22 problems

Quotient-space decay conjecture. The spaces $\mathcal L^{1'}_v$ and $\mathcal L'_v$ possess the same decay properties as $\mathcal L^1_v$ and $\mathcal L_v$, respectively. The sour…

Weighted decay conjecture. There exists $\delta>0$ such that $h_L\in\mathcal L^1_{\tau_\delta}$. If true, this would support identifying $\mathcal L^1$ with a union of the weighted…

Let $\nu$ be a finite positive Borel measure on $[0,1]$, let $T_\nu:L^p(\mathbb{D})\to L^q(\mathbb{D})$ be the associated Shimorin-type operator, and let $\mathcal{C}$ be the bound…

Let $\nu$ be a finite positive Borel measure on $[0,1]$, and let $T_\nu:L^p(\mathbb{D})\to L^q(\mathbb{D})$ be the associated Shimorin-type operator. Define … where $c'$ is the dua…

Let $T$ be the integral operator associated with the interaction kernel defining a conditionally specified stationary multivariate time-series model, and let stationarity mean that…

Open problem 1. For large enough $r$, prove

Let $\mathbb{D}$ be the unit disk, let $\omega$ be a weight on $\mathbb{D}$, and define the Hardy-kernel operator … For an interval $I\subset\mathbb{T}$, let … where $|I|$ is norma…

Hille–Tamarkin conjecture. The extra logarithmic factor in this estimate can be removed, or even replaced by a logarithmic factor with a negative power.

Chalmoukis's conjecture. If $I_{g,a}:H^p\to H^q$ is bounded, then

The representation theorem. For every positive solution $\widetilde{u}$, there exists a Borel measure $\nu$ on $\mathbb{S}^{n-1}$ such that

Spectral conjecture. The spectrum $\sigma(L_{\nu})$ consists of exactly one negative eigenvalue and none or exactly one eigenvalue greater than $4$; more precisely,

Let $\mathcal{R}$ be an integral operator associated with a canonical relation whose left and right projections are denoted by $\pi_L$ and $\pi_R$. Suppose the only singularities o…

Let $L$ be either $L_1(0,1)$ or $L_2(0,1)$. For $\alpha\geq 0$, let $I^\alpha$ be a family of bounded linear operators on $L$ satisfying the interpolation conditions … the index la…

Let $d$ be the dimension and let ${\mathcal K}_r^d$ be the limiting integral operator associated with the normalized random geometric graph on $[-1,1]^d$, for $0<r<2$. Its second l…

Let $(X,d,\mu)$ be a bounded metric measure space with kernel $K$ and associated operator $T_K\colon L^2(X,\mu)\to L^2(X,\mu)$. Let $(X,d,\mu_n)$ have analogous kernel $K_n$ and op…

Exponential decay conjecture. The eigenvalues $\lambda_n$ and coefficients $\pi_n$ have these exponential decay asymptotics, and the asymptotic upper bound for the eigenvalues capt…

The persistence exponent is related to the largest eigenvalue of an integral operator, and the paper establishes Theorem, Theorem, and equations of type for the setting under consi…

Higher-eigenvalue asymptotic conjecture. For every $j=1,2,\ldots$,

Mityagin's quartic-model conjecture. The constant $L$ should coincide with the lowest eigenvalue of

Mityagin's asymptotic constant conjecture. There exists a constant $L>0$ such that

Let $\kappa$ be an irreducible kernel, let $\widehat\chi(\lambda\kappa)$ denote the dual susceptibility, and let $\lambda_{\mathrm{cr}}$ be the critical parameter. Analyticity and…

Let $K_a$ be the operator whose eigenvalues are $E_j(a)=1-\varphi_j(a)$, and suppose the scaled eigenvalue limits satisfy … The heuristic conjecture identifies the operator governi…