arXiv subjects

Xiaodan Zhou

Publications and source records attributed to Xiaodan Zhou.

At least 19 recordsLinked to original sources

arXiv2026

Spatial Causal Tensor Completion for Multiple Exposures and Outcomes: An Application to the Health Effects of PFAS Pollution

Per- and polyfluoroalkyl substances (PFAS) are typically encountered as mixtures of distinct chemicals with distinct effects on multiple health outcomes. Estimating joint causal effects using spatially-dependent observed data is challenging. We propose a spatial causal tensor completion framework that jointly models multiple exposures and outcomes within a low-rank tensor structure, while adjusting for observed confounders and latent spatial confounders. This method combines a low-rank tensor representation to pool information across exposures and outcomes with a spectral adjustment step that incorporates graph-Laplacian eigenvectors to approximate unmeasured spatial confounders, implemented via a projected-gradient descent algorithm. This framework enables causal inference in the presence of unmeasured spatial confounding and pervasive missingness of potential outcomes. We establish theoretical guarantees for the estimator and evaluate its finite-sample performance through extensive simulations. In an application to national PFAS monitoring data, our approach yields more conservative and credible causal relationships between PFOA and PFOS exposure and 13 chronic disease outcomes compared with existing alternatives.

stat.ME↗

arXiv2025

Geometric inequalities related to fractional perimeter: fractional Poincaré, isoperimetric, and boxing inequalities in metric measure spaces

In the setting of a complete, doubling metric measure space $(X,d,μ)$ supporting a $(1,1)$-Poincaré inequality, we show that for all $0<θ<1$, the following fractional Poincaré inequality holds for all balls $B$ and locally integrable functions $u$, $$ \int_{B}|u-u_B|dμ\le C(1-θ)\,\text{rad}(B)^θ\int_{τB}\int_{τB}\frac{|u(x)-u(y)|}{d(x,y)^θμ(B(x,d(x,y)))}dμ(y)dμ(x), $$ where $C\ge 1$ and $τ\ge 1$ are constants depending only on the doubling and $(1,1)$-Poincaré inequality constants. Notably, this inequality features the scaling constant $(1-θ)$ present in the Bourgain-Brezis-Mironescu theory characterizing Sobolev functions via nonlocal functionals. From this inequality, we obtain a fractional relative isoperimetric inequality as well as global and local versions of a fractional boxing inequality, each featuring the same scaling constant $(1-θ)$ and defined in terms of the fractional $θ$-perimeter, and prove equivalences with the above fractional Poincaré inequality. We also show that $(X,d,μ)$ supports a $(1,1)$-Poincaré inequality if and only if the above fractional Poincaré inequality holds for all $θ$ sufficiently close to $1$. Under the additional assumption of lower Ahlfors $Q$-regularity of the measure $μ$, we additionally use the aforementioned results to establish global inequalities, in the form of fractional isoperimetric and fractional Sobolev inequalities, which also feature the scaling constant $(1-θ)$. Moreover, we prove that such inequalities are equivalent with the lower Ahlfors $Q$-regularity condition on the measure.

math.FA↗

arXiv2025

Superposition Property in Disjoint Variables for the Infinity Laplace Equation

We establish a superposition principle in disjoint variables for the inhomogeneous infinity-Laplace equation. We show that the sum of viscosity solutions of the inhomogeneous infinity-Laplace equation in separate domains is a viscosity solution in the product domain. This result has been used in the literature with certain particular choices of solutions to simplify regularity analysis for a general inhomogeneous infinity-Laplace equation by reducing it to the case without sign-changing inhomogeneous terms and vanishing gradient singularities. We present a proof of this superposition principle for general viscosity solutions. We also explore generalization in metric spaces using cone comparison techniques and study related properties for general elliptic and convex equations.

math.AP↗

arXiv2025

Uniqueness and nonuniqueness of $p$-harmonic Green functions on weighted $\mathbf{R}^n$ and metric spaces

We study uniqueness of $p$-harmonic Green functions in domains $Ω$ in a complete metric space equipped with a doubling measure supporting a $p$-Poincaré inequality, with $1<p<\infty$. For bounded domains in unweighted $\mathbf{R}^n$, the uniqueness was shown for the $p$-Laplace operator $Δ_p$ and all $p$ by Kichenassamy--Véron (Math. Ann. 275 (1986), 599-615), while for $p=2$ it is an easy consequence of the linearity of the Laplace operator $Δ$. Beyond that, uniqueness is only known in some particular cases, such as in Ahlfors $p$-regular spaces, as shown by Bonk--Capogna--Zhou (arXiv:2211.11974). When the singularity $x_0$ has positive $p$-capacity, the Green function is a particular multiple of the capacitary potential for $\text{cap}_p(\{x_0\},Ω)$ and is therefore unique. Here we give a sufficient condition for uniqueness in metric spaces, and provide an example showing that the range of $p$ for which it holds (while $x_0$ has zero $p$-capacity) can be a nondegenerate interval. In the opposite direction, we give the first example showing that uniqueness can fail in metric spaces, even for $p=2$.

math.AP↗

arXiv2025

Estimating the Causal Effect of Redlining on Present-day Air Pollution

Recent studies have shown associations between redlining policies (1935-1974) and present-day fine particulate matter (PM$_{2.5}$) and nitrogen dioxide (NO$_2$) air pollution concentrations. In this paper, we reevaluate these associations using spatial causal inference. Redlining policies enacted in the 1930s, so there is very limited documentation of pre-treatment covariates. Consequently, traditional methods fails to sufficiently account for unmeasured confounders, potentially biasing causal interpretations. By integrating historical redlining data with 2010 PM$_{2.5}$ and NO$_2$ concentrations, our study aims to discern whether a causal link exists. Our study addresses challenges with a novel spatial and non-spatial latent factor framework, using the unemployment rate, house rent and percentage of Black population in 1940 U.S. Census as proxies to reconstruct pre-treatment latent socio-economic status. We establish identification of a causal effect under broad assumptions, and use Bayesian Markov Chain Monte Carlo to quantify uncertainty. Our analysis indicates that historically redlined neighborhoods are exposed to notably higher NO$_2$ concentration. In contrast, the disparities in PM$_{2.5}$ between these neighborhoods are less pronounced. Among the cities analyzed, Los Angeles, CA, and Atlanta, GA, demonstrate the most significant effects for both NO$_2$ and PM$_{2.5}$.

stat.AP↗

arXiv2024

Green function in metric measure spaces

We study existence and uniqueness of Green functions for the Cheeger $Q$-Laplacian in metric measure spaces that are Ahlfors $Q$-regular and support a $Q$-Poincaré inequality with $Q>1$. We prove uniqueness of Green functions both in the case of relatively compact domains, and in the global (unbounded) case. We also prove existence of global Green functions in unbounded spaces, complementing the existing results in relatively compact domains proved recently in [BBL20].

math.AP↗

arXiv2023

Hamilton-Jacobi equations in metric spaces

These are lecture notes for our minicourse at OIST Summer Graduate School "Analysis and Partial Differential Equations" on June 12-17, 2023. We give an overview and collect a few important results concerning the well-posedness of Hamilton-Jacobi equations in metric spaces, especially several recently proposed notions of metric viscosity solutions to the eikonal equation. Basic knowledge about metric spaces and a review of viscosity solution theory in the Euclidean spaces are also presented.

math.AP↗

arXiv2023

A second-order operator for horizontal quasiconvexity in the Heisenberg group and application to convexity preserving for horizontal curvature flow

This paper is concerned with a PDE approach to horizontally quasiconvex (h-quasiconvex) functions in the Heisenberg group based on a nonlinear second order elliptic operator. We discuss sufficient conditions and necessary conditions for upper semicontinuous, h-quasiconvex functions in terms of the viscosity subsolution to the associated elliptic equation. Since the notion of h-quasiconvexity is equivalent to the horizontal convexity (h-convexity) of the function's sublevel sets, we further adopt these conditions to study the h-convexity preserving property for horizontal curvature flow in the Heisenberg group. Under the comparison principle, we show that the curvature flow starting from a star-shaped h-convex set preserves the h-convexity during the evolution.

math.AP↗

arXiv2023

BV functions and nonlocal functionals in metric measure spaces

We study the asymptotic behavior of three classes of nonlocal functionals in complete metric spaces equipped with a doubling measure and supporting a Poincaré inequality. We show that the limits of these nonlocal functionals are comparable to the variation $\| Df\|(Ω)$ or the Sobolev semi-norm $\int_Ωg_f^p\, dμ$, which extends Euclidean results to metric measure spaces. In contrast to the classical setting, we also give an example to show that the limits are not always equal to the corresponding total variation even for Lipschitz functions.

math.FA↗

arXiv2023

Discontinuous eikonal equations in metric measure spaces

In this paper, we study the eikonal equation in metric measure spaces, where the inhomogeneous term is allowed to be discontinuous, unbounded and merely $p$-integrable in the domain with a finite $p$. For continuous eikonal equations, it is known that the notion of Monge solutions is equivalent to the standard definition of viscosity solutions. Generalizing the notion of Monge solutions in our setting, we establish uniqueness and existence results for the associated Dirichlet boundary problem. The key in our approach is to adopt a new metric, based on the optimal control interpretation, which integrates the discontinuous term and converts the eikonal equation to a standard continuous form. We also discuss the Holder continuity of the unique solution with respect to the original metric under regularity assumptions on the space and the inhomogeneous term.

math.AP↗

arXiv2023

Horizontally quasiconvex envelope in the Heisenberg group

This paper is concerned with a PDE-based approach to the horizontally quasiconvex (h-quasiconvex for short) envelope of a given continuous function in the Heisenberg group. We provide a characterization for upper semicontinuous, h-quasiconvex functions in terms of the viscosity subsolution to a first-order nonlocal Hamilton-Jacobi equation. We also construct the corresponding envelope of a continuous function by iterating the nonlocal operator. One important step in our arguments is to prove the uniqueness and existence of viscosity solutions to the Dirichlet boundary problems for the nonlocal Hamilton-Jacobi equation. Applications of our approach to the h-convex hull of a given set in the Heisenberg group are discussed as well.

math.AP↗

arXiv2022

Metric quasiconformality and Sobolev regularity in non-Ahlfors regular spaces

Given a homeomorphism $f\colon X\to Y$ between $Q$-dimensional spaces $X,Y$, we show that $f$ satisfying the metric definition of quasiconformality outside suitable exceptional sets implies that $f$ belongs to the Sobolev class $N_{\rm{loc}}^{1,p}(X;Y)$, where $1< p\le Q$, and also implies one direction of the geometric definition of quasiconformality. Unlike previous results, we only assume a pointwise version of Ahlfors $Q$-regularity, which particularly enables various weighted spaces to be included in the theory. Unexpectedly, we can apply this to obtain results that are new even in the classical Euclidean setting. In particular, in spaces including the Carnot groups, we are able to prove the Sobolev regularity $f\in N_{\rm{loc}}^{1,Q}(X;Y)$ without the strong assumption of the infinitesimal distortion $h_f$ belonging to $L^{\infty}(X)$.

math.MG↗

arXiv2022

A characterization of BV and Sobolev functions via nonlocal functionals in metric spaces

We study a characterization of BV and Sobolev functions via nonlocal functionals in metric spaces equipped with a doubling measure and supporting a Poincaré inequality. Compared with previous works, we consider more general functionals. We also give a counterexample in the case $p=1$ demonstrating that unlike in Euclidean spaces, in metric measure spaces the limit of the nonlocal functions is only comparable, not necessarily equal, to the variation measure $\| Df\|(Ω)$.

math.FA↗

arXiv2021

Absolutely continuous mappings on doubling metric measure spaces

Following Malý's definition of absolutely continuous functions of several variables, we consider $Q$-absolutely continuous mappings $f\colon X\to V$ between a doubling metric measure space $X$ and a Banach space $V$. The relation between these mappings and Sobolev mappings $f\in N^{1,p}(X;V)$ for $p\ge Q$ is investigated. In particular, a locally $Q$-absolutely continuous mapping on an Ahlfors $Q$-regular space is a continuous mapping in $N^{1,Q}_{\rm{loc}}(X;V)$, as well as differentiable almost everywhere in terms of Cheeger derivatives provided $V$ satisfies the Radon-Nikodym property. Conversely, though a continuous Sobolev mapping $f\in N^{1,Q}_{\rm{loc}}(X;V)$ is generally not locally $Q$-absolutely continuous, this implication holds if $f$ is further assumed to be pseudomonotone. It follows that pseudomonotone mappings satisfying a relaxed quasiconformality condition are also $Q$-absolutely continuous.

math.FA↗

arXiv2021

Quasiconformal and Sobolev mappings in non-Ahlfors regular metric spaces

We show that a mapping $f\colon X\to Y$ satisfying the metric condition of quasiconformality outside suitable exceptional sets is in the Newton-Sobolev class $N_{\textrm{loc}}^{1,1}(X;Y)$. Contrary to previous works, we only assume an asymptotic version of Ahlfors-regularity on $X,Y$. This allows many non-Ahlfors regular spaces, such as weighted spaces and Fred Gehring's bowtie, to be included in the theory. Unexpectedly, already in the classical setting of unweighted Euclidean spaces, our theory detects Sobolev mappings that are not recognized by previous results.

math.MG↗

arXiv2020

Equivalence of solutions of eikonal equation in metric spaces

In this paper we prove the equivalence between some known notions of solutions to the eikonal equation and more general analogs of the Hamilton-Jacobi equations in complete and rectifiably connected metric spaces. The notions considered are that of curve-based viscosity solutions, slope-based viscosity solutions, and Monge solutions. By using the induced intrinsic (path) metric, we reduce the metric space to a length space and show the equivalence of these solutions to the associated Dirichlet boundary problem. Without utilizing the boundary data, we also localize our argument and directly prove the equivalence for the definitions of solutions. Regularity of solutions related to the Euclidean semi-concavity is discussed as well.

math.AP↗

arXiv2019

Functions of bounded variation on complete and connected one-dimensional metric spaces

In this paper, we study functions of bounded variation on a complete and connected metric space with finite one-dimensional Hausdorff measure. The definition of BV functions on a compact interval based on pointwise variation is extended to this general setting. We show this definition of BV functions is equivalent to the BV functions introduced by Miranda. Furthermore, we study the necessity of conditions on the underlying space in Federer's characterization of sets of finite perimeter on metric measure spaces. In particular, our examples show that the doubling and Poincaré inequality conditions are essential in showing that a set has finite perimeter if the codimension one Hausdorff measure of the measure-theoretic boundary is finite.

math.MG↗

arXiv2019

Horizontal convex envelope in the Heisenberg group and applications to sub-elliptic equations

This paper introduces in a natural way a notion of horizontal convex envelopes of continuous functions in the Heisenberg group. We provide a convexification process to find the envelope in a constructive manner. We also apply the convexification process to show h-convexity of viscosity solutions to a class of fully nonlinear elliptic equations in the Heisenberg group satisfying a certain symmetry condition. Our examples show that in general one cannot expect h-convexity of solutions without the symmetry condition.

math.AP↗