18 problems

A divergence-form elliptic operator has bounded measurable coefficients satisfying a Carleson measure condition of the type arising from changes of variables in smooth elliptic or…

Sharpness conjecture. The constant $e$ is sharp in Uchiyama's lemma, and the constant $2e$ is sharp in the Carleson embedding theorem for the disk.

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…

Dahlberg–Kenig–Pipher conjecture. No smallness assumption on the Carleson norm should be needed for solvability of the regularity problem in $L^p$ for such operators.

Let $H^p$ denote the Hardy space on the unit disk, let $\mu$ be a positive Borel measure on $[0,1)$, and let $\mathcal{DH}_\mu$ denote the derivative-Hilbert operator associated wi…

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…

GLW's conjecture. The description obtained for boundedness of $A_{\mu,s}:H^p\rightarrow L^q(\mathbb{S}_n)$ should hold for all possible values of $p,q,s$.

Carleson-condition necessity conjecture. In general, the Carleson condition is necessary for the boundedness of

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

Carleson packing conjecture. Every intrinsic Lipschitz graph $\Gamma\subset\mathbb{H}$ satisfies this condition for $\eta\geq4$. The conjecture concerns whether vertical oscillatio…

The Carleson embedding conjecture. This condition is necessary and sufficient for Carleson measures on $\mathcal{D}$. Combining the known result of Arcozzi, Rochberg, and Sawyer wi…

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 $\x03bc$ be a measure on $(-1,1)^\x221e$. For $s e(0,ldots)$, define … The measure $\x03bc$ is Carleson for $H^2({\mathbb T}^\x221e)$ if the reproducing-kernel testing conditio…

Converse conjecture. Converses to both the Carleson measure estimate and the $\varepsilon$-approximability theorem should hold, or perhaps a converse to their combination should ho…

Let $G$ be an open subset, with components whose harmonic measures are considered. A positive measure $\mu$ on $G$ is a Carleson measure when it satisfies the relevant Carleson emb…

Let $G$ be a domain conformally equivalent to a circular domain $W$, and let $\phi:G\to W$ be a conformal map. A positive measure on $G$ is a Carleson measure if it satisfies the C…