9 problems

Vitushkin's conjecture. The set $E$ is non-removable for bounded holomorphic functions if and only if its orthogonal projections have positive length in a set of directions of posi…

Let $\Gamma\subset\mathbb{R}^2$ be a rectifiable curve, and let $E\subset\Gamma$ be compact with positive one-dimensional Hausdorff measure, $\mathcal{H}^1(E)>0$. Denjoy's conjectu…

Let $E\subset\mathbb{R}^{n+1}$ be compact. For the $Gamma_{Theta^{1/2}}$ capacity, impose … where $nabla=(\nabla_x,\partial_t)$ is the full gradient. Lebesgue comparability conject…

Let $U$ be a bounded open subset of $\mathbb C$, let $x\in U$, let $t=0$, and let $\phi$ be an admissible function. Let (a) denote the full-area-density condition and (c) the conve…

Let $U$ be a bounded open subset of $\mathbb C$ with $x\in U$, let $Y$ be the closure of $U$, let $t$ be a non-negative integer, and let $\phi$ be an admissible function. Denote by…

Let $E\subset\mathbb{R}^2$ be compact. Its analytic capacity is denoted by $\gamma(E)$; moreover, $\gamma(E)=0$ if and only if $E$ is removable for bounded analytic functions, mean…

Let $K_1$ and $K_2$ be compact subsets of the plane, and let $\alpha$ denote continuous analytic capacity. Vitushkin's quasiadditivity conjecture. There exists a constant $C$ such…

Younsi–Ransford conjecture. For all $z_1,\dots,z_n,w_1,\dots,w_m$, the function $R(r)$ is decreasing on $0<r<\delta/2$.

Monotonicity conjecture. Fix $Z=(z_1,\dots,z_n)$ and $m\in\{1,\dots,n-1\}$. Then $R(Z,r,m)$ is a decreasing function of $r$.