Pointwise convergence of the heat and subordinates of the heat semigroups associated with the Laplace operator on homogeneous trees and two weighted $L^p$ maximal inequalities

TL;DR

This paper investigates pointwise convergence of heat semigroups and their subordinated families on homogeneous trees, characterizing weights for which convergence holds and establishing equivalence with boundedness of associated maximal operators.

Contribution

It provides a characterization of weights ensuring pointwise convergence of heat semigroups on homogeneous trees and links this to maximal operator boundedness in weighted $L^p$ spaces.

Findings

Characterization of weights for pointwise convergence.

Equivalence between convergence and maximal operator boundedness.

Conditions for boundedness of maximal operators on weighted $L^p$ spaces.

Abstract

In this paper we consider the heat semigroup $\{W_t\}_{t>0}$ defined by the combinatorial Laplacian and two subordinated families of $\{W_t\}_{t>0}$ on homogeneous trees $X$. We characterize the weights $u$ on $X$ for which the pointwise convergence to initial data of the above families holds for every $f\in L^{p}(X,\mu,u)$ with $1\le p<\infty$, where $\mu$ represents the counting measure in $X$ . We prove that this convergence property in $X$ is equivalent to the fact that the maximal operator on $t\in (0,R)$, for some $R>0$, defined by the semigroup is bounded from $L^{p}(X,\mu,u)$ into $L^{p}(X,\mu,v)$ for some weight $v$ on $X$.