42 problems

Let $\alpha$ be a continuous $\|\cdot\|_1$-dominating normalized gauge norm, and let $H^{\alpha}$ and $M_{\alpha}(z^n)$ be the associated Hardy space and invariant subspace. Decomp…

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

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…

For $p\text{ in }[1,\text{infinity})$, consider the Jacobian operator … where $\mathscr{H}^p(\mathbb{R}^n)$ is the real Hardy space and $\dot{W}^{1,np}(\mathbb{R}^n,\mathbb{R}^n)$…

Let $p>2$, let $\mathbb{D}$ be the unit disk, and let $f$ be analytic in $\mathbb{D}$. Denote by $\|f\|_{H^p}$ the Hardy-space norm and by $\|f\|_{D_{p/2}}$ the norm of the corresp…

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

Let $H^2(\mathbb{D}^2)$ be the Hardy space on the bidisk, and let $f$ be a polynomial in this space with no zeros in the closed bidisk. For each multidegree, an optimal polynomial…

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

Weak-type level-set conjecture.

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…

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…

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

Let $\mathcal{S}_\mathcal{H}(K)$ be the class of normalized harmonic $K$-quasiconformal mappings, and write $f=h+\overline{g}$. Harmonic K-quasiconformal Hardy-order conjecture. On…

Let $\mathbb D$ be the unit disk, let $\mathcal{S}(\mathbb D)$ denote the holomorphic self-maps of $\mathbb D$, and for $\varphi\in\mathcal{S}(\mathbb D)$ let…

Let $\mu$ be a probability measure satisfying the Blaschke condition, with support generating a discrete subgroup of $PSU(1,1)$. Let $f_\nu(z)$ be a solution of the equation referr…

Critical exponent conjecture. One should have

Weighted maximal operator conjecture. For every $f\in H_1$, the operator $S^{\ast,\nabla}$ is bounded from $H_1$ to $L_1$. Moreover, if $\sup_{k\in\mathbb{N}}|A_{n_k}|=\infty$ and…

Saitoh's conjecture. If $D$ is not simply connected, then

Helson's conjecture. The function $f_\chi$ never has any zeros in its half-plane of convergence.

For $1\leq p<\infty$, let $p'=\frac{p}{p-1}$, let $P_+$ denote the Riesz projection on $L^{p'}(\mathbb T)$, and let $H^{4/p}$ be the Hardy space with exponent $4/p$. Brevig–Ortega-…

Let $H^{\alpha}$ be the Hardy space associated with a continuous rotationally symmetric norm $\alpha$, let $B$ be a finite Blaschke factor of degree $n$, and let…

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…

Fourier multiplier conjecture. The function $\mathcal{F}(\lambda)$ is a Fourier multiplier from $H^1(\mathbb{R}_+^2\times\mathbb{R}_+^2)$ into $L^1$ if and only if there exists a c…

Let $1<p<\infty$, $f\in H^p$, and $n\in\mathbb{N}$. If $f(0)\neq 0$, then the optimal polynomial approximant $q_{n,p}[f,1]$ is the polynomial of degree at most $n$ that best approx…

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…