The Hölder 1/2 a priori estimate conjecture for the Hilbert transport equation

Let $\theta$ be a bounded solution on $[0,T]$ of

$$\partial_t\theta+H\theta\,\partial_x\theta=0,$$

where $H$ is the Hilbert transform. The $C^{1/2}$ Hölder seminorm measures the quotient $|\theta(t,x)-\theta(t,y)|/|x-y|^{1/2}$. The Hölder 1/2 a priori estimate conjecture. There is a universal constant $C$ such that

$$\sup_{x,y\in\mathbb R}\frac{\theta(T,x)-\theta(T,y)}{|x-y|^{1/2}}\leq \frac{C\|\theta\|_{L^\infty}^{1/2}|x-y|^{1/2}}{T^{3/2}}.$$

The conjecture is motivated by the exact cusp profile $-|x|^{1/2}-Ct$ and is intended to describe immediate Hölder regularization. The source also explains that vanishing-viscosity limits are expected to satisfy the same estimate, and that such an estimate would have consequences for critical and supercritical dissipative variants.