Abstract. We extend results of Caffarelli-Silvestre and Stinga-Torrea regarding a characterization of fractional powers of differential operators via an extension problem. Our results apply to generators of integrated families of operators, in particular to infinitesimal generators of bounded C 0 semigroups and operators with purely imaginary symbol. We give integral representations to the extension problem in terms of solutions to the heat equation and the wave equation.
Groups of unbounded operators are approached in the setting of the Esterle quasimultiplier theory. We introduce groups of regular quasimultipliers of growth ω, or ω-groups for short, where ω is a continuous weight on the real line. We study the relationship of ω-groups with families of operators and homomorphisms such as regularized, distribution and integrated groups, holomorphic semigroups, and functional calculi. Some convolution Banach algebras of functions with derivatives to fractional order are needed, which we construct using the Weyl fractional calculus.
We obtain a vector-valued subordination principle for (gα, g β )-regularized resolvent families which unified and improves various previous results in the literature. As a consequence we establish new relations between solutions of different fractional Cauchy problems. To do that, we consider scaled Wright functions which are related to Mittag-Leffler functions, the fractional calculus and stable Lévy processes. We study some interesting properties of these functions such as subordination (in the sense of Bochner), convolution properties, and their Laplace transforms. Finally we present some examples where we apply these results.
Let X be a complex Banach space. The connection between algebra homomorphisms defined on subalgebras of the Banach algebra ℓ 1 (N0) and the algebraic structure of Cesàro sums of a linear operator T ∈ B(X) is established. In particular, we show that every (C, α)-bounded operator T induces -and is in fact characterized -by such an algebra homomorphism. Our method is based on some sequence kernels, Weyl fractional difference calculus and convolution Banach algebras that are introduced and deeply examined. To illustrate our results, improvements to bounds for Abel means, new insights on the (C, α) boundedness of the resolvent operator for temperated α-times integrated semigroups, and examples of bounded homomorphisms are given in the last section.2010 Mathematics Subject Classification. 47C05, 47A35; 44A55, 65Q20.
Abstract.We introduce the notion of almost-distribution cosine functions in a setting similar to that of distribution semigroups defined by Lions. We prove general results on equivalence between almost-distribution cosine functions and α-times integrated cosine functions. E. Marschall considered vector-valued cosine transforms defined by cosine functions ([5]) and he applied them to study spectral properties and the spectral mapping theorem for cosine functions. The present author worked with trigonometric convolution products, cosine functions and sine functions (1-times integrated cosine functions) to define vector-valued cosine and sine transforms ([6]). Almost-distribution cosine function is a new related concept, closer to distribution semigroups defined by J.-L. Lions [4].
Introduction. Integrated cosine functions of operators inEvery α-times integrated cosine function leads to an almost-distribution cosine function of order α. We apply Banach algebras T
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