Improved Hardy Inequality when orthogonal to radial functions?

Question

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 $p$.

Answer 1

Submitting as an answer incase anyone stumbles upon this. I still don't know if this is sharp, but it was sufficient for my needs. Should anyone find a wider answer I'll be happy to accept their answer. Suppose $q$ is an exponent for which the Poincaré inequality holds

$$\int_{\mathbb{S}^d} |u-\bar{u}|^{q}\ d\mu\lesssim_{q,d} \int_{\mathbb{S}^d}|\nabla_{\mathbb{S}^{d}} u|^q\ d\mu$$

where $\bar{u}$ is the average of $u$ on the sphere and $d\mu$ is the typical probability measure on $\mathbb{S}^{d}$. If $u:\mathbb{R}^{d+1}\to \mathbb{C}$ is Schwartz and, in the terminology of the question, $P^{\perp}u=u$, then for each $r>0$ we know that $\theta\mapsto u(r\theta)$ is mean zero on $\mathbb{S}^{d}$. So expanding into polar coordinates we see that $$ \begin{align}\| |x|^{-1}u\|_{L^p(\mathbb{R}^{d+1})}^p &= \int_{0}^{\infty}r^{d-p}\int_{\mathbb{S}^{d}}|u(r\theta)|^{p}\ d\mu dr \\ &\lesssim \int_{0}^{\infty}r^{d-p}\int_{\mathbb{S}^{d}}|r(\nabla_{\mathbb{S}^{d}}u)(r\theta)|^{p}\ d\mu dr\\ &\lesssim || \nabla u||_{L^p(\mathbb{R}^{d+1})}^p\end{align}$$

So now the questions boils down to when do we have a Poincaré inequality for the sphere. The typical Rellich-Kondrachov argument shows that the Poincaré inequality holds on $S^{d}$ for $1<q<d$

$$\int_{\mathbb{S}^{d}} |u - \bar{u}|^q\ d\mu\le \int_{\mathbb{S}^{d}} |\nabla u|^q\ d\mu.$$ The existence of a Green's function for the Laplacian can be used to show that it holds for $q=1$. These details can be found in, for example:

E. Hebey, MR 1481970 Sobolev spaces on Riemannian manifolds, ISBN: 3-540-61722-1.

Now, using say, the results of

Jean Dolbeault, Maria J. Esteban, Michal Kowalczyk, and Michael Loss, MR 3011461 Sharp interpolation inequalities on the sphere: new methods and consequences, Chin. Ann. Math. Ser. B 34 (2013), no. 1, 99--112.

One can show that this inequality holds for $q\in (2, \frac{2d}{d-2}]$ if $d\ge 3$ or $q\in (2,\infty)$ for $d=2$ for functions of mean zero on the sphere.

1. This means that such an estimate holds when $q\in (1,\frac{2d-2}{d-3})$ on $\mathbb{R}^d$, $d\ge 4$.
2. This means that such an estimate holds when $q\in (1,\infty)$ on $\mathbb{R}^3$. The case $q=2$ comes from expanding into harmonic polynomials, and noting that any such function with $P^{\perp}u=u$ is orthogonal to constants.