Sharp $L^p$ estimates for Schrödinger 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 Schrödinger group $e^{itH}$ for $p\in [p_0,p_0^{\prime }]$.

In particular, our results apply to every operator of the form $H=(i\nabla +A{)}^2+V$, with a magnetic potential $A\in L_{\mathrm{loc}}^2({\mathbb{R}}^d,{\mathbb{R}}^d)$ and an electric potential $V$ whose positive and negative parts are in the local Kato class and in the Kato class, respectively.