33 problems

Let $\mathbb{D}^2=\{z\in\mathbb{C}^2:|z_1|<1,\ |z_2|<1\}$, let $\alpha>0$, and let $\mathcal{D}_\alpha(\mathbb{D}^2)$ be the corresponding Dirichlet-type space. For every polynomia…

Let $A^2_\alpha$ be the weighted Bergman space on $\mathbb{D}$, with $\alpha>-1$, and let $M\subseteq A^2_\alpha$ be invariant under the multiplication operator $S f(z)=z f(z)$. De…

For every separable Hilbert space $\mathcal E$, every $B\in H^\infty_1(L(\mathcal E))$, and every $\alpha>-1$, the associated sub-Bergman spaces satisfy…

For weighted Bergman spaces $A_{\alpha}^{p}$, determine a necessary-and-sufficient characterization of the zero-free functions $f\in A_{\alpha}^{p}$ whose reciprocal $1/f$ also bel…

Determine precisely the exponents $0<p<\infty$ for which there exists a constant $C_p<\infty$ such that every Dirichlet polynomial $P$ satisfies…

For each $1\le p<\infty$ and $\alpha>-1$, determine exactly which analytic symbols $g$ make the Volterra-type operators $J_g$ and $I_g$ metrically bounded on the area Nevanlinna sp…

For the Möbius-invariant space $Q_K$ on the unit disc, determine whether the pointwise multiplier algebra satisfies $\mathcal{M}(Q_K)=H^\infty\cap Q_K^{\log}$, where…

Essential positivity conjecture. The operator $T_f$ is essentially positive if and only if

The converse compactness assertion. If the necessary condition (e1) and condition (e2) from Theorem … remains valid throughout this remaining parameter range. The paper presents th…

Let $X$ and $L$ be continuous domains, and let $T$ denote the domain used in the Scott function spaces $[X\to T]$ and $[T\to L]$. The paper considers finite-layer truncation maps a…

Tight fitting conjecture. (a) The embedding $A^p_\alpha\subset A^q_\beta$ is a tight fitting if and only if

Let $v$ be a weight on the unit disk and let $p$ be the parameter appearing in Proposition 1. Assume that the function $x\mapsto v(1-x)x^{\frac{1}{p}-\epsilon}$ is almost increasin…

Let $p>1$, let $\alpha>1$, let $0<s<1$, let $\mu$ be a positive Borel measure on the open unit disc $\mathbb D$, and let $F(p,p\alpha-2,s)$ and ${\mathfrak T}_{s,p}^{\infty}(\mu)$…

Let $p>1$, let $0<s<1$, let $\mu$ be a positive Borel measure on the open unit disc $\mathbb D$, and let $F(p,p-2,s)$ and ${\mathcal T}_{s,p}^p(\mu)$ denote the function and tent s…

Let $0<s<1$ and let $\mu$ be a positive Borel measure on the open unit disc $\mathbb D$. The spaces ${\mathcal Q}_s$ and ${\mathcal T}_{s,2}^2(\mu)$ are linked by the identity map…

Let $G$ and $w\in\mathcal{L}^1(\mathbb{T})$ be as in the core-carrier conjecture, let $\mu$ have the form, and let $\operatorname{core}(w)$ denote the core of $w$. For a singular m…

Let $G$colon $(0,\infty)\to(0,\infty)$ be the function occurring in the measure structure defined by, let $w\in\mathcal{L}^1(\mathbb{T})$ be a non-negative weight, and let … for ev…

Let $D$ be a bounded homogeneous domain in $\mathbb{C}^n$, let $\psi\in H(D)$, and let $\varphi$ be a holomorphic self-map of $D$. Write $\mathcal{B}(D)$ for the Bloch space and…

Let $D$ be a bounded homogeneous domain in $\mathbb{C}^n$, let $\psi$ be a holomorphic function on $D$, and let $\varphi$ be a holomorphic self-map of $D$. Write…

Let $\Omega$ be homogeneous of degree zero on $\mathbb{R}^d$, satisfy the vanishing moment condition, and let $A$ be a function on $\mathbb{R}^d$ whose derivatives of order one bel…

Let $0<p,q<\infty$, let $\omega\in\widehat{\mathcal{D}}$, and let $\nu$ be a positive Borel measure on $\mathbb{D}$. Let $\varphi$ and $\psi$ be analytic self-maps of $\mathbb{D}$.…

Let $k$ and $m$ be parameters satisfying $1\leq k<+\infty$ and $1<m<+\infty$, and let $p,q,p',q'$ satisfy … … with $m\leq q'<q<+\infty$. The weak mixed-norm non-inclusion conjectur…

Asymptotic balance conjecture. As $\epsilon\to0$,

Let ${}_{\Sigma}L_p$ denote the family of spaces associated with the surface $\Sigma$, and let $[\,\ ]_{\theta}$ and $(\,\ )_{\theta,r}$ denote the complex and real interpolation m…

Let $\Sigma$ be a smooth manifold with the $L_p$ restriction property, and let $\Pi_{\Sigma}$ denote the restriction map to $\Sigma$. Write…