Trudinger Type Inequalities Research Articles

A Sharp Moser–Trudinger Type Inequality Involving Adimurthi–Druet Term in Two Dimensional Hyperbolic Space

A Sharp Moser–Trudinger Type Inequality Involving Adimurthi–Druet Term in Two Dimensional Hyperbolic Space

Extremal functions for the critical Trudinger–Moser inequality with logarithmic Kernels

In this paper we study Moser–Trudinger type inequalities for some nonlocal energy functionals in presence of a logarithmic convolution potential, when the domain is a ball. In particular, we perform a blow-up analysis to prove existence of extremal functions in the borderline case of critical growth. Using this, we sharpen the results in [S. Cingolani and T. Weth J. London Math. Soc. 105 (2022) 1897–1935] under critical growth assumptions and give answers to some questions left open in [S. Cingolani and T. Weth J. London Math. Soc. 105 (2022) 1897–1935].

A quantization proof of the uniform Yau–Tian–Donaldson conjecture

Using quantization techniques, we show that the \delta -invariant of Fujita–Odaka coincides with the optimal exponent in a certain Moser–Trudinger type inequality. Consequently, we obtain a uniform Yau–Tian–Donaldson theorem for the existence of twisted Kähler–Einstein metrics with arbitrary polarizations. Our approach mainly uses pluripotential theory, which does not involve Cheeger–Colding–Tian theory or the non-Archimedean language. A new computable criterion for the existence of constant scalar curvature Kähler metrics is also given.

On the global existence of solutions to chemotaxis system for two populations in dimension two

We consider the global existence for the following fully parabolic chemotaxis system with two populations \[\left\{ \begin{array}{@{}ll} \partial_tu_i=\kappa_i\Delta u_i-\chi_i\nabla\cdot(u_i\nabla v),\quad i\in\{1,2\}, & x\in\Omega,\ t>0, \\ v_t=\Delta v-v+u_1+u_2, & x\in\Omega,\ t>0,\\ u_i(x,t=0)=u_{i0}(x),\quad v(x,t=0)=v_0(x), & x\in\Omega, \end{array} \right. \]where $\Omega =\mathbb {R}^2$ or $\Omega =B_R(0)\subset \mathbb {R}^2$ supplemented with homogeneous Neumann boundary conditions, $\kappa _i,\chi _i>0,$$i=1,2$. The global existence remains open for the fully parabolic case as far as the author knows, while the existence of global solution was known for the parabolic-elliptic reduction with the second equation replaced by $0=\Delta v-v+u_1+u_2$ or $0=\Delta v+u_1+u_2$. In this paper, we prove that there exists a global solution if the initial masses satisfy the certain sub-criticality condition. The proof is based on a version of the Moser–Trudinger type inequality for system in two dimensions.

Trudinger-type inequalities on Musielak-Orlicz-Morrey spaces of an integral form

Our aim in this paper is to give Trudinger-type inequalities for Riesz potentials of functions in Musielak-Orlicz-Morrey spaces of an integral form over non-doubling metric measure spaces. Our result are new even for the doubling metric measure setting. As a corollary, we give Trudinger-type inequalities on Musielak-Orlicz-Morrey spaces of an integral form in the framework of double phase functions with variable exponents.

Extremal functions for sharp Moser–Trudinger type inequalities in the whole space [formula omitted

In this paper, we prove the existence of maximizers for the sharp Moser–Trudinger type inequalities in whole space RN, N≥2 with more general nonlinearity. The main key in our proof is a precise estimate of the concentrating level of the Moser–Trudinger functional associated with our inequalities on the normalized concentrating sequences. This estimate solves a heavily non-trivial and open problem related to the sharp Moser–Trudinger inequality. Our method gives an alternative proof of the existence of maximizers for the Moser–Trudinger inequality and singular Moser–Trudinger inequality in whole space RN due to Li and Ruf [30] and Li and Yang [31] without using blow-up analysis argument.

The thresholds of the existence of maximizers for the critical sharp singular Moser–Trudinger inequality under constraints

This paper is addressed to study the existence of maximizers for the singular Moser–Trudinger supremum under constraints in the critical case $$\begin{aligned} MT_{N}(a,\beta ) = \sup _{u\in W^{1,N}({\mathbb {R}}^N),\, \Vert \nabla u\Vert _N^a + \Vert u\Vert _N^N =1} \int _{{\mathbb {R}}^N}\Phi _N\left( (1-\beta /N)\alpha _N |u|^{\frac{N}{N-1}}\right) |x|^{-\beta } dx, \end{aligned}$$where $$a>0$$, $$\beta \in [0,N)$$, $$\Phi _N(t) = e^t -\sum _{k=0}^{N-2} \frac{t^k}{k!}$$, $$\alpha _N = N \omega _{N-1}^{1/(N-1)}$$, and $$\omega _{N-1}$$ denotes the surface area of the unit sphere in $${\mathbb {R}}^N$$. More precisely, we study the effect of the parameter a to the attainability of $$MT_{N}(a,\beta )$$. We will prove that for each $$\beta \in [0,N)$$ there exist the thresholds $$a_*(\beta )$$ and $$a^*(\beta )$$ such that $$MT_{N}(a,\beta )$$ is attained for any $$a \in (a_*(\beta ), a^*(\beta ))$$ and is not attained for $$a a^*(\beta )$$. We also give some qualitative estimates for $$a_*(\beta )$$ and $$a^*(\beta )$$. Our results complete the recent studies on the sharp Moser–Trudinger type inequality under constraints due to do O, Sani and Tarsi (Commun Contemp Math 19:27, 2016), Lam (Proc Am Math Soc 145:4885–4892, 2017; Math Nachr 291(14–15):2272–2287, 2018) and Ikoma, Ishiwata and Wadade (Math Ann 373(1–2):831–851, 2019).

The sharp Sobolev type inequalities in the Lorentz–Sobolev spaces in the hyperbolic spaces

Let W1Lp,q(Hn), 1≤q,p<∞ denote the Lorentz–Sobolev spaces of order one in the hyperbolic spaces Hn. Our aim in this paper is three-fold. First of all, we establish a sharp Poincaré inequality in W1Lp,q(Hn) with 1≤q≤p which generalizes the result in [41] to the setting of Lorentz–Sobolev spaces. Second, we prove several sharp Poincaré–Sobolev type inequalities in W1Lp,q(Hn) with 1≤q≤p<n which generalize the results in [45] to the setting of Lorentz–Sobolev spaces. Finally, we provide the improved Moser–Trudinger type inequalities in W1Ln,q(Hn) in the critical case p=n with 1≤q≤n which generalize the results in [43] and improve the results in [57]. In the proof of the main results, we shall prove a Pólya–Szegö type principle in W1Lp,q(Hn) with 1≤q≤p which maybe is of independent interest.

A Sharp Rearrangement Principle in Fourier Space and Symmetry Results for PDEs with Arbitrary Order

Abstract We prove sharp inequalities for the symmetric-decreasing rearrangement in Fourier space of functions in $\mathbb{R}^d$. Our main result can be applied to a general class of (pseudo-)differential operators in $\mathbb{R}^d$ of arbitrary order with radial Fourier multipliers. For example, we can take any positive power of the Laplacian $(-\Delta )^s$ with $s&amp;gt; 0$ and, in particular, any polyharmonic operator $(-\Delta )^m$ with integer $m \geqslant 1$. As applications, we prove radial symmetry and real-valuedness (up to trivial symmetries) of optimizers for (1) Gagliardo–Nirenberg inequalities with derivatives of arbitrary order, (2) ground states for bi- and polyharmonic nonlinear Schrödinger equations (NLS), and (3) Adams–Moser–Trudinger type inequalities for $H^{d/2}(\mathbb{R}^d)$ in any dimension $d \geqslant 1$. As a technical key result, we solve a phase retrieval problem for the Fourier transform in $\mathbb{R}^d$. To achieve this, we classify the case of equality in the corresponding Hardy–Littlewood majorant problem for the Fourier transform in $\mathbb{R}^d$.

Sharp Adams–Moser–Trudinger type inequalities in the hyperbolic space

The purpose of this paper is to establish some Adams–Moser–Trudinger inequalities, which are the borderline cases of the Sobolev embedding, in the hyperbolic space \mathbb H^n . First, we prove a sharp Adams’ inequality of order two with the exact growth condition in \mathbb H^n . Then we use it to derive a sharp Adams-type inequality and an Adachi–Tanakat-ype inequality. We also prove a sharp Adams-type inequality with Navier boundary condition on any bounded domain of \mathbb H^n , which generalizes the result of Tarsi to the setting of hyperbolic spaces. Finally, we establish a Lions-type lemma and an improved Adams-type inequality in the spirit of Lions in \mathbb H^n . Our proofs rely on the symmetrization method extended to hyperbolic spaces.

Fractional Adams–Moser–Trudinger type inequality on Heisenberg group

The aim of this paper is to establish singular fractional Adams–Moser–Trudinger inequality for both bounded and unbounded domains in the Heisenberg group Hn. We first establish fractional Adams–Moser–Trudinger type inequality on domain Ω⊂Rn with finite measure (Theorem 1.12) and then using this inequality and Hardy–Littlewood–Sobolev inequality adapted to the result of R.O’Neil(1963), we establish singular fractional Adams–Moser–Trudinger type inequality on domain Ω⊂Rn with finite measure (Theorem 1.13). We also establish singular fractional Adams–Moser–Trudinger type inequality in Hn, using the approach of N.Lam and G.Lu (2012, 2013). The main idea of our approach is that any function in higher order fractional Sobolev space in Heisenberg group can be represented in terms of Riesz potential and then using techniques from harmonic analysis and kernel properties of the associated operator, we establish fractional Adams–Moser–Trudinger type inequality in Hn. In this paper, our approach is quite simple and free from symmetrization arguments. As an applications of our theorems, we establish the existence of solution to the following class of problems Tαu=f(ξ,u)|ξ|a+b(ξ)|u|γ−1uinΩ,u=0inHn∖Ω,where Ω is a bounded subset of Hn of class C0,1 with bounded boundary and 0≤a<Q,f satisfies either the subcritical exponential growth or critical exponential growth condition and b is a small L2-perturbation, that is, there exists a small η>0 with 0<‖b‖L2(Ω)<η,0≤γ<1 and α=Q2.

The Hardy–Moser–Trudinger inequality via the transplantation of Green functions

<p style='text-indent:20px;'>We provide a new proof of the Hardy–Moser–Trudinger inequality and the existence of its extremals which are established by Wang and Ye ("G. Wang, and D. Ye, <i>A Hardy–Moser–Trudinger inequality</i>, Adv. Math, <b>230</b> (2012) 294–230.") without using the blow-up analysis method. Our proof is based on the transformation of functions via the transplantation of Green functions. This method enables us to compute explicitly the concentrating level of the Hardy–Moser–Trudinger functional over the normalizing concentrating sequences which is crucial to prove the existence of extremals for the Hardy–Moser–Trudinger inequality. Some comments on the applications of this approach to some other Moser–Trudinger type inequalities are given.

A simple proof of the Adams type inequalities in &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ {\mathbb R}^{2m} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;

We provide the simple proof of the Adams type inequalities in whole space $ {\mathbb R}^{2m} $. The main tools are the Fourier rearrangement technique introduced by Lenzmann and Sok [16], a Hardy–Rellich type inequality for radial functions, and the sharp Moser–Trudinger type inequalities in $ {\mathbb R}^2 $.

Remarks on the Moser–Trudinger type inequality with logarithmic weights in dimension 𝑁

We provide a simpler proof of the Moser–Trudinger type inequality with logarithmic weight w β = ( − ln ⁡ | x | ) β ( N − 1 ) w_\beta = (-\ln |x|)^{\beta (N-1)} , β ∈ [ 0 , 1 ) \beta \in [0,1) in dimension N ≥ 2 N\geq 2 recently established by Calanchi and Ruf. Our proof is based on a suitable change of functions on B B and the classical Moser–Trudinger inequality on B B . We also prove the existence of maximizers for this inequality when β ≥ 0 \beta \geq 0 sufficiently small.

Retraction Note: A sharp Trudinger type inequality for harmonic functions and its applications

The Editors-in-Chief have retracted this paper [1] because its results are invalid. The paper also shows significant overlap with a paper by Yang et al. [2], which was simultaneously under consideration with another journal. Additionally, this paper showed evidence of peer review and authorship manipulation. The authors have not responded to any correspondence with regards to this retraction.

Improved Singular Moser–Trudinger Inequalities and Their Extremal Functions

In this paper, we prove several improvements for the sharp singular Moser–Trudinger inequality. We first establish an improved singular Moser–Trudinger inequality in the spirit of Tintarev (J. Funct. Anal. 266, 55–66, 2014) and prove the existence of extremal functions for this improved singular Moser–Trudinger inequality. Our proof is based on the blow-up analysis method. Due to appearance of the singularity of weights, our proof is more complicated and difficult in dealing with analyzing the asymptotic behavior of the sequence of maximizers in the subcritical case near the blow-up point when comparing with the previous works (see, e.g., Nguyen (Ann. Global Anal. Geom. 54(2), 237–256, 2018), Yang (J. Funct. Anal. 239(1), 100–126, 2006)). To overcome these difficulties, we shall prove a classification result for a singular quasi-linear Liouville equation which maybe is of independent interest. Finally, we derive another improvement of the singular Moser–Trudinger inequality in the sprit of Adimurthi and Druet (Comm. Partial Differential Equations 29, 295–322, 2004). Our results extend many well-known Moser–Trudinger type inequalities to more general setting (e.g., singular weights, higher dimension, any domain, etc).

Existence and non-existence of maximizers for the Moser–Trudinger type inequalities under inhomogeneous constraints

In this paper, we study the existence and non-existence of maximizers for the Moser-Trudinger type inequalities in $\Bbb R^N$ of the form \[ D_{N,\alpha}(a,b):= \sup_{u\in W^{1,N}(\Bbb R^N),\,\|\nabla u\|_{L^N(\Bbb R^N)}^a+\|u\|_{L^N(\Bbb R^N)}^b=1} \int_{\Bbb R^N}\Phi_N\left(\alpha|u|^{N'}\right)dx. \] Here $N\geq 2$, $N'=\frac{N}{N-1}$, $a,b>0$, $\alpha \in (0,\alpha_N]$ and $\Phi_N(t):=e^t-\sum_{j=0}^{N-2}\frac{t^j}{j!}$ where $\alpha_N:= N \omega_{N-1}^{1/(N-1)}$ and $\omega_{N-1}$ denotes the surface area of the unit ball in $\Bbb R^N$. We show the existence of the threshold $\alpha_\ast = \alpha_\ast(a,b,N) \in [0,\alpha_N]$ such that $D_{N,\alpha}(a,b)$ is not attained if $\alpha \in (0,\alpha_\ast)$ and is attained if $ \alpha \in (\alpha_\ast , \alpha_N)$. We also provide the conditions on $(a,b)$ in order that the inequality $\alpha_\ast < \alpha_N$ holds.

An improved Leray–Trudinger inequality

In this paper, we derive the following Leray–Trudinger type inequality on a bounded domain [Formula: see text] in [Formula: see text] containing the origin. [Formula: see text] [Formula: see text] Here, [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], [Formula: see text] for [Formula: see text] This improves an earlier result by Psaradakis and Spector. Also, we prove that for any [Formula: see text] in the place of [Formula: see text], the above inequality is false if we take [Formula: see text].

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

We establish an improved version of the Moser–Trudinger inequality in the hyperbolic space Hn, n≥2. Namely, we prove the following result: for any 0≤λ<n−1nn, then we have supu∈C0∞(Hn)∫Hn|∇gu|gndVolg−λ∫Hn|u|ndVolg≤1∫HnΦn(αn|u|nn−1)dVolg<∞,where αn=nωn−11n−1, ωn−1 denotes the surface area of the unit sphere in Rn and Φn(t)=et−∑j=0n−2tjj!. This improves the Moser–Trudinger inequality in hyperbolic spaces obtained recently by Mancini and Sandeep (2010), by Mancini et al. (2013) and by Adimurthi and Tintarev (2010) for λ=0. In the limiting case λ=(n−1n)n, we prove a Moser–Trudinger inequality with exact growth in Hn, supu∈C0∞(Hn)∫Hn|∇gu|gndVolg−(n−1n)n∫Hn|u|ndVolg≤11∫Hn|u|ndVolg∫HnΦn(αn|u|nn−1)(1+|u|)nn−1dVolg<∞.This improves the Moser–Trudinger inequality with exact growth in Hn established by Lu and Tang (2016). These inequalities are achieved from the comparison of the symmetric non-increasing rearrangement of a function both in the hyperbolic and the Euclidean spaces, and the same inequalities in the Euclidean space. This approach seems to be new comparing with the previous ones.

A note on the Moser–Trudinger inequality in Sobolev–Slobodeckij spaces in dimension one

We discuss some recent results by Parini and Ruf on a Moser–Trudinger type inequality in the setting of Sobolev–Slobodeckij spaces in dimension one. We push further their analysis considering the inequality on the whole \mathbb R and we give an answer to one of their open questions.