Montag, den 04.11.2024 um 17.00 Uhr in B7 - 210
Vortrag von Anatole Gaudin (Clausthal) zum Thema „An introduction to Besov spaces through the (damped) heat equation“.
Abstract: The main goal of this talk is to introduce Besov spaces, which are a natural generalization of Sobolev spaces, allowing more flexibility in the study of PDEs. They provide a finer way to measure integrability and regularity than Sobolev spaces, and they also offer more flexibility for establishing effective non-linear estimates.
To motivate our first definition and use of Besov spaces $B^{s}_{p,q}(\mathbb{R}^n)$, we will investigate the behavior of the linear (damped) heat equation with respect to the initial data, first in the spaces $L^p_{t,x}(\mathbb{R}_+\times\mathbb{R}^n)$ and then in $L^q_{t}(\mathbb{R}_+, L^p_{x}(\mathbb{R}^n))$, with $1\leqslant p,q\leqslant + \infty$. We will first treat the case $p=2$, introducing fractional Sobolev spaces $\H^{s}(\mathbb{R}^n)$.
Equivalent characterizations with different norms, such as through Littlewood-Paley decompositions or finite differences (Sobolev-Slobodeckij/Gagliardo norms), will be discussed.
If time permits, we will describe Besov spaces as traces of Sobolev functions on the half-space (or as the restrictions to a hyperplane of Sobolev functions defined on the whole space).
\textbf{Requirements:} Basic knowledge of Banach spaces, Lebesgue integration, and convolution (and approximation) on $L^p$ spaces, as well as basic Fourier analysis on $L^2$, are assumed. Familiarity with Sobolev spaces of integer order on the whole space is recommended. Knowledge of (tempered) distributions is not required, but would be helpful.