13 problems

Lieb–Solovej conjecture. If $p>2$, then

Let $D^1$ denote the unit disc, let $k(z)=(1+alpha)(1-|z|^2)^\alpha$ with $\alpha>4$, and let $A^2(\mathbb{C},D^1,kd\mu)$ be the corresponding weighted Bergman space. The ARS chara…

Let $\omega$ be a radial weight on the unit disk, let $P_\omega$ be the associated Bergman projection, and let $L^p_\omega$ denote the corresponding weighted $L^p$ space. Let…

Let $(C_t)_{t\geq 0}$ be a composition semigroup on the weighted Bergman space $A^p_\omega$, with infinitesimal generator $\Gamma$, symbols $\varphi_t$, and Denjoy–Wolff point $b$.…

Let $\mathbb{B}_n$ denote the unit ball, and let $A^p_\alpha$ be the weighted Bergman space on $\mathbb{B}_n$, where $0<p<\infty$ and $\alpha$ is real. Zhu's reciprocal conjecture.…

Let $\mathcal{A}_\xi^p$ and $\mathcal{A}_{\xi+p}^p$ be the weighted Bergman spaces appearing in Theorem … can be extended to show that $\widetilde{L}$ extends to an isomorphism bet…

Let $H^2$ be the Hardy space on the unit disk, let $A^p_\beta$ denote the standard weighted Bergman space with exponent $p$ and weight parameter $\beta$, and let the Szegő kernel b…

Holomorphic stability conjecture. For any boundary-accessable measure $\mu$ on $\mathbb{D}$, the pair $(B^2(\mathbb{D},\mu),h^1(\mathbb{D}))$ is holomorphically stable if and only…

Let $A^2_1$ be the weighted Bergman space of holomorphic functions on the unit disk with norm … For $H\in A^2_1$, write … the smallest closed subspace of $A^2_1$ containing $H$ and…

Let $\alpha>0$ and let $(\mathbf c,\mathbf w)\in\Lambda_\alpha$, where $\Lambda_\alpha$ is the parameter set defined in the paper. Let $D_\alpha(\mathbf c,\mathbf w)$ denote the de…

Let $\alpha>1$ and let $f\in H^2$ be a holomorphic function on the unit disc $\mathbb{D}$. For $\beta>1$, define … where…

For $0<p\leq 2$, let $H^p$ be the Hardy space of holomorphic functions on the unit disc $\mathbb{D}$, with norm … and let $A_{2/p}^2$ be the weighted Bergman space with norm … wher…

Let $\omega$ be a radial weight, let $0<p<\infty$, and let $k\in\mathbb{N}$. For a holomorphic function $f\in\mathcal{H}(\mathbb{D})$, write $A^p_\omega$ for the corresponding weig…