Functional Analysis and Applications Seminar 

08/04/2005, 15:00 — 16:00 — Room P3.10, Mathematics Building
Lars Diening, Universität Freiburg, Alemanha

Lebesgue and Sobolev spaces with variable exponent

Lebesgue and Sobolev spaces with variable exponents appear in problems of elasticity, fluid dynamics, calculus of variations, and differential equations with $p(x)$-growth conditions. Unfortunately, these spaces lack some important properties, e.g. translation and convolution are not continuous. Nevertheless, under certain regularity assumptions on $p$ the Hardy-Littlewood maximal operator in continuous on $L_p(·)$. This is the key step in the study of numerous results such as Sobolev embeddings, continuity of singular integrals, extension theorems, and the characterization of the trace spaces. In the talk we summarize the recent developments. The main attention will be on the continuity of the Hardy-Littlewood maximal operator and its applications. We will provide different criteria for the necessary regularity of the exponent $p$.