J. Aust. Math. Soc.  82 (2007), 149-162
Generalized Gaussian estimates and Riesz means of Schrödinger groups

Sönke Blunck

We show that generalized Gaussian estimates for selfadjoint semigroups $(e^{-tA})_{t\in\mathbb{R}_+}$ on $L_2$ imply $L_p$-boundedness of Riesz means and other regularizations of the Schrödinger group $(e^{itA})_{t\in\mathbb{R}}$. This generalizes results restricted to semigroups with a heat kernel, which are due to Sjöstrand, Alexopoulos and more recently Carron, Coulhon and Ouhabaz. This generalization is crucial for elliptic operators A that are of higher order or have singular lower order terms since, in general, their semigroups fail to have a heat kernel.
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