Multivariable Spectral Multipliers and Analysis of Quasielliptic Operators on Fractals

We study multivariable spectral multipliers F (L1, L2) acting on Cartesian product of ambient spaces of two self-adjoint operators L1 and L2. We prove that if F satisfies Hörmander type differentiability condition then the operator F (L1, L2) is of Calderón-Zygmund type. We apply obtained results to the analysis of quasielliptic operators acting on product 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 are continuous on L spaces. This solves the problem posed in [4, (1.3) p. 1363]. I dedicate this paper to the memory of my teachers Andrzej Hulanicki and Tadeusz Pytlik.