Analyse fonctionnelle
Essential differences of potential theories on a tree and on a bi-tree
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 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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DOI : 10.5802/crmath.362
Mozolyako, Pavel 1 ; Volberg, Alexander 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
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Mozolyako, Pavel; Volberg, Alexander. Essential differences of potential theories on a tree and on a bi-tree. Comptes Rendus. Mathématique, Tome 360 (2022) no. G9, pp. 1039-1048. doi : 10.5802/crmath.362. http://www.numdam.org/articles/10.5802/crmath.362/

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