Regularity results for elliptic and parabolic systems of partial differential equations

Abstract 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 $$\mathbb {R}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> . Consequently, we deduce relevant regularity results both in Hölder and Zygmund spaces and in Sobolev and Besov spaces.