Regularity of fractional heat semigroup associated with Schrödinger operators

Abstract

Let L = −∆ + V be a Schrödinger operator, where the potential V belongs to the reverse Hölder class. By the subordinative formula, we introduce the fractional heat semigroup {e −tL α } t>0 , α > 0, associated with L. By the aid of the fundamental solution of the heat equation:we estimate the gradient and the time-fractional derivatives of the fractional heat kernel K L α,t (•, •), respectively. This method is independent of the Fourier transform, and can be applied to the second order differential operators whose heat kernels satisfying Gaussian upper bounds. As an application, we establish a Carleson measure characterization of the Campanato type space BMO γ L (R n ) via {e −tL α } t>0 .

Boundedness of operators generated by fractional semigroups associated with Schrödinger operators on Campanato type spaces via $T1$ theorem

“…By the subordinative formula (4) and Lemma 5, Li et al in [11] proved the following estimates for K L α,t (•, •). Proposition 2 [11, Propositions 3.1 and 3.2] Let 0 < α < 1.…”

“…For the kernels D L α,t (•, •) and D L,m α,t (•, •), m ∈ Z + , t > 0, defined by ( 5), the following regularity estimates were obtained by Li et al in [11].…”

“…where η α t (•) is a continuous function on (0, ∞) satisfying ( 19) below. In [11], the identity (4) was applied to estimate K L α,t (•, •) via the heat kernel K L t (•, •), see Proposition 2. Specially, for α = 1/2, the estimates of K L α,t (•, •) goes back to those of the Poisson kernel P L t (•, •), see [4,Lemma 3.9].…”