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Session I.1 - Multiresolution and Adaptivity in Numerical PDEs

Wednesday, June 14, 17:30 ~ 18:00

Oscillations and Differences in Triebel-Lizorkin-Morrey Spaces

Markus Weimar

JMU Wuerzburg, Germany   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk, we discuss new characterizations of Triebel-Lizorkin-Morrey spaces $\mathcal{E}_{u,p,q}^s$ of positive smoothness $s$ in terms of local oscillations (i.e., local polynomial bestapproximations) as well as integral means of higher order differences. This family of function spaces generalizes the well-established scale of Triebel-Lizorkin spaces $F^s_{p,q}$ which particularly contains the usual $L_p$-Sobolev spaces $H^s_p=F^s_{p,2}$ as special cases. Moreover, there are strong relations to BMO and Campanato spaces. We extend assertions due to Triebel 1992 and Yuan/Sickel/Yang 2010 for spaces on $\mathbb{R}^d$ and additionally consider their restrictions to bounded Lipschitz domains $\Omega$. Furthermore, we indicate possible applications to the regularity theory of quasi-linear elliptic PDEs.

The results to be presented are based on a recent preprint [1] in joint work with Marc Hovemann (Marburg).

[1] M.~Hovemann and M.~Weimar. Oscillations and differences in Triebel-Lizorkin-Morrey spaces. Preprint in preparation, 2023.

Joint work with Marc Hovemann (Marburg).

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