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We propose an approach based on perturbation theory to establish maximal L p -regularity for a class of integro-differential equations. As the left shift semigroup is involved for such equations, we study maximal regularity on Bergman spaces for autonomous and non-autonomous integro-differential equations. Our method is based on the formulation of the integro-differential equations to a Cauchy problems, infinite dimensional systems theory and some recent results on the perturbation of maximal regularity (see [2]). Applications to heat equations driven by the Dirichlet (or Neumann)-Laplacian are considered.

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