Heat Kernel Estimates of Fractional Schrödinger Operators with Negative Hardy Potential

Abstract

We obtain two-sided estimates for the heat kernel (or the fundamental function) associated with the following fractional Schrödinger operator with negative Hardy potentialand λ > 0. The proof is purely analytical but elementary. In particular, for upper bounds of heat kernel we use the Chapman-Kolmogorov equation and adopt self-improving argument.(1.3) 2010 Mathematics Subject Classification. 60G51; 60G52; 60J25; 60J75.

Optimal Hardy inequality for the fractional Laplacian on $L^p$

“…By Lemma 6, ( 27) and (28) we get (36). Since the limit in (36) exists for all u ∈ L p (R d ), we obtain (37) and (38), using dominated convergence.…”

Heat kernel of fractional Laplacian with Hardy drift via desingularizing weights

“…l for all t ą 0, x, y P R d , y ‰ 0. Indeed, taking (11) for granted, and using Proposition 10(i): e ´tP ε i Ñ e ´tΛr strongly in L r for some tε i u Ó 0 and every r Psr c , 8r, we obtain x1 S1 , e ´tΛr 1 S2 y ď Cx1 S1 , e ´tA ϕ t 1 S2 y for all compact S 1 , S 2 Ă R d . Since, by Theorem 2, e ´tΛr is an integral operator for every t ą 0 with integral kernel e ´tΛ px, yq, we obtain by the Lebesgue Differentiation Theorem, possibly after changing e ´tΛ px, yq on a measure zero set in R d ˆRd , that e ´tΛ px, yq ď Ce ´tA px, yqϕ t pyq for all t ą 0, x, y P R d , y ‰ 0, and so Theorem 3 follows.…”

“…on R d for some ε i Ó 0. The upper bound e ´tP ε px, yq ď Ce ´tA px, yqϕ t pyq, see (11), yields ϕe ´tP ε h ď Cϕe ´tA g P L 1 , so we can apply the Dominated Convergence Theorem in the left-hand side of (˛) to obtain that it converges to xgy ´e μ s t xϕe ´tΛ hy as ε i Ó 0. The proof of Proposition 3 is completed.…”

“…In Appendix A, we discuss how the argument of the present paper can be specified to H to yield a purely operator-theoretic proof of the result of [3]. (Concerning the heat kernel estimates for the operator p´Δq α 2 `c|x| ´α, c ą 0, see [7,11]. )…”

Decay of harmonic functions for discrete time Feynman--Kac operators with confining potentials

“…It is known to be a powerful tool which allows one to study various analytic properties of these operators by means of probabilistic methods. Recent contributions include estimates of the heat kernel, heat content and trace [1,2,4,28,36,54], harmonic functions, ground states, eigenfunctions and eigenvalues, and spectral bounds [27,32,35,41,53], intrinsic hyper-and ultracontractivity [15,33,42], to mention just a few of them.…”