10 problems

Let $u(t)$ be a solution of the cubic nonlinear Schrödinger equation on $\mathbb{T}^2$, and let $s>0$ with $s\neq 1$. For every $\epsilon>0$, let $C_\epsilon$ be a constant. Bourga…

Let $x'\in\mathbb R^{d-1}$, let $\alpha$ be a parameter, and let $\gamma(x',\alpha)$ define a line complex. Consider the X-ray transform … Assume that the support of $\chi$ is cont…

Müller–Thiele–Vallarino endpoint conjecture. The Sobolev estimate for solutions to the wave equation on a Damek–Ricci space remains valid at both endpoint regularity exponents abov…

Sharp Sobolev regularity conjecture. The operator $\mathfrak R$ boundedly maps $L^p$ to $L_\alpha^p$ for $\alpha \le \alpha(p)$. The sharp $L^p$ regularity estimate is open in high…

Finite-type Sobolev improving conjecture. The operator $A_{\gamma}$ maps $L^p$ to $L^p_{\alpha}$ for all $p\geq 2$ and

Non-degenerate curve Sobolev improving conjecture. If $\gamma$ is non-degenerate and $2n-2<p<\infty$, then

Finite-type extension conjecture. The conclusion of Theorem ftthm should remain true for every $p>4$, and the assumptions on $\pi_R$ should even be dispensable altogether. The curr…

Consider the operator $R_\beta$ and its adjoint $R_\beta^*$ in the model discussed in the source. For $\beta\ne0$, the right projection $\pi_R$ has only fold singularities, and The…

Let ${\mathcal M}\subset\Omega_L\times\Omega_R$ be a four-dimensional manifold whose projections are submersions, and let $({\mathcal N}^*{\mathcal M})'$ be the corresponding prime…

Holmer–Zworski higher-Sobolev-norm conjecture. The error bounds