Gaussian Estimates and Holomorphy of Semigroups

Abstract

We show that if a self adjoint semigroup T on L2(fi) satisfies a Gaussian estimate \T(t)f\ < MG(bt)\f\, 0 < t < 1 , f e L2(S) (where G = G(f)/>o is the Gaussian semigroup on L2(RN) and Q is an open set of RN ), then T defines a holomorphic semigroup of angle | on Z/(2), 1 < p < oo .We obtain by duality the same result on Co(fl).Applications to uniformly elliptic operators and Schrdinger operators are given. IntroductionLet Q be an open set of RN (with the Lebesgue measure), and consider a selfadjoint semigroup T = T(t)t>o on L2(Q) with generator A. Then T is a holomorphic semigroup with angle , i.e., T can be extended holomorphically to the maximal domain {z, Re z > 0} (see 1 for the precise definition).Assume now that T interpolates on LP(Q), 1 < p < oc ; that is, there exists for each p, a strongly continuous semigroup Tp on LP(Q) with T2 = T and satisfying Tp(t)f = T2(t)f (t > 0) for / D>(Q)nL2(Q).It follows from the Stein interpolation theorem that for 1 < p < oo the semigroup Tp is holomorphic on 1/(0.) with angle 6P > |(1 - -l\) (see Davies [6, p. 23]).However, the case p -1 is more delicate.In the case where A is an elliptic operator of second order (with some smoothness conditions on its coefficients), Amann [2] showed that Tx is holomorphic on L1 (Q) if Q is bounded and smooth; his method is based on duality arguments and the result of Stewart [21,22] on Co(Q).Recently, Arendt and Batty [3] extended the result to an arbitrary open set Q under Dirichlet boundary conditions.We also note that Kato [12] showed the holomorphy on LP(RN), 1 < p < oo, for the Schrdinger operator A = A -V (A is the Laplacian and V = V+ -F_ is a potential).The purpose of this paper is to extend all these results to more general situations.We show the holomorphy on LP(Q), 1 < p < oo, for elliptic operators under more general boundary conditions without regularity on their coefficients and by assuming minimal regularity on Q.More precisely, we show in an abstract setting the following result: