Variable Lebesgue spaces: Harmonic analysis and PDEs

The variable Lebesgue spaces are a generalization of the Lebesgue spaces in which the exponent is allowed to vary over the domain. Intuitively, given a function p : \Omega --> [1,\infty), the space L^p(\Omega) consists of all measurable functions f such that   \int_\Omega |f(x)|^{p(x)} dx < \infty.

With the appropriate modifications this becomes a norm, and L^p(\Omega) becomes a Banach function space. These spaces preserve some but not all of the properties of the classical L^p spaces: for instance, they are neither translation nor rearrangement invariant. The L^p spaces were discovered by Orlicz in the 1930s, and have become the focus of sustained research interest since the early 1990s. Much of the original motivation for their study came from the study of variational problems related to the mathematical modeling of electrorheological fluids, but they are now studied both for their intrinsic interest and for their applications to PDEs and the calculus of variations.

In this talk I will discuss my work on harmonic analysis in the variable Lebesgue space setting, focusing on the boundedness of the Hardy-Littlewood maximal operator and the use of Rubio de Francia extrapolation theory to prove that other operators (such as singular integrals and potential operators) are bounded on L^p(\Omega). I will also discuss some applications to the study of partial differential equations and conclude with some open questions.