9 problems

Let $({\mathcal X},d,\mu)$ be an RD-space, let $f\in H^1_{\operatorname{at}}({\mathcal X},d,\mu)$, and let $g\in \operatorname{BMO}({\mathcal X},d,\mu)$. The product $g\times f$ is…

Let $LM^{p,r;\kappa}(\mathbb R^n)$ denote the Lorentz–Morrey space, and let $M^{p;\kappa}(\mathbb R^n)$ and $\operatorname{BMO}(\mathbb R^n)$ be the Morrey and bounded mean oscilla…

Vasilyev's conjecture. For every positive measurable function $f$ on $\mathbb R^n$, one has $\log f\in BMO(\mathbb R^n)$ if and only if there exist $c>1$, $0<\rho<1$, and a functio…

Consider the scalar vorticity formulation of the two-dimensional Euler system, with vorticity $\omega$ and velocity $v$ satisfying … and initial condition … Here $BMO$ denotes the…

Let $Q\subset\mathbb{R}^d$ be a cube, let $\mathrm{BMO}_{\varepsilon}^{\mathrm{dyad}}(Q)$ denote the dyadic BMO class with parameter $\varepsilon$, and define … An extension…

Let $Q$ be a $d$-dimensional cube and let $f^*$ denote the monotone rearrangement of $f$ on $[0,1]$. Let $c(d)$ be the constant appearing in the dimension-reduction conjecture for…

The dimension-reduction conjecture. There exists $c=c(d)>1$ such that

Exponential Orlicz discrepancy conjecture. The estimate above should hold for suitable $N$-point sets in every dimension $d\geq 3$. This conjecture predicts an improvement over the…

Asymptotic formula conjecture. For every $f\in SBV_{\rm loc}(\mathbb{R}^n)$,