Heat kernel bounds and desingularizing weights

Global heat kernel bounds via desingularizing weights

“…This circumstance attracted great attention (see e.g. [BG,BS,BV,CM,DD,KPS,LS,MS,Se1,Se3,SV]). In the much simpler case of bounded potentials behaving at infinity like bV 0 for bo1 various explicit but not optimal heat kernel estimates were obtained in [DS,Zh].…”

“…and is also necessary for the Gaussian lower bound to be valid [MS,Se1]. Thus any potential V X0 which violates it makes the Gaussian upper bound off the mark.…”

Sharp Gaussian Estimates for Heat Kernels of Schrödinger Operators

“…Simon [23,Theorem B.7.1] resolved them for V in the Kato class, Zhang [28] and Milman and Semenov [20] applied the parabolic Kato class for this purpose. For further discussion we refer the reader to [18], [19], [20], [29] and [6,Lemma 4]. We also refer to Bogdan and Szczypkowski [7,Section 1,4] for a survey of the plain Gaussian bounds for Schrödinger heat kernels along with a streamlined approach, new results and explicit constants based on the so-called 4G inequality.…”

Singular Solutions to the Heat Equations With Nonlinear Absorption and Hardy Potentials

“…In particular, the linear equation does not have the fundamental solution at zero, i.e. the Cauchy problem with δ 0 (x) as the initial datum has no solution, which can be seen from by now well-known two-sided estimates for the corresponding heat kernel p(t, x, y) [18,20,22]:…”

“…To overcome the difficulties described above and to classify solutions to (1.1) satisfying (1.2) we use the technique of transference to the weighted space, which is by now standard in the linear theory and is called the ground-state transform (cf. [18,[20][21][22] Below and further on we use the following notation for the weighted Lebesgue and Sobolev spaces. For a weight ϕ we denote…”