Gaussian estimates and holomorphy of semigroups

Abstract

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