13 problems

Weighted bilinear Calderón–Zygmund conjecture. If the maximal operator satisfies the associated unweighted bilinear inequality, then

Weighted maximal-operator conjecture. If $w\in\mathcal A_{p(\cdot)}$, then $M$ is bounded on $L^{p(\cdot)}(w)$ under these hypotheses, allowing $p_+=\infty$.

Bilinear weighted maximal-operator conjecture. The bilinear maximal operator $\mathcal M$ satisfies

Diening–Hästö's conjecture. The maximal operator $M$ is bounded on $L^{p(\cdot)}(w)$ if and only if it is bounded on $L^{p(\cdot)}$ and $w\in\mathcal{A}_{p(\cdot)}$.

The mathcal{A}infty conjecture. Every reflexive variable Lebesgue space over a space of homogeneous type satisfies the condition $\mathcal{A}_\infty$.

Let $q(3)$ be a variable exponent on $[0,1]$, let $q_*(t)$ and $q^*(t)$ denote its rearranged lower and upper exponent functions, and let $L^{(q_-,4}$ be the corresponding grand Le…

Subunit rearranged embedding conjecture. Some version of Theorem should remain true for $0<4<1$.

Let $T_\Omega$ be the bilinear rough operator in Theorem, and let $p,q,r$ satisfy $1/r=1/p+1/q$. Theorem proves the corresponding weighted estimate in its stated range, while Corol…

Let $p(\cdot),q(\cdot),r(\cdot)$ be variable exponent functions as in Theorem, and let $D^s$ and $J^s$ denote the homogeneous and inhomogeneous fractional differential operators, r…

Sharp spherical-maximal exponent conjecture. The bound $p_+\leq n p_-$ is sharp. Establishing this through extrapolation would follow if the upper bound on the parameter $\delta$ i…

Endpoint weighted fractional-maximal conjecture. The unweighted endpoint estimate should remain valid in the weighted case:

Let ${p(07)}\in\mathcal P$ be a variable exponent and let $w$ be a weight. An $M$-pair $({p(07)},w)$ is a pair for which the maximal operator $M$ is bounded on both…

Massive essential spectrum conjecture. There exists a measurable function $p:\mathbb{T}\to(1,\infty)$ such that $S$ is bounded on $L^{p(\cdot)}(\mathbb{T})$ and…