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Gaussian estimates and holomorphy of semigroups. (English) Zbl 0829.47032
Summary: 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 \(L^2 (\mathbb{R}^N)\) and \(\Omega\) is an open set of \(\mathbb{R}^N\)), then \(T\) defines a holomorphic semigroup of angle \({\pi \over 2}\) on \(L^p (\Omega)\), \(1\leq p< \infty\). We obtain by duality the same result on \(C_0 (\Omega)\). Applications to uniformly elliptic operators and Schrödinger operators are given.

MSC:
47D06 One-parameter semigroups and linear evolution equations
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
47D03 Groups and semigroups of linear operators
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