Sharp $L^p$ estimates for Schr\"odinger groups

Abstract

Consider a non-negative self-adjoint operator $H$ in $L^2(\mathbb{R}^d)$. We suppose that its heat operator $e^{-tH}$ satisfies an off-diagonal algebraic decay estimate, for some exponents $p_0\in[0,2)$. Then we prove sharp $L^p\to L^p$ frequency truncated estimates for the Schr\"odinger group $e^{itH}$ for $p\in[p_0,p'_0]$. In particular, our results apply to every operator of the form $H=(i\nabla+A)^2+V$, with a magnetic potential $A\in L^2_{loc}(\mathbb{R}^d,\mathbb{R}^d)$ and an electric potential $V$ whose positive and negative parts are in the local Kato class and in the Kato class, respectively.