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 ${\int }_{T}{𝕍}_{\epsilon }^{\nu }\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\nu \le C\epsilon |\nu |$ 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 ${\int }_{{T}^{2}}{𝕍}_{\epsilon }^{\nu }\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\nu \le {C}_{\tau }{\epsilon }^{1-\tau }ℰ{\left[\nu \right]}^{\tau }{|\nu |}^{1-\tau }$ is valid on bi-tree with any $\tau >0$. It is proved in [14] and is called improved surrogate maximum principle for potentials on bi-tree. The estimate ${\int }_{{T}^{3}}{𝕍}_{\epsilon }^{\nu }\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\nu \le {C}_{\tau }{\epsilon }^{1-\tau }ℰ{\left[\nu \right]}^{\tau }{|\nu |}^{1-\tau }$ with $\tau =2/3$ holds on tri-tree. We do not know any such estimate with any $\tau <1$ on four-tree. The third counterexample disproves the estimate ${\int }_{{T}^{2}}{𝕍}_{x}^{\nu }\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\nu \le F\left(x\right)$ for any $F$ whatsoever for some probabilistic $\nu$ on bi-tree ${T}^{2}$. On a simple tree $F\left(x\right)=x$ 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.

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
@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
%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/

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

Cited by Sources:

Articles of potential interest

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)