The representation of integers by positive quaternary quadratic forms

Let $f(x,y,x,w)=x^2+y^2+z^2+D\omega^2$, where $D>1$ is an integer such that $D\ne d^2$ and $\sqrt{\mathstrut n}\big/\sqrt{\mathstrut D}=n^{\theta},0<\theta<1/2$. Let $r_f(n)$ be the number of representations of $n$ by $f$. It is proved that
$$ r_f (n)=\pi^2\frac n{\sqrt D}\sigma_f(n)+O\biggl(\frac{n^{1+\varepsilon-c(\theta)}}{\sqrt D}\biggr), $$
where $\sigma_f(n)$ is the singular series, $c(\theta)>0$, and $\varepsilon$ is an arbitrarily small positive constant.