Functional Analysis and Applications Seminar 

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.