  In this article, we consider the Kapustin-Witten equations on a closed
$4$-manifold. We study certain analytic properties of solutions to the
equations on a closed manifold. The main result is that there exists an
$L^{2}$-lower bound on the extra fields over a closed four-manifold satisfying
certain conditions if the connections are not ASD connections. Furthermore, we
also obtain a similar result about the Vafa-Witten equations.
