14 problems

Let $I_{k,\rho}$ denote the discrete fractional integral operator … Here $k\in\mathbb{N}$ and $0<\rho<1$. Discrete fractional integral boundedness conjecture. For $0<\lambda<1$ and…

Cruz–Isralowitz–Moen conjecture. In the scalar case, the conclusion $I_\alpha:L^p(V^{p/q})\to L^q(U)$ remains valid if the hypothesis $\bar{\Psi}\in B_{q'}$ in the stated fractiona…

Generalized support theorem. A similar support theorem should hold for arbitrary, not necessarily radial, functions.

Let $0<\alpha<n$, let $1\leq p<n/\alpha$, and define $q$ by … Let $W\in\mathcal{A}_{p,q}$ be a matrix weight, and let $I_\alpha$ denote the fractional integral operator. The scalar…

Let $R_{+}^{\alpha}$ be the fractional Radon transform considered in the paper, with $(1-n)/2\leq \operatorname{Re}\alpha\leq 0$ and $\alpha\neq 0$. The relevant $L^p$-to-$L^q$ bou…

Fractional integral conjecture. If $\Omega$ is smooth and canceling, then the Riesz potential $\operatorname{I}_{\alpha}$ maps $W_1^\Omega$ continuously into…

Let $P$ be a polynomial of degree $d$, and let $\mathcal I(P)$ denote the set of all triples $(p,q,\lambda)$ for which the inequality … holds if and only if … This conjecture chara…

Bilinear fractional weight conjecture. The condition $\vec{w}\in A_{\vec{p},q}$ should imply

Let $I_\alpha$ be the fractional integral operator, let $u$ be a weight on $\mathbb R^n$, and let $M_{\Phi,\alpha}$ be the Orlicz fractional maximal operator … where…

Strong-type critical-exponent bump conjecture. Then

Let $0<\alpha<n$, let $1<p<\infty$, let $\bar{\Psi}\in B_{p'}$, and suppose $(u,\sigma)\in A_{p',p',\Psi}^\alpha$. Weak-type critical-exponent bump conjecture. Then … This is the e…

Let $0<\alpha<n$, let $1<p<\infty$, and let $I_\alpha$ be the Riesz potential. Cruz–Moen critical-exponent bump conjectures. The weak- and strong-type bump estimates proved for…

Let $0<\alpha<n$, let $1<p<\infty$, let $(u,\sigma)$ be a pair of weights, and let $I_\alpha$ and $M_\alpha$ denote the Riesz potential and fractional maximal operator, respectivel…

Weak-type fractional-integral conjecture. The fractional integral should satisfy