Mathematics Subject Classification: 43A20, 26A33 (main), 44A10, 44A15 We prove equalities in the Banach algebra L1(R+). We apply them to integral transforms and fractional calculus. * Partially supported by Project BFM2001-1793 of the MCYT-DGI and FEDER and Project E-12/25 of D.G.A.
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.
We study the local well-posed integrated Cauchy problem , v(0) = 0, t ∈ [0,κ), with κ > 0, α ≥ 0, and x ∈ X, where X is a Banach space and A a closed operator on X. We extend solutions increasing the regularity in α. The global case (κ = ∞) is also treated in detail. Growth of solutions is given in both cases.
We investigate the weak spectral mapping property (WSMP) , where A is the generator of a ₀-semigroup in a Banach space X, μ is a measure, and μ̂(A) is defined by the Phillips functional calculus. We consider the special case when X is a Banach algebra and the operators , t ≥ 0, are multipliers.
We establish an inversion formula of Post-Widder type for -multiplied vector-valued Laplace transforms (α > 0). This result implies an inversion theorem for resolvents of generators of α-times integrated families (semigroups and cosine functions) which, in particular, provides a unified proof of previously known inversion formulae for α-times integrated semigroups.
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