Comptes Rendus
Functional analysis
Essential differences of potential theories on a tree and on a bi-tree
Comptes Rendus. Mathématique, Volume 360 (2022), pp. 1039-1048.

In this note we give several counterexamples. One shows that small energy majorization on bi-tree fails. The second counterexample shows that energy estimate T 𝕍 ε ν dνCε|ν| always valid on a usual tree by a trivial reason (and with constant C=1) cannot be valid in general on bi-tree with any C whatsoever. On the other hand, a weaker estimate T 2 𝕍 ε ν dνC τ ε 1-τ [ν] τ |ν| 1-τ is valid on bi-tree with any τ>0. It is proved in [14] and is called improved surrogate maximum principle for potentials on bi-tree. The estimate T 3 𝕍 ε ν dνC τ ε 1-τ [ν] τ |ν| 1-τ with τ=2/3 holds on tri-tree. We do not know any such estimate with any τ<1 on four-tree. The third counterexample disproves the estimate T 2 𝕍 x ν dνF(x) for any F whatsoever for some probabilistic ν on bi-tree T 2 . On a simple tree F(x)=x would suffice to make this inequality to hold. The potential theories without any maximum principle are harder than the classical ones (see e.g. [1]), and we prove here that in our potential theories on multi-trees maximum principle must be surrogate.

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Accepted:
Published online:
DOI: 10.5802/crmath.362
Pavel Mozolyako 1; Alexander Volberg 2

1 Department of Mathematics and Computer Science, Saint Petersburg University, Saint Petersburg, 199178, Russia
2 Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA, and Hausdorff Center for Mathematics, University of Bonn, Endenicher allée 60, Bonn 53115, Germany
License: CC-BY 4.0
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Pavel Mozolyako; Alexander Volberg. Essential differences of potential theories on a tree and on a bi-tree. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 1039-1048. doi : 10.5802/crmath.362. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.362/

[1] David R. Adams; Lars I. Hedberg Function Spaces and Potential Theory, Grundlehren der Mathematischen Wissenschaften, 314, Springer, 1999

[2] Nicola Arcozzi; Irina Holmes; Pavel Mozolyako; Alexander Volberg Bi-parameter embedding and measures with restricted energy conditions, Math. Ann., Volume 377 (2020) no. 1-2, pp. 643-674 | DOI | MR | Zbl

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

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

[5] Nicola Arcozzi; Richard Rochberg; Eric Sawyer Carleson measures for analytic Besov spaces, Rev. Mat. Iberoam., Volume 18 (2002) no. 2, pp. 443-510 | DOI | MR | Zbl

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[9] Sun-Yung A. Chang Carleson Measure on the Bi-Disc, Ann. Math., Volume 109 (1979), pp. 613-620 | DOI | MR | Zbl

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[11] Pavel Mozolyako; Georgios Psaromiligkos; Alexander Volberg Counterexamples for multi-parameter weighted paraproducts, C. R. Math. Acad. Sci. Paris, Volume 358 (2020) no. 5, pp. 529-534 | MR | Zbl

[12] Pavel Mozolyako; Georgios Psaromiligkos; Alexander Volberg; Pavel Zorin Kranich Combinatorial property of all positive measures in dimensions 2 and 3, C. R. Math. Acad. Sci. Paris, Volume 358 (2020) no. 6, pp. 721-725 | MR | Zbl

[13] Pavel Mozolyako; Georgios Psaromiligkos; Alexander Volberg; Pavel Zorin-Kranich Carleson embedding on tri-tree and on tri-disc (2001) (https://arxiv.org/abs/2001.02373)

[14] Pavel Mozolyako; Georgios Psaromiligkos; Alexander Volberg; Pavel Zorin-Kranich Improved surrogate bi-parameter maximum principle (2021) (https://arxiv.org/abs/2101.01094)

[15] Camil Muscalu; Jill Pipher; Terence Tao; Christoph Thiele Bi-parameter paraproducts, Acta Math., Volume 193 (2004) no. 2, pp. 269-296 | DOI | MR | Zbl

[16] Camil Muscalu; Jill Pipher; Terence Tao; Christoph Thiele Multi-parameter paraproducts, Rev. Mat. Iberoam., Volume 22 (2006) no. 3, pp. 963-976 | DOI | MR | Zbl

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

[18] Terence Tao Dyadic product H 1 , BMO, and Carleson’s counterexample (Short Stories. available at http://www.math.ucla.edu/~tao/preprints/Expository/product.dvi)

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