Distribution spaces associated with elliptic operators

Abstract

We study complex distribution spaces given over a bounded Lipschitz domain $Ω$ and associated with an elliptic differential operator $A$ with $C^{\infty}$-coefficients on $\overlineΩ$. If $X$ and $Y$ are quasi-Banach distribution spaces over $Ω$, 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 $Ω$, 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 $C^{\infty}(\overlineΩ)$ 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.