16 problems

Lorentz-space approximation and Gelfand number conjecture. Formula…

Let $LM^{p,r;\kappa}(\mathbb R^n)$ denote the Lorentz–Morrey space, and let $M^{p;\kappa}(\mathbb R^n)$ and $\operatorname{BMO}(\mathbb R^n)$ be the Morrey and bounded mean oscilla…

Duality conjecture. Under the framework of Hassell–Sikora, Hassell–Nardulli–Sikora, and Li, the range of $p$ for which the reverse Riesz inequality is bounded should exclude a port…

Local boundedness conjecture. Under these hypotheses, $u\in L^\infty(B_1)$.

Fix $0\leq\alpha<\infty$, and let $\nu$ be a positive, measurable function such that, for … $\mu(x)\,dx$ is a locally finite, doubling Borel measure on $\mathbb R^n$ with doubling…

Lorentz-Morrey Riesz-potential conjecture. If $p_1$ and $q_1$ satisfy

Optimal logarithmic power conjecture. If $K'<\infty$, then there is a constant $C=C(s_1,\dots,s_n,p,n,d,\psi)$ such that every $f\in\mathscr C_0^\infty(\mathbb R^n)$ satisfies

Let $1\leq p_1,\dots,p_m<\infty$ and define $p$ by … Let $w_1,\dots,w_m$ and $\nu$ be weights such that $(w_1,\dots,w_m,\nu)\in A_{\vec P}^{\mathcal R}$, and let $v$ be a weight sa…

Let $1<\alpha<2$, and consider the problem of determining the optimal Lorentz space condition on a real-valued random variable $X$ that guarantees…

Exponential growth conjecture. The ratio $R_{p,n}$ grows exponentially for every $0<p<\infty$.

Weighted Lorentz dispersive estimates conjecture. For every $\alpha\ne0$ and $y\in\mathbb{R}^3$, one has

Konyagin's conjecture. The Lorentz space

Let $Y=L\log L$ denote the Lorentz space, and let a $\mathcal{C}$-space mean a rearrangement-invariant space on which the relevant Carleson operator has the required weak-$L^1$ bou…

Entropy-hull conjecture. Then

Let $X=\dot B^s_{p,a}$ and $Y=L^{q,a}$ be Besov and Lorentz spaces, respectively, in the critical embedding regime … Assume that Assumption 1 refers to the approximation property u…

Let $f=\sum_{n=0}^\infty f_n\chi_n\ge 0$, where $\chi_n$ denotes the radial function supported on words of length $n$, and let $s'$ be the conjugate exponent to $s$, so that…