Given $q>1$, consider the critical Sobolev space $W^{n/q,q}(\mathbb{R}^n)$, which fails to embed in $L^{\infty}(\mathbb{R}^n)$. I'm wondering if we can recover some critical estimate by considering a bilinear expression and by restricting to the radial component. For example, one could ask if the estimate $$ \|f\,\nabla f\|_{L_r^2L_{\theta}^p}\lesssim\|f\|^2_{W^{n/q,q}\,\cap\,H^1} \label{1}\tag{*} $$ holds true for $p<2$, where we write $x\in\mathbb{R}^n$ as $x=r\theta$ with $r\in\mathbb{R}^+$, $\theta\in S^{n-1}$.

Is estimate \eqref{1} actually true? Thank you for any suggestion.


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