type approximation of Sobolev functions.

     

Weierstrass' Theorem in Weighted Sobolev Spaces

We characterize the set of functions which can be approximated by polynomials with the following norm

Full-size image

for a big class of weights w₀, w₁, …, w_k     

A proof of the trace theorem of Sobolev spaces on Lipschitz domains

A complete proof of the trace theorem of Sobolev spaces on Lipschitz domains has not appeared in the literature yet. The purpose of this paper is to give a complete proof of the trace theorem of Sobolev spaces on Lipschitz domains by taking advantage of the intrinsic norm on . It is proved that the trace operator is a linear bounded operator from to for .

     

Combinatorial volume preserving flows and taut foliations

A -injective closed surface in an orientable 3-manifold with a tangentially smooth, transversely C ⁰ taut foliation can be homotoped to an immersed surface which is either transverse to the foliation except at isolated saddle tangencies or mapped into a leaf. Received: November 11, 1997     

A Bernstein type theorem for minimal volume preserving maps

We show that any minimal volume preserving map from the Euclidean plane into itself is a linear diffeomorphism. We derive this from a similar result on minimal diffeomorphisms. We also show that the classical Bernstein theorem on minimal graphs is a corollary of our result.

     

Disjointness preserving operators between little Lipschitz algebras

Given a real number α∈(0,1) and a metric space (X,d), let Lipα(X) be the algebra of all scalar-valued bounded functions f on X such that

     

索伯列夫嵌入不等式的证明

We consider the problem about the space embedded by tile space and the embed-ding inequality. With the Holder inequality and interpolation inequality, we give the proof of the space embedding theorem and the space holder embedding theorem.     

Embedding Theorems for a Certain Function Class of Sobolev Type on Metric Spaces

Given a metric space with a Borel measure , we consider a class of functions whose increment is controlled by the measure of a ball containing the corresponding points and a nonnegative function p-summable with respect to . We prove some analogs of the classical theorems on embedding Sobolev function classes into Lebesgue spaces.     

Sobolev type theorems for an operator with singularity

Spaces of Sobolev type are discussed, which are defined by the operator with singularity: , where and . This operator appears in a one-dimensional harmonic oscillator governed by Wigner's commutation relations. Smoothness of and continuity of () are studied where is in each space of Sobolev type, and results similar to Sobolev's lemma are obtained. The proofs are carried out based on a generalization of the Fourier transform. The results are applied to the Schrödinger equation.

     

On Inequalities with Measures of Sobolev Type Embedding Theorems on Open Sets of the Real Axis

We consider some Sobolev-type spaces and obtain a necessary and sufficient condition for their embedding in a Lebesgue space.     

On the Complex of Sobolev Spaces Associated with an Abstract Hilbert Complex

We consider the complexes of Hilbert spaces whose differentials are closed densely-defined operators. A peculiarity of these complexes is that from their differentials we can construct Laplace operators in every dimension. The Laplace operator together with a sufficiently nice measurable function enables us to define a generalized Sobolev space. There exist pairs of measurable functions allowing us to construct some canonical mappings of the corresponding Sobolev spaces. We find necessary and sufficient conditions for those mappings to be compact. In some cases for a given Hilbert complex we can construct an associated Sobolev complex. We show that the differentials of the original complex are normally solvable simultaneously with the differentials of the associated complex and that the reduced cohomologies of these complexes coincide.     

Weighted Sobolev theorem in Lebesgue spaces with variable exponent

For the Riesz potential operator Iα there are proved weighted estimates

     

Superposition operator in Sobolev spaces on domains

For an arbitrary open set we characterize all functions on the real line such that for all . New element in the proof is based on Maz'ya's capacitary criterion for the imbedding .     

Multiple positive solutions for a semilinear elliptic equation with critical Sobolev exponent

In this paper, we study a class of semilinear elliptic equations with Hardy potential and critical Sobolev exponent. By means of the Ekeland variational principle and Mountain Pass theorem, multiple positive solutions are obtained.     

Characterization of the critical Sobolev space on the optimal singularity at the origin

In the present paper, we investigate the optimal singularity at the origin for the functions belonging to the critical Sobolev space , 1<p<∞. With this purpose, we shall show the weighted Gagliardo-Nirenberg type inequality:

(GN)     

An embedding theorem of Sobolev type for an operator with singularity

We discuss spaces of Sobolev type which are defined by the operator with singularity: , where and . This operator appears in a one-dimensional harmonic oscillator governed by Wigner's commutation relations. We study smoothness of and continuity of () where is in each space of Sobolev type, and obtain a generalization of the Sobolev embedding theorem. On the basis of a generalization of the Fourier transform, the proof is carried out. We apply the result to the Cauchy problems for partial differential equations with singular coefficients.

     

On the asymptotics of a 1-parameter family of infinite measure preserving transformations

We estimate various aspects of the growth rates of ergodic sums for some infinite measure preserving transformations which are not rationally ergodic.Dedicated to R. Mañé     

Sobolev空间上多尺度分析的性质

对Sobolev空间上多尺度分析的性质进行了探讨,给出了稠密性条件的一个特征刻划。