$L^p$ estimates for fractional schrodinger operators with kato class potentials

Gaussian estimates with best constants for higher-order Schrödinger operators with Kato potentials

“…The sharpness of these estimates depends of course on using the right distance function which is not the Euclidean but, rather, a Finsler distance induced by the operator. The sharp constant σ m , also obtained in [18], was first identified by Evgrafov and Postnikov [17] who obtained short time asymptotics of K 0 (t, x, y) for operators with constant coefficients in R n and so-called strongly convex principal symbol (see definition below).…”

“…Under the assumption that H 0 has constant coefficients they prove that estimate (1) is also valid for the heat kernel K(t, x, y) of H 0 + V . In the very recent article [18] the authors consider the operator (−∆) m + V for Kato potentials V and apply the methods of [15] together with Davies' exponential perturbation technique as adpted in [7] in order to obtain estimates such as (1) for K(t, x, y) with the sharp constant c 2 in the Gaussian exponent.…”