A Lorentz invariant sharp Sobolev inequality on the circle

We prove the following sharp Sobolev inequality on the circle $$\int_{\mathbb{S}^1} [4(v')^2 - v^2] \mathrm{d} \theta \geq - \frac{4\pi^2}{\int_{\mathbb{S}^1} v^{-2} \mathrm{d} \theta},$$ with the equality being achieved when $v^{-2} (\theta) = \frac{k\sqrt{1-\alpha^2}}{1+ \alpha \cos(\theta - \theta_o)}$where$k>0$, $\alpha \in (-1,1)$, $\theta_0 \in \mathbb{R}$. If $v$ vanishes somewhere on the circle, then $$4 \int_{\mathbb{S}^1} (v')^2 \mathrm{d} \theta \geq\int_{\mathbb{S}^1} v^2 \mathrm{d} \theta.$$ The basic tools to prove the inequality are the rearrangement inequality on the circle and the variational method. We investigate the variational problem of the functional $\mathcal{F}[v] = \int_{\mathbb{S}^1} [4(v')^2 - v^2] \mathrm{d} \theta$ under the constraint $\int_{\mathbb{S}^1} v^{-2} \mathrm{d} \theta = 2\pi$. An important geometric insight of the functional $\mathcal{F}$ is that it is invariant under the Lorentz group, since $\mathcal{F}[v]$ is the integral of the product of two null expansions of a spacelike curve parameterised by the function $v^{-2}$ in a lightcone in $3$-dim Minkowski spacetime. The global minimiser of $\mathcal{F}$ under the constraint is simply given by the spacelike plane section of the lightcone. We introduce a method which combines the symmetric decreasing rearrangement and the Lorentz transformation. This method isnot confined to the scope of this paper, but is applicable to other Lorentz invariant variational problems on $\mathbb{S}^{n}, n \geq 1$. As an example, we sketch a proof of the sharp Sobolev inequality on $\mathbb{S}^n, n\geq 3$ by this method.