  In this paper, we find a natural four dimensional analog of the moderate
deviation results for the capacity of the random walk, which corresponds to
Bass, Chen and Rosen \cite{BCR} concerning the volume of the random walk range
for $d=2$. We find that the deviation statistics of the capacity of the random
walk can be related to the following constant of generalized
Gagliardo-Nirenberg inequalities, \begin{equation*} \label{eq:maxineq} \inf_{f:
\|\nabla f\|_{L^2}<\infty} \frac{\|f\|^{1/2}_{L^2} \|\nabla f\|^{1/2}_{L^2}}{
[\int_{(\mathbb{R}^4)^2} f^2(x) G(x-y) f^2(y) \text{d}x \text{d}y]^{1/4}}.
\end{equation*}
