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.