Bounded $H_\infty$-calculus for elliptic operators

Abstract

It is shown, in particular, that L p-realizations of general elliptic systems on Rn or on compact manifolds without boundaries possess bounded imaginary powers, provided rather mild regularity conditions are satisfied. In addition, there are given some new perturbation theorems for operators possessing a bounded H00-calculus. 0. Introduction. It is the main purpose of this paper to prove under mild regularity assumptionsthat Lp-realizations of elliptic differential operators acting on vector valued functions over JRn or on sections of vector bundles over compact manifolds without boundaries possess bounded imaginary powers. In fact, we shall prove a more general result guaranteeing that, given any elliptic operator A with a sufficiently large zero order term such that the spectrum of its principal symbol is contained in a sector of the form 8&0 := {z E C; I atgz!::::; eo} U {0} for some 0 e0 E [0, n), and given any bounded holomorphic function f: S& ---7 C for some e E (e0 , n), we can define a bounded linear operator j(A) on Lp, and an estimate of the form llf(A)II.ccLp) ::::; c llflloo is valid. This means that elliptic operators possess a bounded R 00-calculus in the sense of Mcintosh [16]. Choosing, in particular, f(z) :=zit fortE JR, it follows that A possesses bounded imaginary powers ( cf. Section 2 below for more precise statements). There are two main reasons for our interest in this problem. First, it is known (cf. [22], [24]) that the complex interpolation spaces [E, D(A)]& coincide with the domains of the fractional powers A & for 0 < e < 1, provided A is a densely defined linear operator on the Banach space E possessing bounded imaginary powers. Second, by a result of Dore and Venni [1 OJ, the fact that A possesses bounded imaginary powers is intimately connected with 'maximal regularity results' for abstract evolution equations of the form u +Au = f (t). Both these results are of great use in the Received for publication August 1993. AMS Subject Classifications: 35J45, 47F05