Event № 830
Since its establishment, the field of interpolation theory (and interpolation spaces) has proved to be a very useful tool in several branches of Analysis. Crudely speaking, interpolation theory concerns itself with the finding of "intermediate" spaces $X$ lying "between" two given Banach spaces, $X_0$ and $X_1$, with the property that every linear operator which is bounded on both $X_0$ and $X_1$ is also bounded on $X$. One fundamental result in the early history of this field was the work of Riesz and Thorin who showed that for any numbers $p<q<r$, the space $L^q$ is an interpolation space "between" $L^p$ and $L^r$. Later, in 1958, Stein and Weiss extended this result to cases where the various $L^p$ spaces are weighted, with possibly different weights, so that it also makes sense to consider the case where $p=q=r$. It is natural to ask whether analogues of the Stein-Weiss results hold when, instead of weighted $L^p$ spaces, one considers weighted Sobolev spaces on $R^n$. Indeed some authors have obtained such analogous results for special choices of weight functions via Fourier transforms and multiplier theorems. In our talk we will show that, at least when $p=q=r$, the Calderon complex interpolation method enables us to obtain such results for a large class of weight functions which apparently cannot be treated by Fourier methods. We will also briefly discuss how these results can be helpful in the study of time asymptotic behaviour of solutions to evolution equations. This talk is based on joint work with Michael Cwikel. Details can be found in our paper on the arXiv, which is to appear in J. Funct. Analysis.