Complex Powers of a Fourth-Order Operator: Heat Kernels, Green Functions and Lp - Lp1 Estimates

We first construct the minimal and maximal operators of the Hermite operator.Then we apply a classical reslult by Askey and Wainger, to prove that for 4/3 < p < 4. This implies that the Hermite operator is essentially self-adjoint, which means that its minimal and maximal operators coincide. Using the asymptotic behaviour of the Lp-norms of the Hermite functions and essentially the same method as in the proof of 4/3 < p < 4, the same results are true for 1 p . We also compute the spectrum for the minimal and the maximal operator for 4/3 < p < 4. Then we construct a fourth-order operator, called the twisted bi-Laplacian, from the Laplacian on the Heisenberg group, namely, the twisted Laplacian. Using spectral analysis, we obtain explicit formulas for the heat kernel and Green function of the twisted bi-Laplacian. We also give results on the spectral theory and number theory associated with it. We then consider all complex powers of the twisted bi-Laplacian and compute their heat kernels and Green functions, and moreover, we obtain Lp Lp0 estimates for the solutions of the initial value problem for the heat equation and the Poisson equation governed by complex powers of the twisted bi-Laplacian.