Genuinely sharp heat kernel estimates on compact rank-one symmetric spaces, for Jacobi expansions, on a ball and on a simplex

The Laplacian with mixed Dirichlet-Neumann boundary conditions on Weyl chambers

“…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). In Nowak, Sjögren, and Szarek [6] such estimates were proved for the spherical heat kernel and in [7] for heat kernels on all compact rank-one symmetric spaces (and also for Jacobi expansions, and in the context of a ball and a simplex). See also the recent paper by Serafin [9], where estimates precisely describing the exponential behavior of Dirichlet heat kernels in convex domains with C 1,1 boundary were established (clearly, Weyl chambers are not such domains, except, say, half-spaces).…”

“…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).…”

Pointwise monotonicity of heat kernels

“…We have recently been aware of an extension of the results in [4] by the same authors where they provide sharp bounds for the heat kernels on all compact rank-one symmetric spaces, cf. [7].…”

Sharp estimates for Jacobi heat kernels in conic domains

“…Despite hundreds of articles devoted to studying them, it was only very recently that the development of techniques allowed the so-called genuinely sharp estimates to be given in settings other than a few classical ones such as the hyperbolic space H d+1 , see [3]. For the spherical heat kernel genuinely sharp estimates were obtained in [8], while the Jacobi heat kernels for all compact rank-one symmetric spaces, including the classical domain [−1, 1], were investigated in [9], with the aid of some tools elaborated earlier in [7].…”

“…In the proofs we make use of the ideas invented in [9]. Let us also explain that "genuinely sharp" means that the exact expressions which control the heat kernels simultaneously from above and below are given.…”