9 problems

The kinetic Schauder estimate. The displayed estimate asserts a uniform interior Hölder bound for the velocity Hessian of smooth solutions in terms of the solution, the coefficient…

Let $\gamma_1$ and $\gamma_2$ be critical invariant circles in the same universality class, and let $h_{\gamma_i}$ be their conjugacies to the reference dynamics. Write…

Fix $\theta\in\mathbb{R}$, and let $utheta$ be the invariant measure associated with the solution $V^{(\theta)}$ of the SPDE started from the constant initial profile $\theta$. Assu…

Let $p\in(2,\infty)$, let $B_1$ be the unit ball, and let $\Delta_p u\coloneqq\operatorname{div}(|Du|^{p-2}Du)$. A function is locally of class $C^{1,\alpha}$ if it has locally Höl…

Optimal regularity conjecture. Any $K$-finite matrix coefficient of a unitary representation of $U$ is in $C^{(r,\delta)}(U_r)$, and this regularity is optimal.

Kinetic integro-differential Krylov–Safonov conjecture. There exist constants $\alpha>0$ and $C$, depending only on the dimension, $\lambda$, and $\Lambda$, such that

Kinetic Krylov–Safonov conjecture. There exist constants $\alpha>0$ and $C$, depending only on the dimension and $\lambda,\Lambda$, such that

For any $\delta>0$, let $\mathcal{C}_{\mathfrak{s};\mathrm{loc}}^{-\delta}(\mathbf{R}_{>0}\times\mathbf{T}^{2})$ and $\mathcal{C}^{-\delta}(\mathbf{T}^{2})$ denote the correspondin…

Let $\theta$ be a bounded solution on $[0,T]$ of … where $H$ is the Hilbert transform. The $C^{1/2}$ Hölder seminorm measures the quotient $|\theta(t,x)-\theta(t,y)|/|x-y|^{1/2}$.…