\(H_\infty \)-calculus for elliptic operators with non-smooth coefficients. (English) Zbl 0892.47017

The authors consider general systems of elliptic operators on \(R^n\) and on compact manifolds. They prove under minimal regularity assumptions on the coefficients that there exists a bounded holomorphic functional calculus in \(L_p\) spaces which was introduced by A. McIntosh [Operator theory and partial differential equations, Proc. Cent. Math. Anal. Aust. Natl. Univ. 14, 210-231 (1986; Zbl 0634.47016)]. This represents an important extension of the recent results obtained by J. Pruess and H. Sohr [Hiroshima Math. J. 23, No. 1, 161-192 (1993; Zbl 0790.35023)] and by H. Amann, M. Hieber and G. Simonett [Differ. Integral Equ. 7, No. 3-4, 613-653 (1994; Zbl 0799.35060)]. The proofs are based on the use of results and techniques from harmonic analysis.


47A60 Functional calculus for linear operators
47F05 General theory of partial differential operators
35J45 Systems of elliptic equations, general (MSC2000)
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)