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operator-valued fourier multipliers

Proceedings of the Edinburgh Mathematical Society (1)

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Maximal regularity for degenerate differential equations with infinite delay in periodic vector-valued function spaces
Carlos Lizama, Rodrigo Ponce
Journal:

Proceedings of the Edinburgh Mathematical Society / Volume 56 / Issue 3 / October 2013

Published online by Cambridge University Press:

21 August 2013, pp. 853-871
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Let A and M be closed linear operators defined on a complex Banach space X and let a ∈ L1(ℝ+) be a scalar kernel. We use operator-valued Fourier multipliers techniques to obtain necessary and sufficient conditions to guarantee the existence and uniqueness of periodic solutions to the equation
with initial condition Mu(0) = Mu(2π), solely in terms of spectral properties of the data. Our results are obtained in the scales of periodic Besov, Triebel–Lizorkin and Lebesgue vector-valued function spaces.