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

Mozolyako, Pavel; Volberg, Alexander

Comptes Rendus. Mathématique, Tome 360 (2022) no. G9, 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 ${\int }_T{𝕍}_{\epsilon }^{\nu }\mathrm{d}\nu \le C\epsilon \left|\nu \right|$ 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 ${\int }_{T^2}{𝕍}_{\epsilon }^{\nu }\mathrm{d}\nu \le C_{\tau }{\epsilon }^{1-\tau }ℰ{\left[\nu \right]}^{\tau }{\left|\nu \right|}^{1-\tau }$ 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 ${\int }_{T^3}{𝕍}_{\epsilon }^{\nu }\mathrm{d}\nu \le C_{\tau }{\epsilon }^{1-\tau }ℰ{\left[\nu \right]}^{\tau }{\left|\nu \right|}^{1-\tau }$ 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 ${\int }_{T^2}{𝕍}_x^{\nu }\mathrm{d}\nu \le F\left(x\right)$ 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.

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

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

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

[4] 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)

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

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

[7] 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

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

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

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

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

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

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

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

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

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