Journal of Operator Theory

Volume 51, Issue 2, Spring 2004 pp. 335-360.

Gaussian upper bounds for heat kernels of second-order elliptic operators with complex coefficients on arbitrary domains

Authors: El Maati Ouhabaz
Author institution: Lab. Bordelais d'Analyse et Geometrie, CNRS, UMR 5467, Universite Bordeaux I, Cours de la Liberation, 33405 Talence, France

Summary: We consider second-order elliptic operators of the type

$A = -\sum\limits_{k,j} D_{j} ( a_{kj}D_{k}) + \sum\limits_{k} b_{k}D_{k} - D_{k}(c_{k} \,\cdot) + a_{0}$ acting on

$L^{2}(\Om)$

$\Om$ is a domain of $\RR^d$, $d \ge 1$ and subject to various boundary conditions. We allow the coefficients

$a_{kj}, b_{k}, c_{k}$ and

$a_{0}$ to be complex-valued bounded measurable functions. Under a suitable condition on the imaginary parts of the principal coefficients

$a_{kj},$ we prove that for a wide class of boundary conditions, the semigroup

$({\rm e}^{-tA})_{t\ge0}$ is quasi-

$L^p$ -contractive

$1 < p < \infty$ . We show a pointwise domination of

$({\rm e}^{-tA})_{t\ge0}$ by a semigroup generated by an operator with real-valued coefficients and prove a Gaussian upper bound for the associated heat kernel.

Keywords: Elliptic operators, boundary conditions, heat kernels, Gaussian bounds

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