# Sharp $L^p$ estimates for Schrödinger groups

### Piero D'Ancona

Università di Roma La Sapienza, Italy### Fabio Nicola

Politecnico di Torino, Italy

## 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 $p \in [p_0,p'_0]$.

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

## Cite this article

Piero D'Ancona, Fabio Nicola, Sharp $L^p$ estimates for Schrödinger groups. Rev. Mat. Iberoam. 32 (2016), no. 3, pp. 1019–1038

DOI 10.4171/RMI/907