Let $f \in L^1(\mathbf R^N \times \mathbf R^N) \cap L^p(\mathbf R^N \times \mathbf R^N)$. Let $\rho(x) = \int_{\mathbf R^N} f(x,v) dv$. Prove that \begin{equation} \|\rho\|_{L^q(\mathbf R^N)} \le C \|f|v|^2\|_{L^1(\mathbf R^N \times \mathbf R^N)}^\theta \|f\|_{L^p(\mathbf R^N \times \mathbf R^N)}^{1-\theta} \end{equation} where $C$ is some positive constant, $1\le p \le\infty$, $q = \frac{N(p-1)+2p}{N(p-1)+2}$, $\theta = \frac{1}{q}\frac{p-q}{p-1}$.
This inequality appears in this paper at equation (35) without any proof. I have no idea to prove it. Can anyone give some hints? TIA...
Note: From the definition of $q$ and $\theta$, we have $$\frac{1}{q-1}-\frac{1}{p-1} = \frac{N}{2},$$ $$\theta + \frac{1-\theta}{p} = \frac{1}{q}.$$