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Interpolation of cosine operator functions. (English) Zbl 0562.47035

Given a strongly continuous cosine operator function C on \(R^+\) with values in the Banach algebra B(A) of bounded linear operators in a Banach space A and infinitesimal generator \(\Lambda\), we are concerned with the investigation of the intermediate spaces between A and the domain \(D(\Lambda^ r)\), \(r\in N\), as well as with the characterization of the domains of fractional powers \((-\Lambda)^{\alpha}\), \(0<\alpha <r\). Using the K-method we give equivalent characterizations of these intermediate spaces by means of moduli of continuity of C including reduction results both in the case of nonoptimal approximation and saturation. With respect to the regularity of solutions of initial-value problems for second order evolution equations and the reduction of well- posed second order problems to equivalent first order ones we study mapping properties of C and its strong integral S which are the propagators of such problems. As applications, the results of this paper not only provide characterizations of the Besov spaces \(B^{2\alpha,q}\) in lights of a cosine operator functional calculus instead of the well- known semigroup approach via the Weierstrass singular integral but also can be used in the approximate solution of initial-value problems for second order hyperbolic P.D.E.’s by finite difference techniques with special emphasis on the case of non-smooth initial data.

MSC:

47D03 Groups and semigroups of linear operators
46M35 Abstract interpolation of topological vector spaces
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
35D10 Regularity of generalized solutions of PDE (MSC2000)
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