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Huiju Wang

Publications and source records attributed to Huiju Wang.

16 recordsLinked to original sources
arXiv2026

$L^p$-Estimates for maximal averages along mixed homogeneous hypersurfaces in $\mathbb{R}^{3}$

In this paper, we study $L^p$-estimates for maximal averaging operators $\mathcal M$ along hypersurfaces $S$ in $\mathbb{R}^{3}$ which are the graph of a mixed homogeneous function $Φ$ which is analytic away from the origin. The closure of such a surface will pass through the origin, so that the usual transversality condition that had been imposed in many previous works on maximal averages along hypersurfaces will not hold even when $Φ$ is analytic at the origin. As our main result, under mild assumptions which are satisfied for instance for every mixed homogeneous polynomial $Φ,$ we determine the critical Lebesgue exponent $p_c$ for which $\mathcal M$ is $L^p$-bounded for every $p>p_c,$ but unbounded for $p<p_c,$ in terms of multiplicities of the real roots of the Hessian determinant of $Φ.$ It turns out that the study of the contributions by neighborhoods of a certain type of roots is closely related to recent work by Dendrinos, Ikromov and the first and third author on sharp estimates for a maximal averaging operator along a transversal hypersurface of an ``exceptional'' class, whose $L^p$-boundedness had been an open problem for a long time and which has recently been established by means of their new theory of FIO-cone multipliers.

math.CA↗
arXiv2026

A Study on Kakeya Needle Problem for $(n-1)$-Rectifiable Sets

In this article we study the analog of Kakeya needle problem for $(n-1)$-rectifiable sets in $\mathbb{R}^n$ and construct the related Nikodym type sets. The novelty of our approach lies in combining three ingredients: the two-dimensional Venetian blind-type construction for isometries in $\mathbb{R}^n$; the normal geometry of a $(n-1)$-rectifiable set viewed in the projective setting; and measure estimates of moving $(n-1)$-rectifiable sets by isometries in $\mathbb{R}^n$. Together, these ingredients enable us to move $(n-1)$-rectifiable sets along paths of isometries in $\mathbb{R}^n$ and cover a Lebesgue null set.

math.CA↗
arXiv2026

Maximal functions related to homogeneous hypersurfaces in $\mathbb{R}^3$

We study maximal functions related to homogeneous polynomial hypersurfaces in $\mathbb{R}^3$. In a sense made precise in this paper, the region of $(p,q)$ for which we obtain $L^p\rightarrow L^q$ boundedness is optimal up to the endpoints for the corresponding local maximal operators. The boundedness exponents depend explicitly on both the height of the hypersurface and the type of the curve determined by the level set. As a corollary, we obtain $L^p$-estimates and weighted norm inequalities for the associated global maximal functions. Moreover, we also obtain optimal $L^{p}$-estimates for the global maximal operators associated with homogeneous polynomial hypersurfaces without transversality condition in $\mathbb{R}^{3}$.

math.CA↗
arXiv2025

Substitutability-Based Graph Node Pricing

In the era o fdat commodification,the pricing o fgraph data presents unique challenges that differ significantly from traditional data markets. This paper addresses the critical issue of node pricing within graph structures, an area that has been largely overlooked in existing literature. We introduce a novel pricing mechanism based on the concept of substitutability, inspired by economic principles, to better reflect the ntrinsic value of nodes in a graph. Unlike previous studies that assumed known prices for nodes or subgraphs, our approach emphasizes the structural significance of nodes by employing a dominator tree, utilizing the Lengauer-Tarjan algorithm to extract dominance relationships. This innovative framework allows us to derive a more realistic pricing strategy that accounts for the unique connectivity and roles of nodes within their respective networks. Our comparative experiments demonstrate that the proposed method significantly outperforms existing pricing strategies, yielding high-quality solutions across various datasets. This research aims to contribute to the existing literature by addressing an important gap and providing insights that may assist in the more effective valuation of graph data, potentially supporting improved decision-making in data-driven environments.

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arXiv2022

Sharp convergence for sequences of Schrödinger means and related generalizations

For decreasing sequences $\{t_{n}\}_{n=1}^{\infty}$ converging to zero, we obtain the almost everywhere convergence results for sequences of Schrödinger means $e^{it_{n}Δ}f$, where $f \in H^{s}(\mathbb{R}^{N}), N\geq 2$. The convergence results are sharp up to the endpoints, and the method can also be applied to get the convergence results for the fractional Schrödinger means and nonelliptic Schrödinger means.

math.CA↗
arXiv2022

Sparse domination and $L^{p} \rightarrow L^{q}$ estimates for maximal functions associated with curvature

In this paper, we study maximal functions along some finite type curves and hypersurfaces. In particular, various impacts of non-isotropic dilations are considered. Firstly, we provide a generic scheme that allows us to deduce the sparse domination bounds for global maximal functions under the assumption that the corresponding localized maximal functions satisfy the $L^{p}$ improving properties. Secondly, for the localized maximal functions with non-isotropic dilations of curves and hypersurfaces whose curvatures vanish to finite order at some points, we establish the $L^{p}\rightarrow L^{q}$ bounds $(q >p)$. As a corollary, we obtain the weighted inequalities for the corresponding global maximal functions, which generalize the known unweighted estimates.

math.CA↗
arXiv2021

On convergence properties for generalized Schrödinger operators along tangential curves

In this paper, we consider convergence properties for generalized Schrödinger operators along tangential curves in $\mathbb{R}^{n} \times \mathbb{R}$ with less smoothness comparing with Lipschitz condition. Firstly, we obtain sharp convergence rate for generalized Schrödinger operators with polynomial growth along tangential curves in $\mathbb{R}^{n} \times \mathbb{R}$, $n \ge 1$. Secondly, it was open until now on pointwise convergence of solutions to the Schrödinger equation along non-$C^1$ curves in $\mathbb{R}^{n} \times \mathbb{R}$, $n\geq 2$, we obtain the corresponding results along some tangential curves when $n=2$ by the broad-narrow argument and polynomial partitioning. Moreover, the corresponding convergence rate will follow. Thirdly, we get the convergence result along a family of restricted tangential curves in $\mathbb{R} \times \mathbb{R}$. As a consequence, we obtain the sharp $L^p$-Schrödinger maximal estimates along tangential curves in $\mathbb{R} \times \mathbb{R}$.

math.CA↗
arXiv2020

Sharp convergence for sequences of nonelliptic Schrödinger means

We consider pointwise convergence of nonelliptic Schrödinger means $e^{it_{n}\square}f(x)$ for $f \in H^{s}(\mathbb{R}^{2})$ and decreasing sequences $\{t_{n}\}_{n=1}^{\infty}$ converging to zero, where \[{e^{it_{n}\square }}f\left( x \right): = \int_{{\mathbb{R}^2}} {{e^{i\left( {x \cdot ξ+ t_{n}{{ ξ_{1}ξ_{2} }}} \right)}}\widehat{f}} \left( ξ\right)dξ.\] We prove that when $0<s < \frac{1}{2}$, \[\mathop {\lim }\limits_{n \to \infty} {e^{it_{n}\square }}f\left( x \right) = f(x) \hspace{0.2cm} a.e.\hspace{0.2cm} x\in \mathbb{R}^2\] holds for all $f \in {H^s}\left( {{\mathbb{R}^2}} \right)$ if and only if $\{t_{n}\}_{n=1}^{\infty} \in \ell^{r(s), \infty}(\mathbb{N})$, $r(s)=\frac{s}{1-s}$. Moreover, our result remains valid in general dimensions.

math.CA↗
arXiv2020

Pointwise Convergence for sequences of Schrödinger means in $\mathbb{R}^{2}$

We consider pointwise convergence of Schrödinger means $e^{it_{n}Δ}f(x)$ for $f \in H^{s}(\mathbb{R}^{2})$ and decreasing sequences $\{t_{n}\}_{n=1}^{\infty}$ converging to zero. The main theorem improves the previous results of [Sjölin, JFAA, 2018] and [Sjölin-Strömberg, JMAA, 2020] in $\mathbb{R}^{2}$. This study is based on investigating properties of Schrödinger type maximal functions related to hypersurfaces with vanishing Gaussian curvature.

math.CA↗
arXiv2020

A Note on Non-tangential Convergence for Schrödinger Operators

The goal of this note is to establish non-tangential convergence results for Schrödinger operators along restricted curves. We consider the relationship between the dimension of this kind of approach region and the regularity for the initial data which implies convergence. As a consequence, we obtain a upper bound for $p$ such that the Schrödinger maximal function is bounded from $H^{s}(\mathbb{R}^{n})$ to $L^{p}(\mathbb{R}^{n})$ for any $s > \frac{n}{2(n+1)}$.

math.CA↗
arXiv2020

$L^{p} \rightarrow L^{q}$ estimates for maximal functions associated with nonisotropic dilations of hypersurfaces in $\mathbb{R}^3$

The goal of this article is to establish $L^{p} \rightarrow L^{q}$ estimates for maximal functions associated with nonisotropic dilations $δ_t(x)=(t^{a_1}x_1,t^{a_2}x_2,t^{a_3}x_3)$ of hypersurfaces $(x_{1}, x_{2},Φ(x_1,x_2))$ in $\mathbb{R}^3$, where the Gaussian curvatures of the hypersurfaces are allowed to vanish. When $2 α_{2} = α_{3}$, this problem is reduced to study of the $L^{p} \rightarrow L^{q}$ estimates for maximal functions along the curve $γ(x)=(x,x^2(1+ϕ(x)))$ and associated dilations $δ_t(x)=(tx_1,t^2x_2)$. The corresponding maximal function shows features related to the Bourgain circular maximal function, whose $L^{p} \rightarrow L^{q}$ estimate has been considered by [Schlag, JAMS, 1997], [Schlag-Sogge, MRL, 1997] and [Lee, PAMS, 2003]. However, in the study of the maximal function related to the mentioned curve $γ(x)$ and associated dilations, we get the $L^{p} \rightarrow L^{q}$ regularity properties for a family of corresponding Fourier integral operators which fail to satisfy the "cinematic curvature condition" uniformly, which means that classical local smoothing estimates could not be directly applied to our problem. What's more, the $L^{p} \rightarrow L^{q}$ estimates are also new for maximal functions associated with isotropic dilations of hypersurfaces $(x_{1}, x_{2},Φ(x_1,x_2))$ mentioned before.

math.CA↗
arXiv2020

Convergence problems along curves for generalized Schrödinger operators with polynomial growth

In this paper we build the relationship between smoothness of the functions and convergence rate along curves for a class of generalized Schrödinger operators with polynomial growth. We show that the convergence rate depends only on the growth condition of the phase function and regularity of the curve. Our result can be applied to a wide class of operators. In particular, convergence results along curves for a class of generalized Schrödinger operators with non-homogeneous phase functions is built and then the convergence rate is established.

math.CA↗
arXiv2019

A study on a class of generalized Schrödinger operators

In this paper, we consider the pointwise convergence for a class of generalized Schrödinger operators with suitable perturbations, and convergence rate for a class of generalized Schrödinger operators with polynomial growth. We show that the pointwise convergence results remain valid for a class of generalized Schrödinger operators under small perturbations. As applications, we obtain the sharp convergence result for Boussinesq operator and Beam operator in $\mathbb{R}^2$. Moreover, the convergence result for a class of non-elliptic Schrödinger operators with finite-type perturbations is built. Furthermore, we proved that the convergence rate for a class of generalized Schrödinger operators with polynomial growth depends only on the growth condition of their phase functions. This result can be applied to all previously mentioned operators, and more operators.

math.CA↗
arXiv2017

VCExplorer: A Interactive Graph Exploration Framework Based on Hub Vertices with Graph Consolidation

Graphs have been widely used to model different information networks, such as the Web, biological networks and social networks (e.g. Twitter). Due to the size and complexity of these graphs, how to explore and utilize these graphs has become a very challenging problem. In this paper, we propose, VCExplorer, a new interactive graph exploration framework that integrates the strengths of graph visualization and graph summarization. Unlike existing graph visualization tools where vertices of a graph may be clustered into a smaller collection of super/virtual vertices, VCExplorer displays a small number of actual source graph vertices (called hubs) and summaries of the information between these vertices. We refer to such a graph as a HA-graph (Hub-based Aggregation Graph). This allows users to appreciate the relationship between the hubs, rather than super/virtual vertices. Users can navigate through the HA- graph by "drilling down" into the summaries between hubs to display more hubs. We illustrate how the graph aggregation techniques can be integrated into the exploring framework as the consolidated information to users. In addition, we propose efficient graph aggregation algorithms over multiple subgraphs via computation sharing. Extensive experimental evaluations have been conducted using both real and synthetic datasets and the results indicate the effectiveness and efficiency of VCExplorer for exploration.

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