Potential theory on graphs and weighted Hardy operator

Orateur: Pavel MOZOLYAKO
Localisation: Université d'État de Saint-Pétersbourg, Russie
Type: Colloquium Bézout
Site: UGE , 4B 125
Date de début: 15/11/2022 - 10:30
Date de fin: 15/11/2022 - 11:30

Let $\Gamma$ be a finite rooted directed graph without directed cycles, i.e. a finite partially ordered set with a unique maximal element. We introduce a Potential Theory on such graphs via a weighted Hardy operator $\mathbf{I}_w$ and discuss some of its basic properties -- energy, capacity, equilibrium measures and such. Our local aim is to investigate
the boundedness of $\mathbf{I}_w$ as a two-weighted $L^2$-embedding, which is, in turn, connected to several classical problems, for example the description of Carleson measures for different spaces of analytic and/or holomorphic functions on the disc/polydisc; weighted integration operators in $(\mathbb{R}_+)^n$; $L^2$-boundedness of the rectangular maximal functions etc.

We plan to give a short review of known results and (very briefly) discuss some of the aforementioned connections and applications.

Joint work with  N. Arcozzi, A. Volberg and P. Zorin-Kranich