Séminaires :
Séminaire d'Analyse Fonctionnelle
Equipe(s) :
af,
Responsables :
E. Abakoumov - A.Eskenazis - D. Cordero-Erausquin - M. Fathi - O. Guédon - B. Maurey
Email des responsables :
Salle :
salle 13 - couloir 15-16 - 4ème étage
Adresse :
Campus Pierre et Marie Curie
Description
Le Jeudi à 10h30 - IMJ-PRG - 4 place Jussieu - 75005 PARIS
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Orateur(s)
Pavel Mozolyako - University of Bologna,
Titre
Dirichlet space on the polydisc: a discrete approach
Date
12/04/2018
Horaire
10:30 à 14:26
Diffusion
Résume
The Dirichlet space on the polydisc $
D
^d, d \geq 1$, consists of analytic functions satisfying $\displaystyle \|f\|^2_
\mathcal
D
(\mathbb
D
^d)
=\sum_
m_1,\dots,m_d
|\hat
f
(m_1,\dots,m_d)|^2(m_1+1)\cdot\dots\cdot(m_d+1) < +\infty
$. A measure $\mu$ on $\bar
\mathbf
D^d$ is a Carleson measure for $\mathcal
D
(\mathbb
D
^d)$, if the operator $Id\; \mathcal
D
(\mathbb
D
^d) \rightarrow L^2(\bar
\mathbb
D^d,\,d\mu)$ is bounded. In the one dimensional case ($d=1$) Carleson measures were first described by Stegenga ('80) in terms of capacity, further development was achieved in papers by Arcozzi, Rochberg, Sawyer, Wick and others.
Following Arcozzi et al. we consider the equivalent problem in the discrete setting --- characterization of trace measures for the Hardy operator on the polytree $T^d$. We introduce the basics of (poly)logarithmic potential theory on $T^d$, and for $d=2$ we present a description of such measures in terms of bilogarithmic capacity (which, in turn, gives the description of Carleson measures for $\mathcal
D
(\mathbb
D
^2)$ in the sense of Stegenga). We also discuss some arising combinatorial problems.
This talk is based on joint work with N. Arcozzi, K.-M. Perfekt, G. Sarfatti.
Salle
salle 13 - couloir 15-16 - 4ème étage
Adresse
Campus Pierre et Marie Curie
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