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

Abstract

In this paper we consider the heat semigroup {Wt}t>0 defined by the combinatorial Laplacian and two subordinated families of {Wt}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 ∈ L p (X, µ, u) with 1 ≤ p < ∞, where µ 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 ∈ (0, R), for some R > 0, defined by the semigroup is bounded from L p (X, µ, u) into L p (X, µ, v) for some weight v on X.

Asymptotic behavior of solutions to the extension problem for the fractional Laplacian on noncompact symmetric spaces

“…The extension problem has drawn much attention. Since the associated literature is enormous, we shall refer indicatively to [3,9,10,12,16,24,25,27] and the references therein. From a probabilistic point of view, the extension problem corresponds to the property that all symmetric stable processes can be obtained as traces of degenerate Bessel diffusion processes, see [26].…”