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 establish the stability with exact sharp constants 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 with exact optimal constants 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 with exact sharp constant, which is of independent interest and surprisingly absent in the literature. Moreover, we also establish the stability of the sharp stability inequality of the classical Heisenberg Uncertainty Principle proved in [15].

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.

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].

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.

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 ℝ^n+1_+ when k≥ 2 and n≥ 2k+2: ∫_ℝ^n_+|∇^ku|^2dx-∏_i=1^k(2i-1)^2/4∫_ℝ^n_+u^2/x_1^2kdx≥ C_n,k,2n/n-2k(∫_ℝ^n_+|u|^2n/n-2kdx)^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: Poincaré-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≥ 2): P_k(f)-αf=|f|^p-2f at the critical situation α=∏_i=1^k(2i-1)^2/4 when either 2k+2≤ n and p=2n/n-2k or 2k<n and 2<p<2n/n-2k.

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 ℍ^3 : P_2 u = - u^-7, where P_2 denotes the fourth-order Paneitz operator. We prove that any positive solution u on ℍ^3 exhibiting exponential growth at infinity must, up to hyperbolic isometries, be radial and strictly increasing with respect to some point P ∈ℍ^3 . Fourth order equations with negative critical growth on 3-dimensional Euclidean space ℝ^3 have been studied by Choi and Xu in [10], and subsequently by McKenna and Reichel [42] and Xu [45]. 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.

The main purpose of this short note, on the one hand, to is clarify some part of the proof of Theorem 1.3 in [8] in a simple way, and on the other hand, to give an alternative argument from local inequalities to global ones.

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 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|^mv(x) for an appropriately chosen m combined with spherical harmonic decompositions and weighted identities. Moreover, we establish weighted Poincaré inequalities associated with measures of the form e^-δ|x|^τ|x|^β(∏_i=n-k+1^nx_i^2) dx, together with their sharp constants, extremizers and stability estimates, which substantially extend those of the classical Gaussian Poincaré 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.

We employ a Markov semigroup approach combined with the Γ-calculus to establish a generalized Beckner inequality associated with weighted Gaussian measures. As a direct consequence, we derive the corresponding Poincaré inequality in the same setting. Subsequently, by means of a duality argument, we investigate gradient and L^2 stability estimates of the Poincaré inequality. Furthermore, we formulate a scale-dependent version of the Poincaré 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.

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.

We introduce a new family of weighted Gaussian L^2-Poincaré-type inequalities with explicit sharp constants, optimizers, and corresponding sharp L^2-gradient stability estimates. This family substantially extends the classical Gaussian Poincaré inequality. Owing to the singular nature of the weights involved, standard approaches to classical Gaussian Poincaré 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-Poincaré inequalities for all p>1, and derive stability estimates for the L^p-CKN inequalities for p≥ 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.

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→∞. 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.

Let Omega subset of R-4 be a bounded domain with smooth boundary partial derivative Omega. In this paper, we establish the following sharp form of the trace Adams inequality in W-2,W-2(center dot) with zero mean value and zero Neumann boundary condition: S(alpha) = sup(u is an element of W2,2(center dot)\{0}, parallel to 1u parallel to 2 <= 1 integral Omega udx=0, partial derivative u partial derivative nu |partial derivative Omega=0) integral(partial derivative Omega)e alpha u(2) d sigma < infinity holds if and only if alpha <= 12 pi(2). Moreover, we prove a classification theorem for the solutions of a class of nonlinear boundary value problem of biharmonic equations on the half-space R-+(4). With this classification result, we can show that S(12 pi(2)) is attained by using the blow-up analysis and capacitary estimate. As an application, we prove a sharp trace Adams-Onofri-type inequality in general four-dimensional bounded domains with smooth boundary.

Abstract. In this paper, we investigate the perturbed Trudinger–Moser inequalities as follows: [Formula: see text] where [Formula: see text] and [Formula: see text] is a bounded domain in [Formula: see text]. Our results demonstrate that there exists a threshold [Formula: see text] such that [Formula: see text] is attainable if [Formula: see text] but unattainable if [Formula: see text] when [Formula: see text]. For [Formula: see text], however, we show that [Formula: see text] is always attainable for any [Formula: see text]. These results are achieved through a refined blow-up analysis, which allows 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 nonradial solutions of the associated Euler–Lagrange equations. Our study establishes a complete characterization of how [Formula: see text]-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 setting.

We investigate several functional and geometric inequalities on the hyperbolic space ℍ^N, with a primary emphasis on logarithmic Sobolev inequalities, Poincaré 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 Pólya-Szegö 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 Poincaré and logarithmic Sobolev inequalities. Finally, we extend our results beyond hyperbolic space to a class of Riemannian model manifolds 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.

In this paper, we investigate the following critical Trudinger–Moser inequality on ℝ^2 under sharp L^p-perturbations: S(λ,p) := sup_u∈ H^1(ℝ^2) ∫_ℝ^2(|∇ u|^2+|u|^2) dx≤ 1∫_ℝ^2(e^4πu^2-1-λ|u|^p) dx . For 2 λ^∗. 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 , we establish the existence of two finite thresholds λ_∗>-∞ and λ^∗<+∞ such that S(λ,2) is attained when λ_∗<λ<λ^∗, and is not attained when λ<λ_∗ or λ>λ^∗. In contrast, for p>4, we prove that S(λ,p) is attained for all admissible values of λ. 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 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 ℝ^2.

Recently, Dolbeault–Esteban–Figalli–Frank–Loss (Camb J Math 13(2):359–430, 2025) established the optimal stability of the first-order Sobolev inequality with dimension-dependent constant. Subsequently, Chen–Lu–Tang (Adv Math 479:110438, 2025) obtained the optimal stability for the fractional Sobolev inequality of order s when 0

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.

In this paper, we show that nontrivial solutions to a class of higher and fractional order equations with certain nonlinearity are radially symmetric and nonincreasing on geodesic balls in the hyperbolic space Hn ${\mathbb{H}}^{n}$ as well as on the entire Hn ${\mathbb{H}}^{n}$ . Applying the Helgason-Fourier analysis techniques on Hn ${\mathbb{H}}^{n}$ , we develop a moving plane approach for integral equations on Hn ${\mathbb{H}}^{n}$ . We also establish the symmetry to solutions of certain equations with singular terms on Euclidean spaces. Moreover, we obtain the symmetry property of solutions to some semilinear equations involving fractional order derivatives.