Pointwise estimates for densities of stable semigroups of measures

Let ${μ_t}$ be a symmetric α-stable semigroup of probability measures on a homogeneous group N, where 0 < α < 2. Assume that $μ_t$ are absolutely continuous with respect to Haar measure and denote by $h_t$ the corresponding densities. We show that the estimate $h_t(x) ≤ tΩ(x//|x|)|x|^{-n-α}$, x≠0, holds true with some integrable function Ω on the unit sphere Σ if and only if the density of the Lévy measure of the semigroup belongs locally to the Zygmund class LlogL(N╲{e}). The problem turns out to be related to the properties of the maximal function $ℳ f(x) = sup_{t>0} 1/t |\int_{0}^{t} h_{t-s} \ast f \ast h_s(x)ds|$ which, as is proved here, is of weak type (1,1).