Pointwise estimates for densities of stable semigroups of measures

Let ${\displaystyle \left\{{\mu }_t\right\}}$ be a symmetric α-stable semigroup of probability measures on a homogeneous group N, where 0 < α < 2. Assume that ${\displaystyle {\mu }_t}$ are absolutely continuous with respect to Haar measure and denote by ${\displaystyle h_t}$ the corresponding densities. We show that the estimate ${\displaystyle h_t\left(x\right)\le t\Omega (x/\left|x\right|){\left|x\right|}^{-n-\alpha }}$, 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 ${\displaystyle ℳf\left(x\right)={\supset }_{t>0}\frac{1}{t}|{\int }_0^t{h}_{t-s}\ast f\ast h_s\left(x\right)ds|}$ which, as is proved here, is of weak type (1,1).