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Fourier Multipliers and Integro‐Differential Equations in Banach Spaces

Operator‐valued Fourier multiplier theorems are used to establish maximal regularity results for an integro‐differential equation with infinite delay in Banach spaces. Results are obtained under general conditions for periodic solutions in the vector‐valued Lebesgue and Besov spaces. The latter scale includes in particular the Hölder spaces Cα, 0 < α < 1.