- May 21, 2004, Friday - 15h00 (Room P 3.10):
Weighted Sobolev Theorems for Spatial and Spherical Potentials in the Lebesgue Spaces with Variable Exponent
Stefan Samko
(Universidade do Algarve, Faro)
Abstract. 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 R n onto the
unit sphere S n in
R n+1. One of the remarkable
properties of this mapping is that it transforms the distance between two points
x and y in
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
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 R n, which is a
non-compact set (with respect to the usual metrics). The talk is based upon
joint work with Boris Vakulov.
Seminars take place in Lisbon, I.S.T. -
Post Graduation Building
Webpage: http://www.math.ist.utl.pt/funcional
