8 problems

Let $f$ be a smooth uniformly bounded function, and let $\mathbb{U}$ and $\mathbb{V}$ be the two functions associated with $f$ in the paper's Bellman-function formulation.…

Let $f$ be a smooth uniformly bounded function, and let $\mathbb{B}_\varepsilon$ denote the minimal function associated with $f$ on the domain $\Omega_\varepsilon$. $C^{1,1}$ regul…

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

Let $\Omega_0$ be a compact convex subset of $\mathbb{R}^2$ with non-empty interior. Let $\Omega_1$ be a convex open set such that … Set $\Omega=\Omega_0\setminus\Omega_1$. For a b…

Let $f\in C_0^\infty(\mathbb{R}^2)$ and $g\in C_0^\infty(\mathbb{R}^2)$, let $p>2$, and let $p'=p/(p-1)$. Write their heat extensions on $\mathbb{R}^3_+$ using coordinates…

Symmetry conjecture. $C_{p'}=C_p$, or equivalently,