arXiv subjects

Xudong Nie

Publications and source records attributed to Xudong Nie.

2 recordsLinked to original sources

arXiv2025

Sharp variational inequalities for average operators over finite type curves in the plane

The aim of this article is to establish the $L^p(\mathbb{R}^2)$-boundedness of the variational operator associated with averaging operators defined over finite type curves in the plane. Additionally, we present the necessary conditions for the boundedness of these operators in $L^p$. Furthermore, to prove one of these results, we establish a mixed-norm local smoothing estimate from $L^4$ to $L^4(L^2)$ corresponding to a family of Fourier integral operators that do not uniformly satisfy the cinematic curvature condition.

math.CA↗

arXiv2023

Rearrangement inequalities of the one-dimensional maximal functions associated with general measures

We prove a rearrangement inequality for the uncentered Hardy-Littlewood maximal function $M_μ$ associate to general measure $μ$ on $\mathbb{R}$. This inequality is analogous to the Stein's result $cf^{**}(t)\leq(Mf)^{*}(t)\leq C f^{**}(t)$, where $f^*$ is the symmetric decreasing rearrangement function of $f$ and $f^{**}(t)=\int_0^tf^*(x)dx$. Moreover, we compute the best constant of $M_μ$ on $L^{p,\infty}(\mathbb{R},dμ)$.

math.CA↗