Abstract: We present a class of functional spaces that arose from the work of Barton and Mayboroda and called the Z-spaces, of certain functions of two variables $(t,x)\in (0,\infty)\times \R^n$. 

They are suited to characterize Besov spaces defined on \R^n$ and are well-suited to study some elliptic boundary value problems and parabolic evolution problems with rough coefficients. We shall present selected applications. This is based on works with Alex Amenta, Sebastian Bechtel, Moritz Egert, Hedong Hou and Luca Haardt.