arXiv subjects

Xin-han Dong

Publications and source records attributed to Xin-han Dong.

2 recordsLinked to original sources

arXiv2022

Lower bounds for uncentered maximal functions on metric measure space

We show that the uncentered Hardy-Littlewood maximal operators associated with the Radon measure $μ$ on $\mathbb{R}^d$ have the uniform lower $L^p$-bounds (independent of $μ$) that are strictly greater than $1$, if $μ$ satisfies a mild continuity assumption and $μ(\mathbb{R}^d)=\infty$. We actually do that in the more general context of metric measure space $(X,d,μ)$ satisfying the Besicovitch covering property. In addition, we also illustrate that the continuity condition can not be ignored by constructing counterexamples.

math.MG↗

arXiv2017

Non-spectral problem for the planar self-affine measures

In this paper, we consider the non-spectral problem for the planar self-affine measures $μ_{M,D}$ generated by an expanding integer matrix $M\in M_2(\mathbb{Z})$ and a finite digit set $D\subset\mathbb{Z}^2$. Let $p\geq2$ be a positive integer, $E_p^2:=\frac{1}{p}\{(i,j)^t:0\leq i,j\leq p-1\}$ and $\mathcal{Z}_{D}^2:=\{x\in[0, 1)^2:\sum_{d\in D}{e^{2πi\langle d,x\rangle}}=0\}$. We show that if $\emptyset\neq\mathcal{Z}_{D}^2\subset E_p^2\setminus\{0\}$ and $\gcd(\det(M),p)=1$, then there exist at most $p^2$ mutually orthogonal exponential functions in $L^2(μ_{M,D})$. In particular, if $p$ is a prime, then the number $p^2$ is the best.

math.FA↗