Essential differences of potential theories on a tree and on a bi-tree

Pavel Mozolyako; Alexander Volberg

doi:10.5802/crmath.362

Analyse fonctionnelle

Essential differences of potential theories on a tree and on a bi-tree

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

Comptes Rendus. Mathématique, Volume 360 (2022), pp. 1039-1048.

Résumé

In this note we give several counterexamples. One shows that small energy majorization on bi-tree fails. The second counterexample shows that energy estimate $\int_{T} 𝕍_{ε}^{ν} d ν \leq C ε | ν |$ 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 $\int_{T^{2}} 𝕍_{ε}^{ν} d ν \leq C_{τ} ε^{1 - τ} ℰ {[ν]}^{τ} {| ν |}^{1 - τ}$ 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 $\int_{T^{3}} 𝕍_{ε}^{ν} d ν \leq C_{τ} ε^{1 - τ} ℰ {[ν]}^{τ} {| ν |}^{1 - τ}$ 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 $\int_{T^{2}} 𝕍_{x}^{ν} d ν \leq F (x)$ 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.

Reçu le : 2021-10-14
Accepté le : 2022-04-05
Publié le : 2022-09-29

DOI : 10.5802/crmath.362

Affiliations des auteurs :

Pavel Mozolyako ¹ ; Alexander Volberg ²

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2022__360_G9_1039_0,
     author = {Pavel Mozolyako and Alexander Volberg},
     title = {Essential differences of potential theories on a tree and on a bi-tree},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1039--1048},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     year = {2022},
     doi = {10.5802/crmath.362},
     language = {en},
}

TY  - JOUR
AU  - Pavel Mozolyako
AU  - Alexander Volberg
TI  - Essential differences of potential theories on a tree and on a bi-tree
JO  - Comptes Rendus. Mathématique
PY  - 2022
SP  - 1039
EP  - 1048
VL  - 360
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.362
LA  - en
ID  - CRMATH_2022__360_G9_1039_0
ER  -

%0 Journal Article
%A Pavel Mozolyako
%A Alexander Volberg
%T Essential differences of potential theories on a tree and on a bi-tree
%J Comptes Rendus. Mathématique
%D 2022
%P 1039-1048
%V 360
%I Académie des sciences, Paris
%R 10.5802/crmath.362
%G en
%F CRMATH_2022__360_G9_1039_0

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/

Bibliographie
Cité par

[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

[6] Nicola Arcozzi; Richard Rochberg; Eric T. Sawyer; Brett D. Wick The Dirichlet space: a survey, New York J. Math., Volume 17a (2011), pp. 45-86 | MR | Zbl

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

[8] Lennart Carleson A counterexample for measures bounded on $H^{p}$ for the bi-disc, Mittag-Leffler Report, 7, 1974

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

[10] Sun-Yung A. Chang; Robert Fefferman A continuous version of duality of $H^{1}$ with BMO on the bidisc, Ann. Math., Volume 112 (1980) no. 1, pp. 179-201 | DOI | MR | Zbl

[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 $H^{1}$ , $B M O$ , and Carleson’s counterexample (Short Stories. available at http://www.math.ucla.edu/~tao/preprints/Expository/product.dvi)

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Counterexamples for multi-parameter weighted paraproducts

Pavel Mozolyako; Georgios Psaromiligkos; Alexander Volberg

C. R. Math (2020)

Combinatorial property of all positive measures in dimensions $2$ and $3$

Pavel Mozolyako; Georgios Psaromiligkos; Alexander Volberg; ...

C. R. Math (2020)

Riesz capacities of a set due to Dobiński

Nicola Arcozzi; Nikolaos Chalmoukis

C. R. Math (2022)