Trudinger–Moser inequality in the hyperbolic space ℍ N

Though there have been extensive works on best constants for Moser-Trudinger inequalities in Euclidean spaces, Heisenberg groups or compact Riemannian manifolds, much less is known for sharp constants for the Moser-Trudinger inequalities on hyperbolic spaces. Earlier works only include the sharp constant for the Moser-Trudinger inequality on the twodimensional hyperbolic disc. In this paper, we establish best constants for several types of Moser-Trudinger inequalities on high dimensional hyperbolic spaces ℍ n (n ≥ 2). These include sharp constants for the Moser-Trudinger inequalities on both bounded and unbounded domains of the hyperbolic space ℍ n (see Theorems 1.1 and 1.2), sharp constants for the singular Moser-Trudinger inequality on unbounded domains when we impose restrictions only on the gradient norms (Theorem 1.3) or on the full hyperbolic Sobolev norms (Theorem 1.4). Our results are surprisingly general and extend most results in Euclidean spaces to hyperbolic spaces of any dimension. In particular, we have used a rearrangement-free argument in the hyperbolic spaces to establish Theorems 1.3 and 1.4 where symmetrization argument does not work to prove such sharp singular Moser-Trudinger inequalities on the entire hyperbolic space.

Improved Moser–Trudinger type inequalities in the hyperbolic space [formula omitted

Critical and subcritical Trudinger-Moser inequalities on complete noncompact Riemannian manifolds

Wang and Ye [Adv. Math. 230 (2012), pp. 294–320] prove a Hardy–Moser–Trudinger inequality in dimension two which improves both the classical Moser–Trudinger inequality and the classical Hardy inequality in the unit disc. In this paper, we generalize their result both to the higher dimensional unit ball and to the singular weighted cases as well. More precisely, we prove that \[ sup u ∈ W 0 1 , n ( B n ) , ∫ B n | ∇ u | n d x − ( 2 ( n − 1 ) n ) n ∫ B n | u | n ( 1 − | x | 2 ) n d x ≤ 1 ∫ B n e ( 1 − β n ) α n | u | n n − 1 | x | − β d x ≤ C \sup _{\substack {u\in W^{1,n}_0(\mathbb {B}^n),\\ \int _{\mathbb {B}^n} |\nabla u|^n dx -\left (\frac {2(n-1)}n\right )^n \int _{\mathbb {B}^n} \frac {|u|^n}{(1-|x|^2)^n} dx \leq 1}}\int _{\mathbb {B}^n} e^{(1-\frac \beta n)\alpha _n |u|^{\frac n{n-1}}} |x|^{-\beta } dx \leq C \] for any β ∈ [ 0 , n ) \beta \in [0,n) , n ≥ 3 n\geq 3 where α n = n ω n − 1 1 n − 1 \alpha _n = n \omega _{n-1}^{\frac 1{n-1}} and ω n − 1 \omega _{n-1} is the surface area of the n − 1 n-1 dimensional unit sphere. The proof of Wang and Ye is based on the blow-up analysis method which seems not work in the higher dimensions. In this paper, we propose a new approach based on the method of transplantation of Green’s functions to prove our inequality. As a consequence, we obtain a singular Moser–Trudinger inequality in the hyperbolic spaces which confirms affirmatively a conjecture by Mancini, Sandeep and Tintarev [Adv. Nonlinear Anal., 2 (2013), pp. 309–324].

Trudinger–Moser inequalities on hyperbolic spaces under Lorentz norms

The main purpose of this paper is two fold. On the one hand, we review some recent progress on best constants for various sharp Moser-Trudinger and Adams inequalities in Euclidean spaces $$\mathbb{R}^{N}$$ , hyperbolic spaces and other settings, and such sharp inequalities of Lions type. On the other hand, we present and prove some new results on sharp singular Moser-Trudinger and Adams type inequalities with exact growth condition and their affine analogues of such inequalities (Theorems 1.1, 1.2 and 1.3). We also establish a sharpened version of the classical Moser-Trudinger inequality on finite balls (Theorem 1.4).

Adams inequalities for Riesz subcritical potentials

Barthe proved that the regular simplex maximizes the mean width of convex bodies whose John ellipsoid (maximal volume ellipsoid contained in the body) is the Euclidean unit ball; or equivalently, the regular simplex maximizes the ℓ-norm of convex bodies whose Löwner ellipsoid (minimal volume ellipsoid containing the body) is the Euclidean unit ball. Schmuckenschläger verified the reverse statement; namely, the regular simplex minimizes the mean width of convex bodies whose Löwner ellipsoid is the Euclidean unit ball. Here we prove stronger stability versions of these results. We also consider related stability results for the mean width and the ℓ-norm of the convex hull of the support of centered isotropic measures on the unit sphere.

In this paper, we first establish a singular(0<β<n${(0<\beta<n}$) Trudinger–Moser inequality on any bounded domain inℝn${\mathbb{R}^{n}}$with Lorentz–Sobolev norms (Theorem 1.1). Next, we prove the critical singular(0<β<n${(0<\beta<n}$) Trudinger–Moser inequality on any unbounded domain inℝn${\mathbb{R}^{n}}$with Lorentz–Sobolev norms (Theorem 1.2). Then, we set up a subcritical singular(0<β<n${(0<\beta<n}$) Trudinger–Moser inequality on any unbounded domain inℝn${\mathbb{R}^{n}}$with Lorentz–Sobolev norms (Theorem 1.3). Finally, we establish the subcritical nonsingular(β=0${(\beta=0}$) Trudinger–Moser inequality on any unbounded domain inℝn${\mathbb{R}^{n}}$with Lorentz–Sobolev norms (Theorem 1.5). The constants in all these inequalities are sharp. In [9], for the proof of Theorem 1.2 in the nonsingular caseβ=0${\beta=0}$, the following inequality was used (see [17]):u∗⁢(r)-u∗⁢(r0)≤1n⁢wn1/n⁢∫rr0U⁢(s)⁢s1/n⁢d⁢ss,$u^{\ast}(r)-u^{\ast}(r_{0})\leq\frac{1}{nw_{n}^{{1/n}}}\int_{r}^{r_{0}}U(s)s^{% {1/n}}\frac{ds}{s},$whereU⁢(x)${U(x)}$is the radial function built from|∇⁡u|${|\nabla u|}$on the level set ofu, i.e.,∫|u|>t|∇u|dx=∫0|{|u|>t}|U(s)ds.$\int_{|u|>t}\lvert\nabla u|\,dx=\int_{0}^{|\{|u|>t\}|}U(s)\,ds.$The construction of suchUuses the deep Fleming–Rishel co-area formula and the isoperimetric inequality and is highly nontrivial. Moreover, this argument will not work in the singular case0<β<n${0<\beta<n}$. One of the main novelties of this paper is that we can avoid the use of this deep construction of such a radial functionU(see remarks at the end of the introduction). Moreover, our approach adapts the symmetrization-free argument developed in [19, 21, 23], where we derive the global inequalities on unbounded domains from the local inequalities on bounded domains using the level sets of the functions under consideration.

In this paper, we are concerned with a singular version of the Moser-Trudinger inequality with the exact growth condition in the n-dimension hyperbolic space mathbb{H}^{n}. Our result is a natural extension of the work of Lu and Tang in (J. Geom. Anal. 26:837-857, 2016).

Sharp Hardy–Adams inequalities for bi-Laplacian on hyperbolic space of dimension four

In this paper we obtain Hardy, weighted Trudinger-Moser and Caffarelli-Kohn-Nirenberg type inequalities with sharp constants on Riemannian manifolds with non-positive sectional curvature and, in particular, a variety of new estimates on hyperbolic spaces. Moreover, in some cases we also show their equivalence with Trudinger-Moser inequalities. As consequences, the relations between the constants of these inequalities are investigated yielding asymptotically best constants in the obtained inequalities. We also obtain the corresponding uncertainty type principles.

In this paper we survey various results concerning extremal problems related to Loewner chains, the Loewner differential equation, and Herglotz vector fields on the Euclidean unit ball \(\mathbb {B}^n\) in \(\mathbb {C}^n\). First, we survey recent results related to extremal problems for the Caratheodory families \({\mathcal M}\) and \({\mathcal N}_A\) on the Euclidean unit ball \(\mathbb {B}^n\) in \(\mathbb {C}^n\), where \(A\in L(\mathbb {C}^n)\) with m(A) > 0. In the second part of this paper, we present recent results related to extremal problems for the family \(S_A^0(\mathbb {B}^n)\) of normalized univalent mappings with A-parametric representation on the Euclidean unit ball \(\mathbb {B}^n\) in \(\mathbb {C}^n\), where \(A\in L(\mathbb {C}^n)\) with k+(A) < 2m(A). In the last section we survey certain results related to extreme points and support points for a special compact subset of \(S_A^0(\mathbb {B}^n)\) consisting of bounded mappings on \(\mathbb {B}^n\). Particular cases, open problems, and questions will be also mentioned.

Hyperbolic space and hyperbolic embeddings are becoming a popular research field for recommender systems. However, it is not clear under what circumstances the hyperbolic space should be considered. To fill this gap, This paper provides theoretical analysis and empirical results on when and where to use hyperbolic space and hyperbolic embeddings in recommender systems. Specifically, we answer the questions that which type of models and datasets are more suited for hyperbolic space, as well as which latent size to choose. We evaluate our answers by comparing the performance of Euclidean space and hyperbolic space on different latent space models in both general item recommendation domain and social recommendation domain, with 6 widely used datasets and different latent sizes. Additionally, we propose a new metric learning based recommendation method called SCML and its hyperbolic version HSCML. We evaluate our conclusions regarding hyperbolic space on SCML and show the state-of-the-art performance of hyperbolic space by comparing HSCML with other baseline methods.

Some sharp Schwarz-Pick type estimates and their applications of harmonic and pluriharmonic functions