Mittwoch, den 29.01.2025 um 11.00 Uhr in B7 - 210
Vortrag von Anatole Gaudin (Clausthal) zum Thema: „Besov Spaces as Spaces of Traces“.
Abstract: In this presentation, we continue the study of Besov spaces, now considering them as spaces of traces of Sobolev functions. Indeed, while Sobolev (or Besov) functions on $\mathbb{R}^n$ might not have a continuous representative, one can make sense of their restriction (\textit{i.e.}, taking a trace) with respect to one or several variables, and such a restriction yields a function that lies in an appropriate Besov space with a non-integer amount of regularity. First, we show some basic trace results. We will then exhibit that such a procedure is possible for $\L^p$-based Sobolev and Besov spaces at the cost of a $1/p$ derivative, and that the optimal range of the trace operator in the said Besov space is actually sharp. Since this procedure can be extended to the half-space, by a localization argument, a rotation, and a flattening procedure of the boundary, the procedure can be proven to hold for taking the restriction on the boundary, in some sense, for Sobolev functions defined on a domain with a "minimally smooth" boundary. If time permits, we will discuss the optimality and applications of such sharp results.