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\(H_\infty \)-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.

MSC:

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.)
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