Inductive methods for Hardy inequality on trees

Abstract

We study the Hardy inequality on at most countable rooted tree. The main known criteria in the lower--triangle case for this inequality are two Arcozzi--Rochberg--Sawyer criteria and the capacity criterion. In the survey we show that these two criteria are connected with the criteria for the Hardy inequalities for the sequences, for the Hardy inequality on an interval of the real axis and for the trace inequalities with the Riesz potentials. We provide the examples from the literature, when the trace inequality or another statement is characterized in terms of the validity of the Hardy inequality on the tree. We simplify two known proofs of the Arcozzi--Rochberg--Sawyer criteria, which are based on the Marcinkiewicz interpolation theorem and on the capacity criterion. We provide new proofs for Arcozzi--Rochberg--Sawyer criteria, which are based on the induction in the tree, the inductive formula for the capacity and the formula of integration by parts. The latter of the proofs is written for the Hardy inequality on the tree with a boundary and for the Hardy inequality over the family of all binary cubes. In the diagonal case this proof provides an optimal constant $p$, which coincides with the
Bennett constant in the Hardy inequality for the sequences. In the general case we provide a few new inductive criteria for the validity of the Hardy inequality in terms of the existence of a family of functions satisfying an inductive relation. One of these criteria is applied in the proof of a theorem containing additional equivalent conditions for the validity of the Hardy inequality on the trees in the diagonal case.