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\(H^\infty\)-calculus for cylindrical boundary value problems. (English) Zbl 1277.47021
The authors consider an \({\mathcal{R}}\)-bounded \({\mathcal{H}}^\infty\)-calculus for linear operators associated to “cylindrical” boundary value problems. Their results are based on an abstract result on operator-valued functional calculus by N. Kalton and L. Weis [Math. Ann. 321, No. 2, 319–345 (2001; Zbl 0992.47005)], and “cylindrical” means that both domain and differential operator possess a certain cylindrical structure. The approach employed seems less technical and provides short proofs in comparison to standard methods (e.g., localization procedures). Besides this, the authors are able to deal with some classes of equations on rough domains. For instance, they extend the well-known (and in general sharp) range for \(p\) such that the (weak) Dirichlet Laplacian admits an \({\mathcal{H}}^\infty\)-calculus on \(L^p(\Omega),\) from \((3+\varepsilon)'<p<3+\varepsilon\) to \((4+\varepsilon)'<p<4+\varepsilon\) for 3D bounded or unbounded Lipschitz cylinders \(\Omega.\)

47A60 Functional calculus for linear operators
35J40 Boundary value problems for higher-order elliptic equations
35K52 Initial-boundary value problems for higher-order parabolic systems
35K35 Initial-boundary value problems for higher-order parabolic equations
Full Text: Euclid