Essential differences of potential theories on a tree and on a bi-tree
Pavel Mozolyako, Alexander L'vovich Vol'berg · Comptes Rendus Mathématique · 2022
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 d ν ≤ C ε | 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 d ν ≤ C τ ε 1 - τ ℰ [ 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 d ν ≤ C τ ε 1 - τ ℰ [ 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 d ν ≤ F ( x ) for any F whatsoever for some probabilistic ν on bi-tree T 2