On solvability of parabolic functional differential equations in Banach spaces (II)

In this paper, a parabolic functional differential equation is considered in the spaces $C(0, T; H^s_p (Q))$ for $s$ close to $1$ and $p$ close to $2$. The transformations of the space argument are supposed to be bounded in the spaces $H^s_p (Q)$ with small smoothness exponent and $p$ close to $2$. The corresponding resolvent estimate of the elliptic part of the operator is obtained in order to show that it generates a strongly continuous semigroup.