In this note we give several counterexamples. One shows that small energy majorization on bi-tree fails. The second counterexample shows that energy estimate always valid on a usual tree by a trivial reason (and with constant ) cannot be valid in general on bi-tree with any whatsoever. On the other hand, a weaker estimate is valid on bi-tree with any . It is proved in [14] and is called improved surrogate maximum principle for potentials on bi-tree. The estimate with holds on tri-tree. We do not know any such estimate with any on four-tree. The third counterexample disproves the estimate for any whatsoever for some probabilistic on bi-tree . On a simple tree 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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Pavel Mozolyako 1; Alexander Volberg 2
@article{CRMATH_2022__360_G9_1039_0, author = {Pavel Mozolyako and Alexander Volberg}, title = {Essential differences of potential theories on a tree and on a bi-tree}, journal = {Comptes Rendus. Math\'ematique}, pages = {1039--1048}, publisher = {Acad\'emie des sciences, Paris}, volume = {360}, year = {2022}, doi = {10.5802/crmath.362}, language = {en}, }
TY - JOUR AU - Pavel Mozolyako AU - Alexander Volberg TI - Essential differences of potential theories on a tree and on a bi-tree JO - Comptes Rendus. Mathématique PY - 2022 SP - 1039 EP - 1048 VL - 360 PB - Académie des sciences, Paris DO - 10.5802/crmath.362 LA - en ID - CRMATH_2022__360_G9_1039_0 ER -
Pavel Mozolyako; Alexander Volberg. Essential differences of potential theories on a tree and on a bi-tree. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 1039-1048. doi : 10.5802/crmath.362. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.362/
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