Weak Compactness in Wk,∞

We characterize weak compactness in the Sobolev space Wk,∞(Ω). For non-reflexive spaces like Wk,∞, criteria beyond boundedness are required. By exploiting the von Neumann algebra structure of L∞ via Gelfand duality, we establish a unified theory. Our main result is a necessary and sufficient condition: a subset is relatively weakly compact if and only if it is bounded and its weak derivatives up to order k have uniformly small oscillation on a finite measurable partition of Ω. This provides a tool for analyzing nonlinear problems in these spaces.