“…Bounds of heat kernels in various settings were extensively investigated using both, probabilistic and analytic methods; see, for instance, [2], [3], [8], [12], [13], and references therein. It was noted in Nowak, Sjögren, and Szarek [7] (see also [6]) that "Compared with qualitatively sharp estimates, genuinely sharp heat kernel bounds are in general harder to prove and appear rarely in the literature." See, for instance, Ma lecki and Serafin [4], where genuinely sharp estimates in the setting of the Dirichlet Laplacian on the Euclidean ball were established (see also the references therein for other settings).…”