As shown in [Strauss: Existence of solitary waves in higher dimensions][1], Strauss introduces the Stauss lemma. Precisely speaking, we have the following theorem:

**Theorem** Let $N \ge 2$, every radial function $u \in H^1(\mathbb{R}^N)$ is almost everywhere equal to a function $U(x)$, continuous for $x \not = 0$, such that
\begin{equation}
|x|^{\frac{N-1}{2}} U(x) \lesssim \| u \|_{H^1},
\end{equation}
where the constant only depends on $N$.

In fact, following the argument above, we can get more precise estimate. Concretely, by the equation
\begin{equation}
-(r^{N-1} u^2)_r = -(N-1) r^{N-2} u^2 - 2 r^{N-1} u u_r \le -2 r^{N-1} uu_r,
\end{equation}
we integrate over $[r,+\infty)$ to obtain that
\begin{equation}
r^{N-1}u^2(r) \lesssim \int_r^\infty s^{N-1} |u(s)| |u_s(s)| ds \lesssim \| u \|_{L^2} \| \nabla u \|_{L^2},
\end{equation}
thus $|x|^{\frac{N-1}{2}} U(x) \lesssim \| u \|_{L^2}^\frac{1}{2} \| \nabla u \|_{L^2}^\frac{1}{2}$.

My question is that intuitively speaking, it seems that the weight $|x|^{\frac{N-1}{2}}$ on the LHS can be controlled by the "half-gradient" on RHS. However, if the power of weight $|x|^{\alpha}$ becomes smaller, can we expect the less gradient on the right? Precisely speaking, if $\alpha <\frac{N-1}{2}$, can we have
\begin{equation}
|x|^{\alpha} U(x) \lesssim \| u \|_{L^2}^{1-\beta} \| \nabla u \|_{L^2}^\beta, \; |x| \ge 1
\end{equation}
where  $\beta<\frac{1}{2}$?


  [1]: https://Existence%20of%20solitary%20waves%20in%20higher%20dimensions