12 problems

Wang's conjecture. The range of $\beta$ in Proposition can be extended to

Let $n\geq2$ and $\frac12<s<\frac n2$. Let $(f_j)_j$ be an orthonormal system in $H^s(\mathbb{R}^n)$, and let $\beta$ be the exponent in the maximal estimate denoted by … holds for…

Let $n\geq2$, $q\geq \frac{2(n+1)}{n-1}$, and let $(f_j)_j$ be an orthonormal system in the homogeneous Sobolev space $\dot{H}^s(\mathbb{R}^n)$. For a sequence of coefficients…

Let $n\geq2$, let $(1/p,1/q)$ lie on the line segment $(O,A]$, and set $s$ by … For an orthonormal family $\{f_j\}_{j\in J}\subset\dot{H}^{s}(\mathbb{R}^n)$ and coefficients…

Let $n\geq1$, and let $A=(1/p,1/q)=((n-1)/(n+1),n/(n+1))$. For an orthonormal family $\{f_j\}_{j\in J}\subset L^2(\mathbb{R}^n)$ and a sequence…

Let $n=1$. Let $A$ denote the endpoint in the $(1/p,1/q)$-region for the frequency-global estimate, and let $\alpha$ be the summability exponent for the orthonormal system. Endpoin…

Let $2\le q<p<\infty$, and let $\Phi$ be a finite, uniformly bounded orthonormal system. For a finite collection $\Theta$ of functions, define $K_p^*(\Theta)$ to be the smallest co…

Frank–Sabin endpoint conjecture. The estimate should hold for some $\beta\in(1,2)$.

Let $L^2_{[0,1]}$ be the space of square-integrable functions on $[0,1]$. A sequence $\{\varphi_n\}_{n=1}^{\infty}$ is a complete Euclidean system if it is complete and orthonormal…

Restricted-type endpoint conjecture. The estimate

Bounded orthonormal systems recovery conjecture. A similar recovery estimate should hold for bounded orthonormal systems, allowing a substantially larger number of sparse corruptio…

Structured-basis conjecture. As far as the topics addressed here are concerned, any basis with a useful structure and uniform boundedness should behave similarly to the Fourier and…