Maximal regularity for time-fractional integro-differential equations with infinite delay on periodic Triebel–Lizorkin spaces

We study a linear evolution equation on the torus in which the time derivative is fractional of order between zero and one, the leading term is a closed operator on a complex Banach space, and an infinite-delay convolution acts on the operator part of the solution. The equation is posed as a periodic boundary value problem and is analysed on the scale of vector-valued periodic Triebel–Lizorkin spaces of functions with vanishing mean, the time derivative being the periodic Caputo–Weyl derivative. The delay term is defined spectrally, through the sequence of Fourier symbols of its kernel. This admits kernels that are not integrable at infinity — the Abel kernel in particular — which lie outside the usual Laplace transform framework. Under a hypothesis on the symbol sequence combining a uniform gap condition with discrete smoothness up to order three, we prove that the problem has maximal regularity precisely when an explicit sequence of complex numbers, built from the fractional symbol and from the kernel symbol, lies in the resolvent set of the operator, and the associated resolvent family is uniformly bounded. Because the underlying operator-valued multiplier theorem is valid on an arbitrary Banach space, the characterization requires neither the UMD property nor R -boundedness. We verify the hypothesis explicitly for the regularized Abel kernel and record a consequence for sectorial operators. When the order of the derivative equals one, the symbol reduces to the one appearing in earlier work on first-order integro-differential equations with infinite delay.