Multivariable Spectral Multipliers and Analysis of Quasielliptic Operators on Fractals

We study multivariable spectral multipliers F(L-1, L-2) acting on the Cartesian product of ambient spaces of two self-adjoint operators L-1 and L-2. We prove that if F satisfies Hormander type differentiability condition then the operator F (L-1, L-2) is of Calderon-Zygmund type. We apply obtained results to the analysis of quasielliptic operators acting on products of some fractal spaces. The existence and surprising properties of quasielliptic operators have been recently observed in works of Bockelman, Drenning and Strichartz. This paper demonstrates that Riesz type operators corresponding to quasielliptic operators art, continuous on L-P spaces. This solves the problem posed in [4, (1.3) p. 1363].