11 problems

Let ${\rm {\mathcal{D}}}$ denote the classical Dirichlet space, let $f\in {\rm {\mathcal{D}}}$ be outer, and let $\mathcal{Z}(f)$ be its boundary zero set. Writing $c_m$ for the ca…

Let $\omega$ be a positive superharmonic weight on $\mathbb{D}$, and let $f\in\mathcal{D}_\omega$. Write $[f]_{\mathcal{D}_\omega}$ for the smallest closed shift-invariant subspace…

Let $w_n$ be given by equation $(15)$, let ${\mathcal H}_{\mathbf w}$ be the corresponding Hilbert space, and let $Mult({\mathcal H}_{\mathbf w})$ denote its multiplier algebra. A…

Let $\mathbb B_n$ be the unit ball in $\mathbb C^n$, let $\mathbb S_n$ be its boundary sphere, and let $\mathcal{Z}(p)$ denote the zero set of a polynomial $p$. A polynomial is cyc…

Extension conjecture. The same conclusion as Corollary should hold for $0<a\le 1/2$. The conjecture extends the known Carleson-sequence dichotomy to the presently untreated paramet…

Let $f$in the classical Dirichlet space $\mathcal{D}$ be outer, and let $Z(f)$ denote its boundary zero set. Suppose that $Z(f)$ has logarithmic capacity $0$. Brown and Shields con…

Weak Bakry–Émery conjecture. On a non-compact metric graph with finitely many edges, the weak Bakry–Émery curvature condition holds for every $t>0$.

Wu's conjecture. The conclusion of Theorem B remains true for $2<p<\infty$.

Let $(M,d, u)$ be the metric-measure space and let $V$ and $V{epsilon}(t)$ be as defined above. Assume that $textup{Vol}(V; alpha1, alpha2)$ and…

Let $\mathcal D^p_{p-1}$ denote the Dirichlet type space of analytic functions on the unit disk whose derivatives belong to $A^p_{p-1}$. A measure $\mu$ is a Carleson measure for a…

Let $f_a=(1-z)^a$, where $a\in\mathbb{C}$ has positive real part, and let the optimal approximants to $1/f_a$ be defined in the same setting as in the formula for $f_N=(1-z)^N$. Th…