HORMANDER TYPE FUNCTIONAL CALCULUS AND SQUARE FUNCTION ESTIMATES

We investigate Hormander spectral multiplier theorems as they hold on X = L-P(Omega), 1 < p < infinity, for many self-adjoint elliptic differential operators A including the standard Laplacian on R-d. A strengthened matricial extension is considered, which coincides with a completely bounded map between operator spaces in the case that X is a Hilbert space. We show that the validity of the matricial Hormander theorem can be characterized in terms of square function estimates for imaginary powers A(it), for resolvents R(lambda, A), and for the analytic semigroup exp(-zA). We deduce Hormander spectral multiplier theorems for semigroups satisfying generalized Gaussian estimates.