We develop a new method to obtain symmetrization inequalities of Sobolev type. Our approach leads to new inequalities and considerable simplification in the theory of embeddings of Sobolev spaces based on rearrangement invariant spaces.

Abstract: This talk is on some new characterizations of Sobolev spaces which are based on non-local functionals whose roots are from definition of the fractional Sobolev spaces. New types of the Poincare inequality, the Sobolev inequality, and the Reillich Kondrachov compactness theorem will also be discussed. Possible applications for image processing will be mentioned. This is partially joint...

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 consider the Laplace equation under mixed boundary conditions on a polygonal domain Ω. Regularity estimates in terms of Sobolev norms of fractional order for this type of problem are proved. The analysis is based on new interpolation results and multilevel representation of norms on the Sobolev spaces Hα(Ω). The Fourier transform and the construction of extension operators to Sobolev spaces ...

We study Sobolev spaces on Lie manifolds, which we define as a class of manifolds described by vector fields (see Definition 1.2). The class of Lie manifolds includes the Euclidean spaces Rn, asymptotically flat manifolds, conformally compact manifolds, and manifolds with cylindrical and polycylindrical ends. As in the classical case of Rn, we define Sobolev spaces using derivatives, powers of ...

It is shown that if the Boyd indices of the rearrangement invariant Banach function space Lρ(R) are strictly between 0 and 1, ρ is an absolutely continuous function norm, Ω is a domain from R satisfying the restricted cone condition, denoting by ω the restriction of ρ to Ω, there exists an extension operator for the abstract Sobolev space W Lω(Ω). This is a generalization to abstract Sobolev sp...

In 1994, Lions, Perthame and Tadmor conjectured the maximal smoothing effect for multidimensional scalar conservation laws in Sobolev spaces. For strictly smooth convex flux and the one-dimensional case we detail the proof of this conjecture in the framework of Sobolev fractional spaces W s,1, and in fractional BV spaces: BV s. The BV s smoothing effect is more precise and optimal. It implies t...

In this paper we give a simple characterization of weighted Sobolev spaces (with piecewise monotonous weights) such that the multiplication operator is bounded: it is bounded if and only if the support of μ0 is large enough. We also prove some basic properties of the appropriate weighted Sobolev spaces. To have bounded multiplication operator has important consequences in Approximation Theory: ...

In this paper, we consider composition operators on Hardy-Sobolev spaces in connections with BMO-quasiconformal mappings. Using the duality of Hardy and BMO-spaces, prove that mappings generate bounded from Hardy–Sobolev to Sobolev spaces.