Gaussian estimates and holomorphy of semigroups

Ouhabaz, El-Maati

doi:10.1090/S0002-9939-1995-1232142-3

Gaussian estimates and holomorphy of semigroups
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Proc. Amer. Math. Soc. 123 (1995), 1465-1474 Request permission

Abstract:

We show that if a selfadjoint semigroup T on ${L^2}(\Omega )$ satisfies a Gaussian estimate $|T(t)f| \leq MG(bt)|f|,0 \leq t \leq 1,f \in {L^2}(\Omega )$ (where $G = G{(t)_{t \geq 0}}$ is the Gaussian semigroup on ${L^2}({R^N})$ and $\Omega$ is an open set of ${R^N}$), then T defines a holomorphic semigroup of angle $\frac {\pi }{2}$ on ${L^p}(\Omega )$ . We obtain by duality the same result on ${C_0}(\Omega )$. Applications to uniformly elliptic operators and Schrödinger operators are given.

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