article

Title: Multivariable spectral multipliers and analysis of quasielliptic operators on fractals

Authors: Adam Sikora

Issue: Volume 58 (2009), Issue 1, 317-334

Abstract:

$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 H\"ormander type differentiability condition then the operator $F(L_1,L_2)$ is of Calder\'on-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 are continuous on $L^p$ spaces. This solves the problem posed in [B. Bockelman and R.S. Strichartz, \emph{Partial differential equations on products of Sierpin\'nski gaskets}, Indiana Univ. Math. J. \textbf{56} (2007), (1.3) p. 1363].$