Bounded holomorphic functional calculus for non-divergence form differential operators

Abstract

Let $L$ be a second-order elliptic partial differential operator of non-divergence form acting on ${\bf R^n}$ with bounded coefficients. We show that for each $1 < p_0 <2, L$ has a bounded $H_{\infty}$-functional calculus on $L^p({\bf R^n})$ for $p_0 <p <\infty$ if the $BMO$ norm of the coefficients is sufficiently small.