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

Abstract

In this note we give several counterexamples. One shows that small energy majorization on bi-tree fails. The second counterexample shows that energy estimate $\int _T {\mathbb{V}}^\nu _\varepsilon \, \mathrm{d}\nu \le C \varepsilon |\nu |$ 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 $\int _{T^2} {\mathbb{V}}^\nu _\varepsilon \, \mathrm{d}\nu \le C_\tau \varepsilon ^{1-\tau } {\mathcal{E}}[\nu ]^{\tau } |\nu |^{1-\tau }$ 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 $\int _{T^3} {\mathbb{V}}^\nu _\varepsilon \, \mathrm{d}\nu \le C_\tau \varepsilon ^{1-\tau } {\mathcal{E}}[\nu ]^{\tau } |\nu |^{1-\tau }$ 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 $\int _{T^2} {\mathbb{V}}^\nu _x \, \mathrm{d}\nu \le F(x)$ for any $F$ whatsoever for some probabilistic $\nu $ 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.