Revista Matemática Iberoamericana

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Volume 32, Issue 3, 2016, pp. 1019–1038
DOI: 10.4171/RMI/907

Published online: 2016-10-03

Sharp $L^p$ estimates for Schrödinger groups

Piero D'Ancona[1] and Fabio Nicola[2]

(1) Università di Roma La Sapienza, Italy
(2) Politecnico di Torino, Italy

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.

Keywords: Spectral multipliers, Schrödinger group, heat kernel

D'Ancona Piero, Nicola Fabio: Sharp $L^p$ estimates for Schrödinger groups. Rev. Mat. Iberoamericana 32 (2016), 1019-1038. doi: 10.4171/RMI/907