Regularity results for elliptic and parabolic systems of partial differential equations

We study Cauchy problems associated to elliptic operators acting on vector-valued functions and coupled up to the first-order. We prove pointwise estimates for the spatial derivatives of the semigroup associated to these problems in the space of bounded and continuous functions over ${\mathbb{R}}^d$. Consequently, we deduce relevant regularity results both in Hölder and Zygmund spaces and in Sobolev and Besov spaces.