A formulation of Hardy's Inequality on $\mathbb{R}^d$ states that when $0\le s<d$ and $1< p<\tfrac{d}{s}$ that

$$ \|\frac{u}{|x|^s}\|_{L^p(\mathbb{R}^d)}\lesssim_{s,p,d} \||\nabla|^s u\|_{L^p(\mathbb{R}^d)}$$

when, say, $u:\mathbb{R}^d\to \mathbb{C}$ is Schwartz.

- When $d\ge 3$, $s=1$, and $p=2$ the sharp constant is known: $$\frac{(d-2)^2}{4}\int_{\mathbb{R}^d}\frac{|u|^2}{|x|^2}\ dx\le \int_{\mathbb{R}^d} |\nabla u|^2\ dx$$ for all $u$ Schwartz and holds for no smaller constant.
- However, we can make improvements if we restrict our focus. Let $P:L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)$ denote the orthogonal projection onto spherically symmetric functions, given by $$Pu(x)=\frac{1}{\omega_{d}}\int_{S^{d-1}}u(|x|\theta)d\sigma(\theta).$$ If we define $P^{\perp}:=I-P$, then $$ \frac{d^2}{4}\int_{\mathbb{R}^d} \frac{|P^{\perp}u|^2}{|x|^2}\ dx\le \int_{\mathbb{R}^d} |\nabla P^{\perp}u|^2\ dx$$ for all Schwartz $u$, as proved by T. Ekholm and R.L. Frank in
T. Ekholm, R. L. Frank, On Lieb-Thirring inequalities for Schrodinger operators with virtual level, Commun. Math. Phys. 264 (2006), 725–740

My question, if the community could provide any references or insight, is whether the range of admissible exponents in Hardy's inequality is improved for those Schwartz functions $u$ so that $P^{\perp}u=u$. Namely, with $d\ge 3$ and, say, $s=1$, for which $p$ does $$ \| \frac{u}{|x|}\|_{L^p(\mathbb{R}^d)}\lesssim_{p,d} \| |\nabla| u\|_{L^p(\mathbb{R}^d)} $$ hold for all Schwartz $u$ with $P^{\perp}u=u$. Restricting to this class of functions is certainly enough to break through the sharp constant in the Hardy inequality above, I'm curious to figure out if it's been asked or answered if this also affects the range of permissible parameters as well, as the known examples that break Hardy's inequality outside its range of permissible parameters seem to all be radial.

My attempts so far have fallen short, insofar as I only know how to prove Hardy's inequality for $p\ne 2$ by something like Schur's test with weights (for which I can't correctly identify the effect that $P^{\perp}$ has on the integral kernel). The proof by Ekholm and Frank uses, from what I understand, fundamentally $L^2$-based methods, and I don't know how to push them to other values of $p$.