Global heat kernel bounds via desingularizing weights

Abstract

We obtain global heat kernel bounds for semigroups which need not be ultracontractive by transferring them to appropriately chosen weighted spaces where they become ultracontractive. Our construction depends upon two assumptions: the classical Sobolev imbedding and a ''desingularizing'' ðL 1 ; L 1 Þ bound on the weighted semigroup.

Self-similar asymptotics of solutions to heat equation with inverse square potential

“…Using a new weighted version of the Hardy-Poincaré inequality, they showed [13,Theorem 10.3] the convergence of some solutions toward radially symmetric, explicit self-similar solution. Here, we prove a result on the large-time behavior of solutions to problem (1.1)-(1.2) which is closely related to the one by Vázquez and Zuazua [13] and which is based on the estimates of the kernel of the Schrödinger operator Hu = − u − λ |x| 2 u, obtained recently by Liskevich and Sobol [8], Milman and Semenov [9], and Moschini and Tesei [10] (see Remark 2.4 for more details).…”

“…Moreover, the operator −H = u + λ |x| 2 generates a semigroup of linear operators on L 2 (R n ) (see Proposition 3.4 below) and the solution agrees with a semigroup solution. In addition, due to the results from [8,9] (see also [10,14,15]), this solution has the integral representation…”

“…To state our result, we recall a weight function, which is systematically used in the works [8][9][10]:…”

Sobolev spaces adapted to the Schrödinger operator with inverse-square potential

“…This is a result of [20] that we reproduce here in Theorem 2.1. For further discussion of the peculiarities and subtleties of the heat equation in the endpoint case see [28], as well as the references therein.…”

Heat kernel on manifolds with ends

“…This and (8.3) lead to the desired estimates of q (t, x, y). A particularly interesting case, which received attention in literature is when Φ (x) = c |x| −2 for large x (see, e.g., [27], [49], [50], [65]). In this case, as we will see below, the exponent of the long time decay of q (t, x, y) depends on c. In R n , n 2, this problem is actually easier and the result is simpler than in R 1 because R n satisfies (RCA) and the gluing techniques are not necessary (see [36,Section 10.4]).…”