99 problems

Let $L^2_{+}(\mathbb{T})$ and $L^2_{+}(\mathbb{R})$ denote the positive-frequency Hardy subspaces of $L^2(\mathbb{T})$ and $L^2(\mathbb{R})$, respectively. Consider the defocusing…

Krzyż's conjecture. For every $f\in\Omega$ and every $n\geq 1$,

Let $H^2$ be the Hardy space on the unit disk $\mathbb D$, let $f,g\in H^2$, and let $T_f$ and $T_{\overline g}$ denote the corresponding Toeplitz operators on $H^2$. For…

Let $p\in[1,\infty]$ and let $q$ be its conjugate. For $\Omega=\mathbb{B}^{n}$ or $\mathbb{H}^{n}$, suppose that $u=\mathcal{P}_{\Omega}[\phi]$ with…

Let $\mathbb D$ be the unit disc and $H^2(\mathbb D)$ its Hardy space. For an analytic self-map $\varphi$ of $\mathbb D$, define the composition operator $C_\varphi$ by … Let…

Let $H^p$ denote a product Hardy space, and call a function a rectangle atom when it satisfies the relevant support, size, and separate cancellation conditions on a product of ball…

For $1<p<\infty$, let $P$ denote the Riesz projection from $L^p$ onto the classical Hardy space $H^p$, and let $\mathcal{B}(L^p)$ be the bounded operators on $L^p$. Gohberg–Krupnik…

Gilbert's conjecture. There exist $\eta$ and $\mathfrak{H}_0$ such that

Let $X$ be a separable Banach function space over the unit circle, and let $H[X]$ be the corresponding abstract Hardy space. Denote by $P$ the Riesz projection and by…

Let $1\leq p<\infty$, let $\varphi$ and $\psi$ be analytic self-maps of the unit disk $\mathbb{D}$, and let $C_\varphi f=f\circ\varphi$ and $C_\psi f=f\circ\psi$ denote the associa…

Let $({\mathcal X},d,\mu)$ be an RD-space, let $f\in H^1_{\operatorname{at}}({\mathcal X},d,\mu)$, and let $g\in \operatorname{BMO}({\mathcal X},d,\mu)$. The product $g\times f$ is…

Let ${\rvert E,b^\pm\rangle}={\rvert E^\pm\rangle}\in\Phi^\times_\mp$ be the Lippmann–Schwinger kets, and let the corresponding energy wave functions be … They are smooth functions…

Let $D$ be a bounded domain in ${\mathbb R}^n$ with boundary of class $C^{1/p}$, where $0<p\leq 1$. Let $G(f)$ and $N(f)$ denote the solutions of the Dirichlet and Neumann problems…

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 $H_p$ be the Hardy space, where $1\leqslant p<\infty$ and $p\ne 2$. A norm one projection on $H_p$ is a bounded projection $P:H_p\to H_p$ with norm one. Wojtaszczyk's conjectur…

Let $I$ be an ideal of polynomials in several variables, viewed with the relative topology induced by the Hardy space of the unit polydisk. Let $V(I)$ denote the variety of $I$, an…

Endpoint boundedness conjecture. If $g\in\Lambda(p,1/p)$, then

Near-isometric duality conjecture. For every $n\times n$ matrix $A$ with $n>2$,

Bagchi's criterion. The Riemann Hypothesis holds if and only if $E$ belongs to the closed linear span of the functions $G_k$, $k\geq 2$, in $H^2(\mathbb{C}_{1/2})$. This is an equi…

Let $H^2(\mathbb D^2)$ be the Hardy space on the bidisk and let $M\subset H^2(\mathbb D^2)$ be a submodule. For each integer $k\geq 0$, let $\Sigma_k(M)$ denote the numerical invar…

Let $H^p$ denote the Hardy space on the unit disk, let $d54b$ be the unit circle, and let $f\in H^p$ satisfy $f(0)\neq 0$. An inner function is a bounded analytic function on the u…

Virtanen's conjecture. The Toeplitz operator $T_a$ on $H^1(\mathbb{T})$ should be Fredholm.

Let $0<p<1$, and let $P:H^p(\mathbb T)\to H^p(\mathbb T)$ be a contractive projection satisfying $P\mathbf 1=\mathbf 1$. Boundedness conjecture. The operator $P$ should automatical…

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

Let $H^p$ be the Hardy space, let $g$ be a symbol, and define the generalized Hilbert operator by … For $p>1$, let $\Lambda(p,1/p)$ denote the corresponding mean Lipschitz space. G…