An Allard type regularity theorem for varifolds with H\"older continuous generalized normal

We prove that Allard's regularity theorem holds for rectifiable $n$-dimensional varifolds $V$ assuming a weaker condition on the first variation. This, in the special case when $V$ is a smooth manifold translates to the following: If $\omega_n^{-1}\rho^{-n}{\rm area}(V\cap B_\rho(x))$ is sufficiently close to $1$ and the unit normal of $V$ satisfies a $C^{0,\alpha}$ estimate, then $V\cap B_{\rho/2}(x)$ is the graph of a $C^{1,\alpha}$ function with estimates. Furthermore, a similar boundary regularity theorem is true.