Searcharxiv⌕ Search

arXiv subjects

Guozhen Lu

Publications and source records attributed to Guozhen Lu.

At least 19 recordsLinked to original sources
arXiv2026

Sharp $L^2$-Caffarelli--Kohn--Nirenberg and weighted Poincar\'e inequalities on half-spaces and orthants and their stability

Though the sharp $L^{2}$-Caffarelli--Kohn--Nirenberg (CKN) inequalities have been extensively studied in the entire Euclidean spaces, the corresponding problem on domains whose boundary contains the origin remains largely unexplored. We investigate the sharp $L^{2}$-CKN inequalities on half-spaces and orthants $\mathbb R^{n}_{k,+}$ by computing explicitly the optimal constants, determining all possible extremal functions, and establishing exact identities for the deficits. Since the singular weights $|x|^{-2b}$ rule out the lifting argument that is available for the simpler Heisenberg Uncertainty Principle, we develop an approach based on the transformations $u(x)=|x|^{m}v(x)$ for an appropriately chosen $m$ combined with spherical harmonic decompositions and weighted identities. Moreover, we establish weighted Poincar\'e inequalities associated with measures of the form \[ e^{-\delta|x|^{\tau}}|x|^{\beta}\bigl(\prod_{i=n-k+1}^{n}x_i^{2}\bigr)\,dx, \] together with their sharp constants, extremizers and stability estimates, which substantially extend those of the classical Gaussian Poincar\'e inequality. On the full orthant, the linear modes cease to be admissible competitors, since all odd spherical harmonics are annihilated by the lifting; the first non-radial mode is then of degree two, and both the sharp constant and the manifold of optimizers change accordingly. Finally, we establish several stability estimates, and second-order stability estimates, of the CKN inequalities on the half-spaces and orthants throughout the full parameter range.

math.AP↗
arXiv2026

A complete characterization of the existence of extremals for the Trudinger-Moser inequality on $\mathbb{R}^2$ under sharp $L^p$-perturbations

In this paper, we investigate the following critical Trudinger--Moser inequality on $\mathbb R^2$ under sharp $L^p$-perturbations: $$ S(\lambda,p) := \sup_{\substack{u\in H^{1}(\mathbb R^{2})\\ \int_{\mathbb R^2}(|\nabla u|^2+|u|^2)\,dx\le 1}} \int_{\mathbb R^2} \left(e^{4\pi u^2}-1-\lambda |u|^p\right)\,dx . $$ For $2 \lambda^{\ast}$. Moreover, we show that the nonattainment in this range is caused by a vanishing phenomenon. For $p=2$, combining our analysis with the nonexistence results for $L^2$-perturbed Trudinger--Moser inequalities obtained in \cite{Chenluzhu}, we establish the existence of two finite thresholds $\lambda_{\ast}>-\infty$ and $\lambda^{\ast}<+\infty$ such that $S(\lambda,2)$ is attained when $\lambda_{\ast}<\lambda<\lambda^{\ast}$, and is not attained when $\lambda<\lambda_{\ast}$ or $\lambda>\lambda^{\ast}$. In contrast, for $p>4$, we prove that $S(\lambda,p)$ is attained for all admissible values of $\lambda$. Our results indicate that, in the whole-space setting, the $L^p$-perturbation term affects the existence and nonexistence of extremals through either concentration or vanishing phenomena, which is fundamentally different from the bounded-domain case, where existence or nonexistence is governed solely by concentration phenomena. These results provide a complete characterization of how sharp $L^p$ perturbations determine the existence and nonexistence of extremals for critical Trudinger--Moser inequalities on the entire $\mathbb R^2$. The resulting existence and nonexistence theory exhibits a threshold structure with respect to the $L^p$ pertubation reminiscent of the classical Brezis--Nirenberg phenomenon in the whole space $\mathbb R^2$.

math.AP↗
arXiv2026

Caffarelli-Kohn-Nirenberg and Weighted Gaussian Poincar\'e Inequalities: a complete characterization of sharp $L^2$ stability and $L^p$ extensions

We introduce a new family of weighted Gaussian $L^2$-Poincar\'e-type inequalities with explicit sharp constants, optimizers, and corresponding sharp $L^2$-gradient stability estimates. This family substantially extends the classical Gaussian Poincar\'e inequality. Owing to the singular nature of the weights involved, standard approaches to classical Gaussian Poincar\'e inequalities do not apply. To overcome this difficulty, we develop a new method based on generalized Laguerre polynomial expansions, spherical harmonic decompositions, and a Kelvin-type transform. As an application, we completely characterize the stability of the $L^2$-Caffarelli--Kohn--Nirenberg (CKN) inequalities by establishing sharp stability estimates, together with the stability of the stability inequality results, throughout the entire parameter range. Previous results were available only in a few special cases. We further establish weighted $L^p$-Poincar\'e inequalities for all $p>1$, and derive stability estimates for the $L^p$-CKN inequalities for $p\geq 2$ throughout the full parameter regime in which sharp constants and optimizers are known. In contrast, earlier $L^p$ results were restricted to highly limited parameter ranges.

math.AP↗
arXiv2026

Stability for Critical Points of the Hardy--Littlewood--Sobolev Inequality and a Dual Stability Framework

Although quantitative stability for critical points of the Sobolev and fractional Sobolev inequalities has been extensively studied, the corresponding stability theory for critical points of the Hardy--Littlewood--Sobolev (HLS) inequality remains largely unexplored. A major difficulty is that the natural stability problem for HLS critical points involves a non-Hilbertian distance, so the classical orthogonal decomposition methods used in Hilbert-space settings are no longer available. In this paper, we develop a weak-decomposition--strong-stability method tailored to the stability structure of HLS critical points and establish the corresponding stability inequality. Our approach also yields an explicit lower bound for the stability of Palais--Smale sequences of the HLS integral equation. To the best of our knowledge, this appears to be the first quantitative stability result for Palais--Smale sequences of a variational functional measured in a non-Hilbertian distance. We further introduce a duality framework connecting Struwe-type decompositions and stability inequalities for critical points of the Sobolev inequality with their HLS counterparts. As a consequence, we derive Struwe-type decomposition and stability results for critical points of the fractional Sobolev inequality for general functions, thereby removing the nonnegativity assumption imposed in [26].

math.AP↗
arXiv2026

Log-Sobolev and Beckner inequalities and stability of Poincar\'e inequality with weighted Gaussian measures

We employ a Markov semigroup approach combined with the $\Gamma$-calculus to establish a generalized Beckner inequality associated with weighted Gaussian measures. As a direct consequence, we derive the corresponding Poincar\'e inequality in the same setting. Subsequently, by means of a duality argument, we investigate gradient and $L^2$ stability estimates of the Poincar\'e inequality. Furthermore, we formulate a scale-dependent version of the Poincar\'e inequality for homogeneous Gaussian-type measures and apply it to analyze the stability of the Heisenberg Uncertainty Principle with homogeneous weights. Finally, we establish a Logarithmic Sobolev inequality for weighted Gaussian measures and utilize it to derive the Euclidean Logarithmic Sobolev inequality with homogeneous log-concave weights.

math.FA↗
arXiv2026

Conformally invariant equations with negative critical exponents on the three dimensional hyperbolic space

We establish a symmetry result for positive entire solutions with a prescribed growth rate to the following fourth order equation on the 3-dimensional hyperbolic space $\mathbb{H}^3$: \[ P_2 u = - u^{-7}, \] where $P_2$ denotes the fourth-order Paneitz operator. We prove that any positive solution $u$ on $\mathbb{H}^3$ exhibiting exponential growth at infinity must, up to hyperbolic isometries, be radial and strictly decreasing with respect to some point $P \in \mathbb{H}^3$. Fourth order equations with negative critical growth on 3-dimensional Euclidean space $\mathbb{R}^3$ has been studied by Choi and Xu in \cite{CX09 }, and subsequently by McKenna and Reichel \cite{MR03} and Xu \cite{Xu05}. Unlike the Euclidean case, the behavior of the Green's function of $P_2$ is substantially different, which prevents us from using the moving plane (sphere) method directly.

math.AP↗
arXiv2026

Heisenberg Uncertainty Principle on half spaces and Orthants: Best constants, Optimizers and Stability

Though the sharp Heisenberg Uncertainty Principle has been extensively studied in the entire Euclidean spaces, the counterpart on the half spaces or more general orthants has been missing in the literature. We investigate the sharp Heisenberg Uncertainty Principle on orthants by computing explicitly the optimal constant and determining all possible extremal functions. Moreover, we establish several stability estimates of the Heisenberg Uncertainty Principle on the half spaces and orthants.

math.AP↗
arXiv2026

Logarithmic Sobolev, Poincar\'e and Beckner Inequalities on Hyperbolic Spaces and Riemannian Manifolds

We investigate several functional and geometric inequalities on the hyperbolic space $\mathbb{H}^N$, with a primary emphasis on logarithmic Sobolev inequalities, Poincar\'e inequalities, and Beckner-type inequalities, all studied within the framework of the AB program. The main analytical tool employed throughout this paper is symmetrization. More precisely, our approach relies on an improved version of the P\'olya-Szeg\"o inequality on the hyperbolic space, obtained through a careful comparison of the gradient norms of rearranged functions in the hyperbolic and Euclidean settings. For Beckner-type inequalities, we adopt a semigroup approach based on sharp estimates for the heat semigroup, leading to refined interpolation inequalities between Poincar\'e and logarithmic Sobolev inequalities. Finally, we extend our results beyond hyperbolic space to a class of Riemannian model manifolds $\mathbb{M}^N$ satisfying the centered isoperimetric inequality. This shows that the inequalities and methods developed in this work are robust and rely mainly on geometric and isoperimetric properties, rather than on the specific structure of hyperbolic space itself.

math.AP↗
arXiv2026

Radial Sobolev embeddings on spherically symmetric Riemannian manifolds

We study Sobolev spaces of radial functions on spherically symmetric Riemannian manifolds. Using geodesic polar coordinates, we give a sharp one-dimensional reduction: a radial function belongs to the Sobolev space on the manifold if and only if its radial representation lies in an associated weighted Sobolev space on an interval, with weights determined explicitly by the metric. This characterization allows us to prove optimal Sobolev-type embeddings for radial functions into weighted Lebesgue spaces on both bounded and unbounded spherically symmetric manifolds. As further consequences, we establish new radial lemmas and decay estimates that capture the precise behaviour of radial Sobolev functions near the origin and at infinity. Our results unify and extend the classical radial embeddings in Euclidean and hyperbolic spaces.

math.AP↗
arXiv2026

Existence and symmetry of extremals for the high order Hardy-Sobolev-Maz'ya inequalities

In this article, we establish the existence of an extremal function for the k-th order critical Hardy-Sobolev-Maz'ya (HSM) inequalities on the upper half space $\mathbb{R}^{n+1}_{+}$ when $k\ge 2$ and $n\geq 2k+2$: $$\int_{\mathbb{R}^{n}_{+}}|\nabla^{k}u|^2dx-\prod_{i=1}^{k}\frac{\left(2i-1\right)^2}{4}\int_{\mathbb{R}^{n}_{+}}\frac{u^2}{x_1^{2k}}dx\geq C_{n,k,\frac{2n}{n-2k}} \left(\int_{\mathbb{R}^{n}_{+}}|u|^{\frac{2n}{n-2k}}dx\right)^{\frac{n-2k}{n}}. $$ The analysis of this extremal problem is challenging due to the presence of the higher order derivatives, the lack of translation invariance, the inapplicability of rearrangement techniques on the upper half-space, and the presence of a Hardy singularity along the boundary. To overcome these difficulties, instead of directly considering the HSM inequality on the upper half space, we establish the existence of an extremal for its equivalent version: Poincar\'e-Sobolev inequality on the hyperbolic space. We develop a novel duality theory of the minimizing sequences, the concentration-compactness principle for radial functions in the hyperbolic setting, which combines with the Helgason-Fourier analysis and the Riesz rearrangement inequality on the hyperbolic space, to resolve the lack of compactness issue. As an application, we also obtain the existence of positive symmetric solutions for the high order Brezis-Nirenberg equation on the entire hyperbolic space associated with the GJMS operators $P_k$ (i.e., when $k\ge 2$): $$ P_{k}\left(f\right)-\alpha f=|f|^{p-2}f $$ at the critical situation $\alpha=\prod\limits_{i=1}^{k}\frac{\left(2i-1\right)^2}{4}$ when either $2k+2\leq n$ and $p=\frac{2n}{n-2k}$ or $2k<n$ and $2<p<\frac{2n}{n-2k}$.

math.AP↗
arXiv2025

Asymptotically sharp stability of Sobolev inequalities on the Heisenberg group with dimension-dependent constants

In this paper, we are concerned with the optimal asymptotic lower bound for the stability of Sobolev inequality on the Heisenberg group. We first establish the optimal local stability of Sobolev inequality on the CR sphere through bispherical harmonics and complicated orthogonality technique ( see Lemma 3.1). The loss of rearrangement inequality in the CR setting makes it impossible to use any rearrangement flow technique (either differential rearrangement flow or integral rearrangement flow) to derive the optimal stability of Sobolev inequality on the CR sphere from corresponding optimal local stability. To circumvent this, we will use the CR Yamabe flow to establish the optimal stability of Sobolev inequality on the Heisenberg group with the dimension-dependent constants (see Theorem 1.1). As an application, we also establish the optimal stability of the Hardy-Littlewood-Sobolev (HLS) inequality for special conformal index with the dimension-dependent constants (see Theorem 1.3). Our approach is rearrangement-free and can be used to study the optimal stability problem for fractional Sobolev inequality or HLS inequality on the Heisenberg group once the corresponding continuous flow is established.

math.AP↗
arXiv2025

Existence and Nonexistence of Extremals for Trudinger-Moser inequalities with $L^p$ type perturbation on any bounded planar domains

In this study, we investigate the perturbed Trudinger-Moser inequalities as follows:\[ S_\Omega(\lambda,p)=\sup_{u\in H_{0}^{1}(\Omega),\Vert\nabla u\Vert _{L^{2}\left( \Omega\right) }\leq 1}\int_{\Omega}\left( e^{4\pi u^{2}}-\lambda|u|^{p}\right) dx, \] where $1\leq p<\infty$ and $\Omega$ is a bounded domain in $\mathbb{R}^2$. Our results demonstrate that there exists a threshold $\lambda^{\ast}(p)>0$ such that $S_\Omega(\lambda,p)$ is attainable if $\lambda<\lambda^{\ast}(p)$, but unattainable if $\lambda>\lambda^{\ast}(p)$ when $p\in[1,2]$. For $p>2$, however, we show that $S_\Omega(\lambda,p)$ is always attainable for any $\lambda\in \mathbb{R}$. These results are achieved through a refined blow-up analysis, which allow us to establish a sharp Dirichlet energy expansion formula for sequences of solutions to the corresponding Euler-Lagrange equations. The asymmetric nature of our problem poses significant challenges to our analysis. To address these, we will establish an appropriate comparison principle between radial and non-radial solutions of the associated Euler-Lagrange equations. Our study establishes a complete characterization of how $L^p$-type perturbations influence the existence of extremals for critical Trudinger-Moser inequalities on any bounded planar domains, this extends the classical Brezis-Nirenberg problem framework to the two-dimensional settings.

math.AP↗
arXiv2025

Sharp stability of the Heisenberg Uncertainty Principle: Second-Order and Curl-Free Field Cases

Using techniques from harmonic analysis, we derive several sharp stability estimates for the second order Heisenberg Uncertainty Principle. We also present the explicit lower and upper bounds for the sharp stability constants and compute their exact limits when the dimension $N\rightarrow\infty$. Our proofs rely on spherical harmonics decomposition and Fourier analysis, differing significantly from existing approaches in the literature. Our results substantially improve the stability constants of the second order Heisenberg Uncertainty Principle recently obtained in [27]. As direct consequences of our main results, we also establish the sharp stability, with exact asymptotic behavior of the stability constants, of the Heisenberg Uncertainty Principle with curl-free vector fields and a sharp version of the second order Poincar\'{e} type inequality with Gaussian measure.

math.AP↗
arXiv2025

Stability of Gaussian Poincar\'{e} inequalities and Heisenberg Uncertainty Principle with monimial weights

We use the Bakry-\'{E}mery curvature-dimension criterion and $\Gamma$-calculus to establish the Poincar\'{e} inequality with monomial Gaussian measure, and then apply the duality approach to study its improvements and its gradient stability. We also set up the scale-dependent Poincar\'{e} inequality with monomial Gaussian type measure and use it to inspect the stability of the Heisenberg Uncertainty Principle with monomial weight. Finally, we apply the improved versions of the monomial Gaussian Poincar\'{e} inequality to investigate the improved stability of the Heisenberg Uncertainty Principle with monomial weight. As special cases of our main results, we obtain the gradient stability of the classical Gaussian Poincar\'{e} inequality, which is of independent interest. Moreover, we also establish the stability of the sharp stability inequality of the classical Heisenberg Uncertainty Principle proved in [15].

math.AP↗
arXiv2024

Quantization analysis of Moser-Trudinger equations in the Poincar\'e disk and applications

In this paper, we first establish the quantitative properties for positive solutions to the Moser-Trudinger equations in the two-dimensional Poincar\'e disk $\mathbb{B}^2$: \begin{equation*}\label{mt1} \left\{ \begin{aligned} &-\Delta_{\mathbb{B}^2}u=\lambda ue^{u^2},\ x\in\mathbb{B}^2, &u\to0,\ \text{when}\ \rho(x)\to\infty, &||\nabla_{\mathbb{B}^2} u||_{L^2(\mathbb{B}^2)}^2\leq M_0, \end{aligned} \right. \end{equation*} where $0<\lambda<\frac{1}{4}=\inf\limits_{u\in W^{1,2}(\mathbb{B}^2)\backslash\{0\}}\frac{\|\nabla_{\mathbb{B}^2}u\|_{L^2(\mathbb{B}^2)}^2}{\|u\|_{L^2(\mathbb{B}^2)}^2}$, $\rho(x)$ denotes the geodesic distance between $x$ and the origin and $M_0$ is a fixed large positive constant (see Theorem 1.1). Furthermore, by doing a delicate expansion for Dirichlet energy $\|\nabla_{\mathbb{B}^2}u\|_{L^2(\mathbb{B}^2)}^2$ when $\lambda$ approaches to $0,$ we prove that there exists $\Lambda^\ast>4\pi$ such that the Moser-Trudinger functional $F(u)=\int_{\mathbb{B}^2}\left(e^{u^2}-1\right) dV_{\mathbb{B}^2}$ under the constraint $\int_{\mathbb{B}^2}|\nabla_{\mathbb{B}^2}u|^2 dV_{\mathbb{B}^2}=\Lambda$ has at least one positive critical point for $\Lambda\in(4\pi,\Lambda^{\ast})$ up to some M\"{o}bius transformation. Finally, when $\lambda\rightarrow 0$, by doing a more accurate expansion for $u$ near the origin and away from the origin, applying a local Pohozaev identity around the origin and the uniqueness of the Cauchy initial value problem for ODE,Cauchy-initial uniqueness for ODE, we prove that the Moser-Trudinger equation only has one positive solution when $\lambda$ is close to $0.$ During the process of the proofs, we overcome some new difficulties which involves the decay properties of the positive solutions, as well as some precise expansions for the solutions both near the origin and away from the origin.

math.AP↗
arXiv2024

Scale-Dependent Poincar\'{e} inequalities, log-Sobolev inequality and the stability of the Heisenberg Uncertainty Principle on the hyperbolic space

We establish a general scale-dependent Poincar\'{e}-Hardy type identity involving a vector field on the hyperbolic space. By choosing suitable parameter, potential and vector field in this identity, we can recover, as well as derive new versions of and substantially improve several Poincar\'{e} type, Hardy type and Poincar\'{e}-Hardy type inequalities in the literature. We also investigate weighted Poincar\'{e} inequalities on hyperbolic space, where the weight functions depend on a scaling parameter. This leads to a new family of scale-dependent Poincar\'{e} inequalities with Gaussian type measure on the hyperbolic space which is of independent interest. As a result, we derive both scale-dependent and scale-invariant $L^{2}$-stability results for the Heisenberg uncertainty principle in this setting. Finally, we study the logarithmic Sobolev inequality with Gaussian measure on the hyperbolic spaces, that is still missing in the literature.

math.AP↗
arXiv2024

Sobolev interpolation inequalities with optimal Hardy-Rellich inequalities and critical exponents

We establish a new family of the critical higher order Sobolev interpolation inequalities for radial functions as well as for non-radial functions. These Sobolev interpolation inequalities are sharp in the sense that they use the optimal quadratic forms of the sharp Hardy-Rellich inequalities and cover the Sobolev critical exponents. Our results extend those studied by Dietze and Nam in [15] for the first order derivative case to higher order setting. The well-known P\'{o}lya-Szeg\"{o} symmetrization principle and the nonlinear ground state representation play an important role in the work of [15]. To overcome the absence of the P\'{o}lya-Szeg\"{o} principle and the nonlinear ground state representation in the higher order case, our proofs rely on the Fourier analysis and a higher order verion of the Talenti comparison principle. We also study a new version of the critical Hardy-Sobolev interpolation inequality involving the critical quadratic form of the Hardy inequality and Lorentz norms. Our critical Hardy-Sobolev interpolation inequality complements the result of Dietze and Nam in [15].

math.AP↗