Analyse fonctionnelle

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

Mozolyako, Pavel¹ ; Volberg, Alexander²

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, Tome 360 (2022) no. G9, 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 ${\int }_T{𝕍}_{\epsilon }^{\nu }\mathrm{d}\nu \le C\epsilon \left|\nu \right|$ 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.

Détail

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

DOI : 10.5802/crmath.362

Licence : CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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