no. 6
Analyse, Analyse harmonique
Combinatorial property of all positive measures in dimensions 2 and 3
Mozolyako, Pavel1, 2 ; Psaromiligkos, Georgios2 ; Volberg, Alexander2  ; Kranich, Pavel Zorin3
1 Department of Mathematics and Computer Science, Saint Petersburg State University, Saint Petersburg, 199178, Russia
2 Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
3 Department of Mathematics, University of Bonn, Bonn, 51011, Germany
Comptes Rendus. Mathématique, Tome 358 (2020) no. 6, pp. 721-725.

We prove multi-parameter dyadic embedding theorem for Hardy operator on the multi-tree. We also show that for a large class of Dirichlet spaces of holomorphic functions in bi-disc and tri-disc this proves the embedding theorem of those spaces on bi- and tri-disc. We completely describe the Carleson measures for such embeddings.

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DOI : 10.5802/crmath.90
Mozolyako, Pavel 1, 2 ; Psaromiligkos, Georgios 2 ; Volberg, Alexander 2 ; Kranich, Pavel Zorin 3

1 Department of Mathematics and Computer Science, Saint Petersburg State University, Saint Petersburg, 199178, Russia
2 Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
3 Department of Mathematics, University of Bonn, Bonn, 51011, Germany 
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     title = {Combinatorial property of all positive measures in dimensions $2$ and $3$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {721--725},
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Mozolyako, Pavel; Psaromiligkos, Georgios; Volberg, Alexander; Kranich, Pavel Zorin. Combinatorial property of all positive measures in dimensions $2$ and $3$. Comptes Rendus. Mathématique, Tome 358 (2020) no. 6, pp. 721-725. doi : 10.5802/crmath.90. http://www.numdam.org/articles/10.5802/crmath.90/

[1] Arcozzi, Nicola; Holmes, Irina; Mozolyako, Pavel; Volberg, Alexander Bi-parameter embedding and measures with restriction energy condition (2018) (https://arxiv.org/abs/1811.00978) | Zbl

[2] Arcozzi, Nicola; Mozolyako, Pavel; Perfekt, Karl-Mikael; Sarfatti, Giulia Carleson measures for the Dirichlet space on the bidisc (2018) (https://arxiv.org/abs/1811.04990)

[3] Arcozzi, Nicola; Mozolyako, Pavel; Psaromiligkos, Georgios; Volberg, Alexander; Zorin-Kranich, Pavel Bi-parameter Carleson embeddings with product weights (2019) (https://arxiv.org/abs/1906.11150)

[4] Arcozzi, Nicola; Mozolyako, Pavel; Psaromiligkos, Georgios; Volberg, Alexander; Zorin-Kranich, Pavel Carleson embedding on tri-tree and on tri-disc (2020) (https://arxiv.org/abs/2001.02373)

[5] Arcozzi, Nicola; Rochberg, Richard; Sawyer, Eric T.; Wick, Brett D. Potential theory on trees, graphs and Ahlfors-regular metric spaces, Potential Anal., Volume 41 (2014) no. 2, pp. 317-366 | DOI | MR | Zbl

[6] Nazarov, Fedor; Treil, Sergei; Volberg, Alexander The Bellman functions and two-weight inequalities for Haar multipliers, J. Am. Math. Soc., Volume 12 (1999) no. 4, pp. 909-928 | DOI | MR | Zbl

[7] Sawyer, Eric T. Weighted inequalities for the two-dimensional Hardy operator, Stud. Math., Volume 82 (1985) no. 1, pp. 1-16 | DOI | MR | Zbl

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