Lp estimates for fractional Schrödinger operators with Kato class potentials
Abstract
Let α > 0, H =(- Δ) α + V (x), V (x) belongs to the higher order Kato class K2α (Rn). For 1 ≤ p ≤ ∞, we prove a polynomial upper bound of ‖e-itH(H + M) - β ‖ Lp ,Lp in terms of time t for all integers α and 2 α ≥ [n/2] + 1 if α is not an integer. Both the smoothing exponent β and the growth order in t are almost optimal compared to the free case. The main ingredients in our proof are pointwise heat kernel estimates for the semigroup e-tH. We obtain a Gaussian upper bound with sharp coefficient for integral α and a polynomial decay for fractional α.
- Publication:
-
Journal of Differential Equations
- Pub Date:
- November 2018
- DOI:
- 10.1016/j.jde.2018.06.004
- Bibcode:
- 2018JDE...265.4181H
- Keywords:
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- L<SUP>p</SUP> estimates;
- Heat kernel estimates;
- Fractional Schrödinger equations