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Guozhen Lu

Publications and source records attributed to Guozhen Lu.

At least 37 records · Page 2Linked to original sources

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 Poincaré type inequality with Gaussian measure.

math.AP↗

arXiv2025

Quantization property of n-Laplacian mean field equation and sharp Moser-Onofri inequality

In this paper, we are concerned with the following $n$-Laplacian mean field equation \[ \left\{ {\begin{array}{*{20}{c}} { - Δ_n u = λe^u} & {\rm in} \ \ Ω, \\ {\ \ \ \ u = 0} &\ {\rm on}\ \partial Ω, \end{array}} \right. \] \[\] where $Ω$ is a smooth bounded domain of $\mathbb{R}^n \ (n\geq 2)$ and $- Δ_n u =-{\rm div}(|\nabla u|^{n-2}\nabla u)$. We first establish the quantization property of solutions to the above $n$-Laplacian mean field equation. As an application, combining the Pohozaev identity and the capacity estimate, we obtain the sharp constant $C(n)$ of the Moser-Onofri inequality in the $n$-dimensional unit ball $B^n:=B^n(0,1)$, $$\mathop {\inf }\limits_{u \in W_0^{1,n}(B^n)}\frac{1}{ n C_n}\int_{B^n} | \nabla u|^n dx- \ln \int_{B^n} {e^u} dx\geq C(n),$$ which extends the result of Caglioti-Lions-Marchioro-Pulvirenti in \cite{Caglioti} to the case of $n$-dimensional ball. Here $C_n=(\frac{n^2}{n-1})^{n-1} ω_{n-1}$ and $ω_{n-1}$ is the surface measure of $B^n$. For the Moser-Onofri inequality in a general bounded domain of $\mathbb{R}^n$, we apply the technique of $n$-harmonic transplantation to give the optimal concentration level of the Moser-Onofri inequality and obtain the criterion for the existence and non-existence of extremals for the Moser-Onofri inequality.

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:\[ S_Ω(λ,p)=\sup_{u\in H_{0}^{1}(Ω),\Vert\nabla u\Vert _{L^{2}\left( Ω\right) }\leq 1}\int_Ω\left( e^{4πu^{2}}-λ|u|^{p}\right) dx, \] where $1\leq p<\infty$ and $Ω$ is a bounded domain in $\mathbb{R}^2$. Our results demonstrate that there exists a threshold $λ^{\ast}(p)>0$ such that $S_Ω(λ,p)$ is attainable if $λ<λ^{\ast}(p)$, but unattainable if $λ>λ^{\ast}(p)$ when $p\in[1,2]$. For $p>2$, however, we show that $S_Ω(λ,p)$ is always attainable for any $λ\in \mathbb{R}$. 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

Scale-Dependent Poincaré inequalities, log-Sobolev inequality and the stability of the Heisenberg Uncertainty Principle on the hyperbolic space

We establish a general scale-dependent Poincaré-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 Poincaré type, Hardy type and Poincaré-Hardy type inequalities in the literature. We also investigate weighted Poincaré inequalities on hyperbolic space, where the weight functions depend on a scaling parameter. This leads to a new family of scale-dependent Poincaré 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↗

arXiv2025

Optimal stability of Hardy-Littlewood-Sobolev and Sobolev inequalities of arbitrary orders with dimension-dependent constants

Recently, Dolbeault-Esteban-Figalli-Frank-Loss [20] established the optimal stability of the first-order $L^2$-Sobolev inequality with dimension-dependent constant. Subsequently, Chen-Lu-Tang [18] obtained the optimal stability for the $L^2$ fractional Sobolev inequality of order $s$ when $0<s<1$.This paper considers the remaining case $1<s<\frac{n}{2}$. Our strategy is to first establish the optimal stability for the HLS inequality directly without using the stability of the Sobolev inequality. The main difficulty lies in establishing the optimal local stability of HLS inequality when $1<s<\frac{n}{2}$. The loss of the Hilbert structure of the distance appearing in the stability of the HLS inequality brings challenge in establishing the desired stability. To achieve our goal, we develop a new strategy based on the $H^{-s}-$decomposition instead of $L^{\frac{2n}{n+2s}}-$decomposition to obtain the local stability of the HLS inequality with $L^{\frac{2n}{n+2s}}-$distance. However, new difficulties arise to deduce the global stability from the local stability because of the non-uniqueness and non-continuity of $\|r\|_{\frac{2n}{n+2s}}$ for the rearrangement flow. As an important application of the optimal stability of the HLS inequality together with the duality theory of the stability developed initially by Carlen [11] and further improved in [17], we deduce the optimal stability of the $L^2$-Sobolev inequality of order s when $1\le s<\frac{n}{2}$ and the non-Hilbertian $L^{\frac{2n}{n+2s}}$-Sobolev inequality with the dimension-dependent constants. As another application, we can derive the optimal stability of Beckner's [5] restrictive Sobolev inequality on the flat sub-manifold $\mathbb{R}^{n-1}$ and the sphere $\mathbb{S}^{n-1}$ with dimension-dependent constants.

math.AP↗

arXiv2025

Stability of Gaussian Poincaré inequalities and Heisenberg Uncertainty Principle with monimial weights

We use the Bakry-Émery curvature-dimension criterion and $Γ$-calculus to establish the Poincaré 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 Poincaré 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 Poincaré 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 Poincaré 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↗

arXiv2025

Sharp Sobolev and Adams-Trudinger-Moser inequalities for symmetric functions without boundary conditions on hyperbolic spaces

Embedding theorems for symmetric functions without zero boundary condition have been studied on flat Riemannian manifolds, such as the Euclidean space. However, these theorems have only been established on hyperbolic spaces for functions with zero boundary condition. In this work, we focus on sharp Sobolev and Adams-Trudinger-Moser embeddings for radial functions in hyperbolic spaces, considering both bounded and unbounded domains. One of the main features of our approach is that we do not assume boundary zero condition for symmetric functions on geodesic balls or the entire hyperbolic space. Our main results include Theorems 1.2, 1.3, and 1.4, which establish weighted Sobolev embedding theorems, and Theorems 1.5 together with 1.6, which present Adams-Trudinger-Moser type of embedding theorems. In particular, a key result is Theorem 1.1 which is a highly nontrivial comparison result between norms of the higher order covariant derivatives and higher order derivatives of the radial functions. Higher order asymptotic behavior of radial functions on hyperbolic spaces are established to prove our main theorems. This approach includes novel radial lemmata and decay properties of higher order radial Sobolev functions defined in hyperbolic space.

math.AP↗

arXiv2025

The Method of Moving Spheres on the Hyperbolic Space and the Classification of Solutions and the prescribed Q-curvature problem

The classification of solutions to semilinear partial differential equations, as well as the classification of critical points of the corresponding functionals, have wide applications in the study of partial differential equations and differential geometry. The classical moving plane method and the method of moving sphere on the Euclidean space $\mathbb{R}^n$ provide an effective approach to capture the symmetry of solutions. As far as we know, the moving sphere method has yet to be developed on the hyperbolic space $\mathbb{H}^n$. In the present paper, we focus on the following equation \begin{equation*} P_k u = f(u) \end{equation*} on hyperbolic spaces $\mathbb{H}^n$, where $P_k$ denotes the GJMS operators on $\mathbb{H}^n$ and $f : \mathbb{R} \to \mathbb{R}$ satisfies certain growth conditions. We develop a moving sphere approach on $\mathbb{H}^n$ to obtain the symmetry propertyas well as the classification of positive solutions to the above equation. Our methods also rely on the Helgason-Fourier analysis and Hardy-Littlewood-Sobolev inequalities on hyperbolic space together with a Kelvin transform we introduce on the hyperbolic space in this paper. We also present applications to the higher order prescribed $Q$-curvature problem on the hyperbolic space.

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 Pólya-Szegö symmetrization principle and the nonlinear ground state representation play an important role in the work of [15]. To overcome the absence of the Pólya-Szegö 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↗

arXiv2024

Optimal asymptotic lower bound for stability of fractional Sobolev inequality and the stability of Log-Sobolev inequality on the sphere

We establish the optimal asymptotic lower bound for the stability of fractional Sobolev inequality: \begin{equation}\label{Sob sta ine} \left\|(-Δ)^{s/2} U \right\|_2^2 - \mathcal S_{s,n} \| U\|_{\frac{2n}{n-2s}}^2\geq C_{n,s} d^{2}(U, \mathcal{M}_s), \end{equation} where $\mathcal{M}_s$ is the set of maximizers of the fractional Sobolev inequality of order $s$, $s\in (0, 1)$ and $C_{n,s}$ denotes the optimal lower bound of stability. We prove that the optimal lower bound $C_{n,s}$ behaves asymptotically at the order of $\frac{1}{n}$ when $n\rightarrow +\infty$ for any fixed $s\in (0,1)$. This extends the work by Dolbeault-Esteban-Figalli-Frank-Loss [19] on the stability of the first order Sobolev inequality and quantify the asymptotic behavior for lower bound of stability of fractional Sobolev inequality established by the current author's previous work in [15] in the case of $s\in (0, 1)$. Moreover, $C_{n,s}$ behaves asymptotically at the order of $s$ when $s\rightarrow 0$ for any given dimension $n$. (See Theorem 1.1.) As an application of this asymptotic estimate as $s\to 0$ and through the end-point differentiation method, we also derive the global stability for the log-Sobolev inequality on the sphere established by Beckner in [3,4] with the optimal asymptotic lower bound on the sphere. (see Theorem 1.6). This sharpens the earlier work by the authors in [14] where only the local stability for the log-Sobolev inequality on the sphere was proved. We also obtain the asymptotically optimal lower bound for the Hardy-Littlewood-Sobolev inequality when $s\to 0$ for fixed dimension $n$ and when $n\to \infty$ for fixed $s\in (0, 1)$ (See Theorem 1.4 and the subsequent Remark 1.5).

math.AP↗

arXiv2024

Conformally Covariant Boundary Operators and Sharp Higher Order CR Sobolev Trace Inequalities on the Siegel Domain and Complex Ball

We first introduce an appropriate family of conformally covariant boundary operators associated to the Siegel domain ${\mathcal U}^{n+1}$ with the Heisenberg group $\mathbb{H}^{n}$ as its boundary and the complex ball $\mathbb{B}_{\mathbb{C}}^{n+1}$ with the complex sphere $\mathbb{S}^{2n+1}$ as its boundary. We provide the explicit formulas of these conformally covariant boundary operators. Second, we establish all higher order extension theorems of Caffarelli-Silvestre type for the Siegel domain and complex ball. Third, we prove all higher order CR Sobolev trace inequalities for the Siegel domain ${\mathcal U}^{n+1}$ and the complex ball $\mathbb{B}_{\mathbb{C}}^{n+1}$.In particular, we generalize the Sobolev trace inequalityfor $γ\in (0, 1)$ in the CR setting by Frank-González-Monticelli-Tan to the case for all $γ\in (0, n+1)\backslash \mathbb{N}$. The family of higher order conformally covariant boundary operators we define are naturally intrinsic to the higher order Sobolev trace inequalities on both the Siegel domain ${\mathcal U}^{n+1}$ and complex ball $\mathbb{B}_{\mathbb{C}^{n+1}}$. Finally, we give an explicit solution to the scattering problem on the complex hyperbolic ball. More precisely, we obtain an integral representation and an expansion in terms of special functions for the solution to the scattering problem.

math.AP↗

arXiv2023

Stability of Hardy-Littlewood-Sobolev inequalities with explicit lower bounds

In this paper, we establish the stability for the Hardy-Littlewood-Sobolev (HLS) inequalities with explicit lower bounds. By establishing the relation between the stability of HLS inequalities and the stability of fractional Sobolev inequalities, we also give the stability of the fractional Sobolev inequalities with the lower bounds. This extends the stability of Sobolev inequalities with the explicit lower bounds established by Dolbeault, Esteban, Figalli, Frank and Loss in [16] to the fractional order case. Our proofs are based on the competing symmetries, the continuous Steiner symmetrization inequality for the HLS integral and the dual stability theory.

math.AP↗

arXiv2023

Explicit Formulas of Fractional GJMS operators on hyperbolic spaces and sharp fractional Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities

Using the scattering theory on the hyperbolic space $\mathbb{H}^n$, we give the explicit formulas of the fractional GJMS operators $P_γ$ for all $γ\in(0,\frac{n}{2})\setminus\mathbb{N}$ on $\mathbb{H}^n$.These $P_γ$ for $γ\in(0,\frac{n}{2})\setminus\mathbb{N}$ are neither conformal to the fractional Laplacians on $\mathbb{R}^n_{+}$ nor on $\mathbb{B}^n$ in $\mathbb{R}^{n}$ though $P_γ$ are conformal to $(-Δ)^γ$ via half space model and ball model of hyperbolic spaces when $γ\in\mathbb{N}$. To circumvent this, we introduce another family of fractional operators $\tilde{P}_γ$ on $\mathbb{H}^n$ which are conformal to the fractional Laplacians on $\mathbb{R}^n_{+}$ and $\mathbb{B}^n$. It is worthwhile to note that $\tilde{P}_γ\not =P_γ$ unless $γ$ is an integer. We establish the fractional Poincaré-Sobolev inequalities associated with both $P_γ$ and $\tilde{P}_γ$ on $\mathbb{H}^n$. In particular, when $n\geq 3$ and $\frac{n-1}{2}\leq γ<\frac{n}{2}$, we prove that the sharp constants in the $γ$-th order of Poincaré-Sobolev inequalities on the hyperbolic space associated with $P_γ$ and $\tilde{P}_γ$ coincide with the best $γ$-th order Sobolev constant in the $n$-dimensional Euclidean space $\mathbb{R}^n$. We also establish fractional Hardy-Sobolev-Maz'ya inequality on $\mathbb{R}^{n}_+$ and $\mathbb{B}^n$ and prove that the sharp constants in the $γ$-th order Hardy-Sobolev-Maz'ya inequalities on half space $\mathbb{R}^{n}_+$ and unit ball $\mathbb{B}^n$ are the same as the best $γ$-th order Sobolev constants in $\mathbb{R}^n$ when $n\geq 3$ and $\frac{n-1}{2}\leq γ<\frac{n}{2}$. Our methods crucially rely on the Helgason-Fourier analysis on hyperbolic spaces and delicate analysis of special functions.

math.AP↗

arXiv2023

Conformally Covariant Boundary Operators and Sharp Higher Order Sobolev Trace Inequalities on Poincaré-Einstein Manifolds

In this paper we introduce conformally covariant boundary operators for Poincaré-Einstein manifolds satisfying a mild spectral assumption. Using these boundary operators we set up higher order Dirichlet problems whose solutions are such that, when applied to by our boundary operators, they recover the fractional order GJMS operators on the conformal infinity of the manifold. We moreover obtain all related higher order trace Sobolev inequalities on these manifolds. In conjunction with Beckner's fractional Sobolev inequalities on the sphere, we obtain as an application the sharp higher order Sobolev trace inequalities on the ball.

math.DG↗

arXiv2023

$L^p$-Hardy identities and inequalities with respect to the distance and mean distance to the boundary

Firstly, this paper establishes useful forms of the remainder term of Hardy-type inequalities on general domains where the weights are functions of the distance to the boundary. For weakly mean convex domains we use the resulting identities to establish nonexistence of extremizers for and improve known sharp Hardy inequalities. Secondly, we establish geometrically interesting remainders for the Davies-Hardy-Tidblom inequalities for the mean distance function, as well as generalize and improve several Hardy type inequalities in the spirit of Brezis and Marcus and spectral estimates of Davies. Lastly, we apply our results to obtain Sobolev inequalities for non-regular Riemannian metrics on geometric exterior domains.

math.AP↗

arXiv2023

Trudinger-Moser and Hardy-Trudinger-Moser inequalities for the Aharonov-Bohm Magnetic field

The main results of this paper concern sharp constant of the Trudinger-Moser inequality in $\mathbb{R}^{2}$ for Aharonov-Bohm magnetic fields. This is a borderline case of the Hardy type inequalities for Aharonov-Bohm magnetic fields in $\mathbb{R}^2$ studied by A. Laptev and T. Weidl. As an application, we obtain the exact asymptotic estimates on best constants of magnetic Hardy-Sobolev inequalities. In order to achieve our goal, we introduce a new operator $T_{a}$ on the unit circle $\mathbb{S}^{1}$ and give the asymptotic estimates of the heat kernel $e^{tT_{a}}$ via the Poisson summation formula. Finally, we show that such Trudinger-Moser inequalities in the unit ball $\mathbb{B}^{2}$ can be improved via subtraction of an additional Hardy term to derive a Hardy-Trudinger-Moser inequality.

math.AP↗

arXiv2023

A new approach to weighted Hardy-Rellich inequalities: improvements, symmetrization principle and symmetry breaking

We investigate necessary and sufficient conditions on the weights for the Hardy-Rellich inequalities to hold, and propose a new way to use the notion of Bessel pair to establish the optimal Hardy-Rellich type inequalities. Our results sharpened earlier Hardy-Rellich and Rellich type inequalities in the literature. We also study several results about the symmetry and symmetry breaking properties of the Rellich type and Hardy-Rellich type inequalities, and then partially answered an open question raised by Ghoussoub and Moradifam. Namely, we will present conditions on the weights such that the Rellich type and Hardy-Rellich type inequalities hold for all functions if and only if the same inequalities hold for all radial functions.

math.AP↗