We define Sobolev space J V ' , ~ for 1 < p < (x, on an arbitrary metric space with finite diameter and equipped with finite, positive Bore1 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. Mathem...

We give a characterization of Sobolev spaces of bivariate periodic functions with dominating smoothness properties in terms of Sobolev spaces of univariate functions. The mixed Sobolev norm is proved to be a uniform crossnorm. This property can be used as a powerful tool in approximation theory. x1. Introduction Beside the approximation of functions from the usual isotropic periodic Sobo-lev sp...

We prove that every Sobolev function de ned on a metric space coincides with a Holder continuous function outside a set of small Hausdor content or capacity. Moreover, the Holder continuous function can be chosen so that it approximates the given function in the Sobolev norm. This is a generalization of a result of Mal y [Ma1] to the Sobolev spaces on metric spaces [H1].

We prove the equivalence of Nash type and super log Sobolev inequalities for Dirichlet forms. We also show that both inequalities are equivalent to Orlicz-Sobolev type inequalities. No ultracontractivity of the semigroup is assumed. It is known that there is no equivalence between super log Sobolev or Nash type inequalities and ultracontractivity. We discuss Davies-Simon’s counterexample as bor...

The equality sign holds in (1) i] u has the Jorm: (3) u(x) = [a + btxI,~',-'] 1-~1~ , where Ix[ = (x~ @ ...-~x~) 1⁄2 and a, b are positive constants. Sobolev inequalities, also called Sobolev imbedding theorems, are very popular among writers in part ial differential equations or in the calculus of variations, and have been investigated by a great number of authors. Nevertheless there is a ques...

Hardy-Sobolev spaces and interpolation N. Badr Institut Camille Jordan Université Claude Bernard Lyon 1 UMR du CNRS 5208 F-69622 Villeurbanne Cedex [email protected] F. Bernicot Laboratoire de Mathématiques Université de Paris-Sud UMR du CNRS 8628 F-91405 Orsay Cedex [email protected] October 19, 2010 Abstract The purpose of this work is to describe an abstract theory of Ha...

Let M be a compact, COO CR manifold of dimension 3 over R. Associated to the CR structure is a first-order differential operator, Db' on M. We study the regularity properties, in terms of L P Sobolev and Holder norms, of the equation Db u = f. M is said to be CR if there is given a COO sub-bundle, denoted T I .o M , of the complexified tangent bundle TM, such that each fiber T~'o M is of dimens...

We describe a generalized Levenberg-Marquardt method for computing critical points of the Ginzburg-Landau energy functional which models superconductivity. The algorithm is a blend of a Newton iteration with a Sobolev gradient descent method, and is equivalent to a trust-region method in which the trustregion radius is defined by a Sobolev metric. Numerical test results demonstrate the method t...