  We study complex distribution spaces given over a bounded Lipschitz domain
$\Omega$ and associated with an elliptic differential operator $A$ with
$C^{\infty}$-coefficients on $\overline{\Omega}$. If $X$ and $Y$ are
quasi-Banach distribution spaces over $\Omega$, then the space $X(A,Y)$ under
study consists of all distributions $u\in X$ such that $Au\in Y$ and is endowed
with the graph quasi-norm. Assuming $X$ to be an arbitrary Besov space or
Triebel--Lizorkin space over $\Omega$, we find sufficient conditions for $Y$
under which the interpolation between the spaces $X(A,Y)$ preserves their
structure, these spaces are separable, and the set
$C^{\infty}(\overline{\Omega})$ is dense in them. We then explicitly describe
the spaces obtained by the real, complex, and $\pm$ interpolation between the
spaces under study. We apply these spaces to general elliptic problems with
rough boundary data by proving the Fredholm property for bounded operators
induced by these problems and defined on certain spaces $X(A,Y)$. Specifically,
we establish the maximal regularity of solutions to some elliptic problems with
Gaussian white noise in boundary conditions. Quasi-Banach distribution spaces
are involved in the concept of $X(A,Y)$ for the first time. Our results are new
even for inner product Sobolev spaces of integer-valued order.
