Functional Analysis and Applications Seminar 

21/05/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building
Stefan Samko, Universidade do Algarve, Faro

Weighted Sobolev Theorems for Spatial and Spherical Potentials inthe Lebesgue Spaces with Variable Exponent

One of the open problems in the "variable exponent business" was related to embeddings in the Sobolev spaces with variable exponent in the case of unbounded domains, in particular, in the case of the whole Euclidean space. As is known, such embeddings are related to mapping properties of potential type operators. In this talk there are presented weighted results on the boundedness of the Riesz potential operator from the generalized Lebesgue space over Euclidean space, with variable exponent \(p(x)\), to a similar space with the Sobolev limiting exponent \(q(x)\).

Spherical potential operators are also treated in a similar setting in the corresponding spaces with variable exponent on the unit sphere in the Euclidean space. Stereographical projection is used for this purpose, which maps the Euclidean space \(\mathbb{R}^n\) onto the unit sphere \(S^n\) in \(\mathbb{R}^{n+1}\). One of the remarkable properties of this mapping is that it transforms the distance between two points \(x\) and \(y\) in \(\mathbb{R}^n\) exactly into the difference between their images \(s(x)\) and \(s(y)\) on \(S^n\) multiplied by the power weight functions fixed to infinity. This property allows to derive many results for various types of operators, known for \(\mathbb{R}^n\) to similar types of spherical operators on the sphere, and, to the contrary, from what may be obtained on the compact set \(S^n\), one may derive results for operators on \(\mathbb{R}^n\), which is a non-compact set (with respect to the usual metrics). The talk is based upon joint work with Boris Vakulov.