  We consider one-dimensional inhomogeneous parabolic equations with
higher-order elliptic differential operators subject to periodic boundary
conditions. In our main result we show that the property of continuous maximal
regularity is satisfied in the setting of periodic little-H\"older spaces,
provided the coefficients of the differential operator satisfy minimal
regularity assumptions. We address parameter-dependent elliptic equations,
deriving invertibility and resolvent bounds which lead to results on generation
of analytic semigroups. We also demonstrate that the techniques and results of
the paper hold for elliptic differential operators with operator-valued
coefficients, in the setting of vector-valued functions.
