Heat Kernels of Second Order Complex Elliptic Operators and Applications

We study the heat kernels of second order elliptic operators in divergence form with complex bounded measurable coefficients on Rn. We obtain Gaussian bounds without further assumption ifn⩽2, and when the principal part has Hölder continuous coefficients ifn⩾3. A thorough study of the boundedness properties of the Green operator is made in dimension 1. We construct the fundamental solution of these operators in dimension 2. Boundedness results of the maximal accretive square roots onLpSobolev spaces and Hölder spaces are obtained whenn⩾2 under Hölder continuity assumption on the coefficients. BoundedH∞functional calculi onLp, 1<p<+∞, are also discussed.