arXiv subjects

Xiaosheng Lin

Publications and source records attributed to Xiaosheng Lin.

9 recordsLinked to original sources

arXiv2026

$\Gamma$-Convergence of Weak-Type Nonlocal Functionals on Bounded Domains

Let $N\ge1$, $p\in[1,\infty)$, $\gamma\in(0,\infty)$, and $\Omega\subset\mathbb R^N$ be a bounded open interval when $N=1$ or a bounded Lipschitz domain when $N\ge2$. For any $\lambda\in(0,\infty)$ and any measurable function $u$, consider the weak-type nonlocal functional \begin{align*} G_{\lambda,p,\gamma}(u;\Omega) :=\lambda\iint_{\Omega\times\Omega} \mathbf 1_{\left\{(x,y)\in\Omega\times\Omega:\ x\neq y,\ \frac{|u(x)-u(y)|^p}{|x-y|^{p+\gamma}}\geq\lambda\right\}} |x-y|^{\gamma-N}\,dx\,dy. \end{align*} In this article, we prove that, as $\lambda\to\infty$, the family $G_{\lambda,p,\gamma}$ converges, in the sense of $\Gamma$-convergence in $L^p(\Omega)$, to the functional \begin{align*} \Psi_{p,\gamma}^{\mathrm{cell}}(u;\Omega):= \begin{cases} C_{N,p,\gamma}^{\mathrm{cell}}\displaystyle\int_\Omega|\nabla u|^p\,dx, &p\in(1,\infty)\ \hbox{and}\ u\in W^{1,p}(\Omega),\\[2mm] C_{N,1,\gamma}^{\mathrm{cell}}|Du|(\Omega), &p=1\ \hbox{and}\ u\in BV(\Omega),\\[1mm] \infty,&\hbox{otherwise}, \end{cases} \end{align*} where the positive constants $C_{N,p,\gamma}^{\mathrm{cell}}$ are independent of $\Omega$ and characterized by a cell formula. This gives an affirmative answer to the problem posed by Brezis [Open Problem~9.3, Rend. Lincei Mat. Appl. 2023].

math.CA↗

arXiv2026

Local Hardy Spaces Associated with Operators and Ball Quasi-Banach Function Spaces on Doubling Metric Measure Spaces and Their Applications

Let $(\mathcal{X},d,\mu)$ be a doubling metric measure space, $X$ a ball quasi-Banach function space on $\mathcal{X}$, and $L$ a non-negative self-adjoint operator on $L^2(\mathcal{X})$ whose heat kernel satisfies a Gaussian upper bound. In this article, we study the local Hardy space $h_{X,L}(\mathcal{X})$ associated with both $X$ and $L$. We first establish the atomic and molecular characterizations of $h_{X,L}(\mathcal{X})$. As applications of these characterizations, we obtain the relations between $h_{X,L}(\mathcal{X})$ and the global Hardy spaces $H_{X,L}(\mathcal{X})$ and $H_{X,L+mI}(\mathcal{X})$. We also establish the radial and non-tangential maximal function characterizations of $h_{X,L}(\mathcal{X})$. Using these maximal function characterizations, under the additional assumptions that the heat kernel satisfies the conservation property and a H\"older regularity estimate, we further show that $h_{X,L}(\mathcal{X})$ coincides with the local atomic Hardy space $h_{X,\mathrm{at}}^p(\mathcal{X})$ with equivalent quasi-norms in the corresponding range of indices. Finally, we apply the above results to Lebesgue spaces, Orlicz spaces, weighted Lebesgue spaces, and variable Lebesgue spaces.

math.FA↗

arXiv2026

Compactness and Its Applications of Sobolev Spaces Associated with Ball Banach Function Spaces

Let $N\in\mathbb{N}\cap[2,\infty),$ $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^N$, and $X(\Omega)$ be a ball Banach function space on $\Omega.$ In this article, under some mild assumptions, we establish a compactness theorem for Sobolev spaces associated with $X(\Omega)$. Different from the fractional Sobolev space, our proof is based on an elaborate decomposition of bounded Lipschitz domains and its corresponding weighted fractional Poincar\'e inequality on each piece. As applications, we obtain the fractional Poincar\'e inequality in $X(\Omega)$ that, for any $s\in (s_0,1)$ and $f\in X(\Omega)$, \begin{align*} \|f-f_\Omega\|_{X(\Omega)} \lesssim(1-s)^{\frac 1q} \left\|\left[\int_\Omega \frac{|f(\cdot)-f(y)|^q}{|\cdot-y|^{N+sq}}\,dy \right]^{\frac{1}{q}} \right\|_{X(\Omega)}, \end{align*} where $s_0$ is a given positive constant and the implicit positive constant is independent of $s$ and $f$. Using this, we further establish the well-posedness of a weighted Triebel--Lizorkin type nonlocal variational problem. These results are of wide generality and, even when they are applied to Morrey spaces, weighted Lebesgue spaces, mixed-norm Lebesgue spaces, variable Lebesgue spaces, Orlicz spaces, and Orlicz-slice spaces, the obtained results are also new.

math.FA↗

arXiv2026

A Counterexample to the Necessity of the Vanishing Carleson Condition for VMO Poisson Kernels

In [Problem 3.2.23, CBMS Regional Conference Series in Mathematics 83, 1994], Kenig asked whether the vanishing Carleson condition is the necessary and sufficient for the logarithm of the Poisson kernel of a perturbation of the Laplacian on the unit ball in $\mathbb{R}^n$ belonging to the space VMO. The sufficiency was proved by Escauriaza [Israel J. Math. 1996] and extended by Milakis, Pipher, and Toro [Contemp. Math. 2014] to more general setting. In this article, using the technique of bi-Lipschitz mappings, we construct a counterexample to show that the vanishing Carleson condition is not necessary and hence give a negative answer to the aforementioned problem.

math.AP↗

arXiv2026

A Counterexample to Kenig's Interpolation Problem for Sobolev Spaces with Zero Boundary Conditions

Let $n\in \mathbb N\cap[2,\infty)$. In this article, we show that there exists a bounded $C^1$ domain $\Omega\subset \mathbb R^n$ such that, for any given $s\in(1,2)\setminus\{\frac32\}$, \begin{align*} \left[H_0^1(\Omega),H^2(\Omega)\cap H_0^1(\Omega)\right]_{s-1} =H^s(\Omega)\cap H_0^1(\Omega)=H_0^s(\Omega) \end{align*} with equivalent norms, but \begin{align*} \left[H_0^1(\Omega),H^2(\Omega)\cap H_0^1(\Omega)\right]_{\frac12} \subsetneqq H^{\frac32}(\Omega)\cap H_0^1(\Omega), \end{align*} which provides a counterexample to Problem 3.3.19 of Kenig in [CBMS Regional Conf. Ser. in Math. 83, 1994]. As applications, we prove that for such a domain $\Omega$ \begin{align*} H^2(\Omega)\cap H_0^1(\Omega)\subsetneqq D(-\Delta_D) \end{align*} (the domain of the Dirichlet Laplacian operator $-\Delta_D$ on $\Omega$) and construct a solution of the homogeneous heat equation with zero Dirichlet boundary condition, which does not belong to $L^2((0,T);H^2(\Omega)\cap H_0^1(\Omega))$ for any given $T\in(0,\infty)$.

math.AP↗

arXiv2023

Hardy Spaces Associated with Non-Negative Self-Adjoint Operators and Ball Quasi-Banach Function Spaces on Doubling Metric Measure Spaces and Their Applications

Let $(\mathcal{X},d,μ)$ be a doubling metric measure space in the sense of R. R. Coifman and G. Weiss, $L$ a non-negative self-adjoint operator on $L^2(\mathcal{X})$ satisfying the Davies--Gaffney estimate, and $X(\mathcal{X})$ a ball quasi-Banach function space on $\mathcal{X}$ satisfying some mild assumptions. In this article, the authors introduce the Hardy type space $H_{X,\,L}(\mathcal{X})$ by the Lusin area function associated with $L$ and establish the atomic and the molecular characterizations of $H_{X,\,L}(\mathcal{X}).$ As an application of these characterizations of $H_{X,\,L}(\mathcal{X})$, the authors obtain the boundedness of spectral multiplies on $H_{X,\,L}(\mathcal{X})$. Moreover, when $L$ satisfies the Gaussian upper bound estimate, the authors further characterize $H_{X,\,L}(\mathcal{X})$ in terms of the Littlewood--Paley functions $g_L$ and $g_{λ,\,L}^\ast$ and establish the boundedness estimate of Schrödinger groups on $H_{X,\,L}(\mathcal{X})$. Specific spaces $X(\mathcal{X})$ to which these results can be applied include Lebesgue spaces, Orlicz spaces, weighted Lebesgue spaces, and variable Lebesgue spaces. This shows that the results obtained in the article have extensive generality.

math.FA↗

arXiv2022

Brezis--Van Schaftingen--Yung Formulae in Ball Banach Function Spaces with Applications to Fractional Sobolev and Gagliardo--Nirenberg Inequalities

Let $X$ be a ball Banach function space on ${\mathbb R}^n$. In this article, under some mild assumptions about both $X$ and the boundedness of the Hardy--Littlewood maximal operator on the associate space of the convexification of $X$, the authors prove that, for any locally integrable function $f$ with $\|\,|\nabla f|\,\|_{X}<\infty$, $$\sup_{λ\in(0,\infty)}λ\left \|\left|\left\{y\in{\mathbb R}^n:\ |f(\cdot)-f(y)| >λ|\cdot-y|^{\frac{n}{q}+1}\right\}\right|^{\frac{1}{q}} \right\|_X\sim \|\,|\nabla f|\,\|_X$$ with the positive equivalence constants independent of $f$, where the index $q\in(0,\infty)$ is related to $X$ and $|\{y\in{\mathbb R}^n:\ |f(\cdot)-f(y)| >λ|\cdot-y|^{\frac{n}{q}+1}\}|$ is the Lebesgue measure of the set under consideration. In particular, when $X:=L^p({\mathbb R}^n)$ with $p\in [1,\infty)$, the above formulae hold true for any given $q\in (0,\infty)$ with $n(\frac{1}{p}-\frac{1}{q})<1$, which when $q=p$ are exactly the recent surprising formulae of H. Brezis, J. Van Schaftingen, and P.-L. Yung, and which in other cases are new. This generalization has a wide range of applications and, particularly, enables the authors to establish new fractional Sobolev and new Gagliardo--Nirenberg inequalities in various function spaces, including Morrey spaces, mixed-norm Lebesgue spaces, variable Lebesgue spaces, weighted Lebesgue spaces, Orlicz spaces, and Orlicz-slice (generalized amalgam) spaces, and, even in all these special cases, the obtained results are new. The proofs of these results strongly depend on the Poincaré inequality, the extrapolation, the exact operator norm on $X'$ of the Hardy--Littlewood maximal operator, and the exquisite geometry of $\mathbb{R}^n.$

math.CA↗

arXiv2022

Maximal Function and Riesz Transform Characterizations of Hardy Spaces Associated with Homogeneous Higher Order Elliptic Operators and Ball Quasi-Banach Function Spaces

Let $L$ be a homogeneous divergence form higher order elliptic operator with complex bounded measurable coefficients on $\mathbb{R}^n$ and $X$ a ball quasi-Banach function space on $\mathbb{R}^n$ satisfying some mild assumptions. Denote by $H_{X,\, L}(\mathbb{R}^n)$ the Hardy space, associated with both $L$ and $X$, which is defined via the Lusin area function related to the semigroup generated by $L$. In this article, the authors establish both the maximal function and the Riesz transform characterizations of $H_{X,\, L}(\mathbb{R}^n)$. The results obtained in this article have a wide range of generality and can be applied to the weighted Hardy space, the variable Hardy space, the mixed-norm Hardy space, the Orlicz--Hardy space, the Orlicz-slice Hardy space, and the Morrey--Hardy space, associated with $L$. In particular, even when $L$ is a second order divergence form elliptic operator, both the maximal function and the Riesz transform characterizations of the mixed-norm Hardy space, the Orlicz-slice Hardy space, and the Morrey--Hardy space, associated with $L$, obtained in this article, are totally new.

math.FA↗

arXiv2021

Poincaré Inequality Meets Brezis--Van Schaftingen--Yung Formula on Metric Measure Spaces

Let $(\mathcal{X}, ρ, μ)$ be a metric measure space of homogeneous type which supports a certain Poincaré inequality. Denote by the symbol $\mathcal{C}_{\mathrm{c}}^\ast(\mathcal{X})$ the space of all continuous functions $f$ with compact support satisfying that $\operatorname{Lip} f:=\limsup _{r \rightarrow 0} \sup_{y\in B(\cdot, r)} |f(\cdot)-f(y)|/r$ is also a continuous function with compact support and $\operatorname{Lip} f=\lim _{r \rightarrow 0} \sup_{y\in B(\cdot, r)} |f(\cdot)-f(y)|/r$ converges uniformly. Let $p \in[1,\infty)$. In this article, the authors prove that, for any $f\in\mathcal{C}_{\mathrm{c}}^\ast(\mathcal{X})$, \begin{align*} &\sup_{λ\in(0,\infty)}λ^p\int_{\mathcal{X}} μ\left(\left\{y\in \mathcal{X}:\ |f(x)-f(y)|>λρ(x,y) [V(x,y)]^{\frac 1p}\right\}\right)\, dμ(x)\\ &\quad\sim \int_{\mathcal{X}} [\operatorname{Lip}f(x)]^p \,dμ(x) \end{align*} with the positive equivalence constants independent of $f$, where $V(x,y):=μ(B(x,ρ(x,y)))$. This generalizes a recent surprising formula of H. Brezis, J. Van Schaftingen, and P.-L. Yung from the $n$-dimensional Euclidean space ${\mathbb R}^n$ to $\mathcal{X}$. Applying this generalization, the authors establish new fractional Sobolev and Gagliardo--Nirenberg inequalities in $\mathcal{X}$. All these results have a wide range of applications. Particularly, when applied to two concrete examples, namely, ${\mathbb R}^n$ with weighted Lebesgue measure and the complete $n$-dimensional Riemannian manifold with non-negative Ricci curvature, all these results are new. The proofs of these results strongly depend on the geometrical relation of differences and derivatives in the metric measure space and the Poincaré inequality.

math.FA↗