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

TL;DR

This paper establishes $L^p$ estimates for fractional Schrödinger operators with potentials in the Kato class, providing near-optimal bounds on the evolution operator's behavior over time.

Contribution

It introduces new $L^p$ bounds for fractional Schrödinger operators with Kato class potentials, extending previous results to fractional orders and fractal dimensions.

Findings

Polynomial upper bounds for the evolution operator in $L^p$ spaces.

Almost optimal smoothing and growth exponents compared to free case.

Heat kernel estimates with Gaussian bounds and polynomial decay.

Abstract

Let $\alpha>0$, $H=(-\triangle)^{\alpha}+V(x)$, $V(x)$ belongs to the higher order Kato class $K_{2\alpha}(\mathbbm{R}^n)$. For $1\leq p\leq \infty$, we prove a polynomial upper bound of $\|e^{-itH}(H+M)^{-\beta}\|_{L^p, L^p}$ in terms of time $t$. Both the smoothing exponent $\beta$ 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 $\alpha$ and a polynomial decay for fractal $\alpha$.