arXiv subjects

Chenjian Wang

Publications and source records attributed to Chenjian Wang.

4 recordsLinked to original sources

arXiv2026

Weighted mixed-norm estimates for circular averages and exceptional set estimates for the wave equation

We prove mixed-norm estimates for circular averages with respect to $α$-dimensional fractal measures on $\mathbb{R}^2$, using circle tangency bounds when $α\in (0,1]$ and a $δ$-discretized slicing lemma for fractals when $α\in (1,2]$. The former estimate is sharp, while the latter improves previous results for $α\in (\frac{3}{2},2]$. These estimates can be viewed as X-ray-type extensions of Wolff's and Bourgain's circular maximal functions. As applications, we obtain new exceptional set estimates for the radial integrability of functions in Lebesgue spaces, as well as for the Hölder regularity in time of solutions to the linear wave equation on $\mathbb{R}^2$. The latter results are the first of their kind.

math.CA↗

arXiv2025

Pinned patterns and density theorems in $\mathbb R^d$

For integers $k\geq 3,d\geq 2,$ we consider the abundance property of pinned $k$-point patterns occurring in $E\subseteq \mathbb R^d$ with positive upper density $δ(E)$. We show that for any fixed $k$-point pattern $V$, there is a set $E$ with positive upper density such that $E$ avoids all sufficiently large affine copies of $V$, with one vertex fixed at any point in $E$. However, we obtain a positive quantitative result, which states that for any fixed $E$ with positive upper density, there exists a $k$-point pattern $V,$ such that for any $x\in E$, the pinned scaling factor set \begin{equation*} D_x^V(E):=\{r> 0: \exists \text{ isometry } O \text{ such that }x+r\cdot O(V)\subseteq E\}, \end{equation*} has upper density $\geq \tilde \varepsilon>0$, where constant $\tilde \varepsilon$ depends on $k,d$ and $δ(E)$.

math.CO↗

arXiv2025

Pinned distances and density theorems in $\mathbb R^d$

We study a pinned variant of Bourgain's theorem, concerning the occurrence of affine copies of $k$-point patterns in $\mathbb{R}^d$. Focusing on the case $k=2$, which corresponds to pinned distances, we show that the classical conclusion does not extend to the pinned setting: there exist sets of positive upper density in $\mathbb{R}^d$, $d \geq 2$, such that no single pinned point determines all sufficiently large distances. However, we establish a weaker quantitative result: for every point $x$ in such a set, the pinned distance set at $x$ has (one-dimensional) positive upper density. We also construct an example demonstrating the sharpness of this bound. These findings highlight a structural distinction between global and pinned configurations.

math.CA↗

arXiv2025

A note on maximal operators for moment curves

We consider a type of maximal operators associated to moment curves in $\mathbb R^d, d\geq 3.$ We derive $L^p$ mapping properties for these operators. In a special case, the estimate is sharp.

math.CA↗