We derive local estimates of positive solutions to the conformal $Q$-curvature equation $$ (-��)^m u = K(x) u^{\frac{n+2m}{n-2m}} ~~~~~~ in ~ ��\backslash ��$$ near their singular set $��$, where $��\subset \mathbb{R}^n$ is an open set, $K(x)$ is a positive continuous function on $��$, $��$ is a closed subset of $\mathbb{R}^n$, $2 \leq m < n/2$ and $m$ is an integer. Under certain flatness conditions at critical points of $K$ on $��$, we prove that $u(x) \leq C [{dist}(x, ��)]^{-(n-2m)/2}$ when the upper Minkowski dimension of $��$ is less than $(n-2m)/2$.

57 pages