06/02/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building
Stefan Samko, Universidade do Algarve, Faro
Hardy-Littlewood-Stein-Weiss Inequality in the Lebesgue Spaces with
Variable Exponent
The Hardy-Littlewood inequality, for one-dimensional fractional
integrals for Lebesgue spaces in the case of power weights and the
limiting exponent, was generalized to potential type operators by
Stein and Weiss. In the rapidly developing "variable exponent
business" there was an open problem to prove such an inequality for
potentials of variable order in the weighted Lebesgue spaces with
variable \(p(x)\), that is to prove the boundedness of the
potential type operator of order \(m\) from the weighted Lebesgue
space of order \(p(x)\) to a weighted space of order \(q(x)\) with
\(1/q(x) = 1/p(x)-m/n\).
The solution of this problem is presented for the case of
bounded domains in the Euclidean space. It is based on a technique
of estimation of weighted norms in the Lebesgue spaces with
variable exponent of powers of distances \(|y-x|\) truncated to the
exterior of the ball of radius r centered at the point
x of the Euclidean space, and on Hedberg's approach of
comparison of potentials with maximal functions.
One of the main points in the result obtained is that the bounds
for the weight exponents are exactly related to the values of the
Lebesgue exponent \(p(x)\) at the points to which the weight is
fixed. As a corollary, imbeddings for the Sobolev spaces with
varying \(p(x)\) are derived.