TY - JOUR

T1 - R-boundedness, pseudodifferential operators, and maximal regularity for some classes of partial differential operators

AU - Denk, Robert

AU - Krainer, Thomas

PY - 2007/11

Y1 - 2007/11

N2 - It is shown that an elliptic scattering operator A on a compact manifold with boundary with operator valued coefficients in the morphisms of a bundle of Banach spaces of class (HT) and Pisier's property (α) has maximal regularity (up to a spectral shift), provided that the spectrum of the principal symbol of A on the scattering cotangent bundle avoids the right half-plane. This is accomplished by representing the resolvent in terms of pseudodifferential operators with R-bounded symbols, yielding by an iteration argument the R-boundedness of λ(A-λ)-1 in R for some R. To this end, elements of a symbolic and operator calculus of pseudodifferential operators with R-bounded symbols are introduced. The significance of this method for proving maximal regularity results for partial differential operators is underscored by considering also a more elementary situation of anisotropic elliptic operators on R with operator valued coefficients.

AB - It is shown that an elliptic scattering operator A on a compact manifold with boundary with operator valued coefficients in the morphisms of a bundle of Banach spaces of class (HT) and Pisier's property (α) has maximal regularity (up to a spectral shift), provided that the spectrum of the principal symbol of A on the scattering cotangent bundle avoids the right half-plane. This is accomplished by representing the resolvent in terms of pseudodifferential operators with R-bounded symbols, yielding by an iteration argument the R-boundedness of λ(A-λ)-1 in R for some R. To this end, elements of a symbolic and operator calculus of pseudodifferential operators with R-bounded symbols are introduced. The significance of this method for proving maximal regularity results for partial differential operators is underscored by considering also a more elementary situation of anisotropic elliptic operators on R with operator valued coefficients.

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U2 - 10.1007/s00229-007-0131-1

DO - 10.1007/s00229-007-0131-1

M3 - Article

AN - SCOPUS:35548972974

SN - 0025-2611

VL - 124

SP - 319

EP - 342

JO - Manuscripta Mathematica

JF - Manuscripta Mathematica

IS - 3

ER -