Let $d \ge 1$, $p \ge d$, and let $Ω$ be a smooth bounded open subset of $\mathbb{R}^d$. We prove some exponential integrability in the spirit of Moser-Trudinger's inequalities for measurable functions $u$ defined in $Ω$ such that $$ \mathop{\int_Ω \int_Ω}_{|u(x) - u(y)| > δ} \frac{1}{|x-y|^{d+p}} \, dx \, dy < + \infty, $$ for some $δ> 0$. This double integral appeared in characterizations of Sobolev spaces and involved in improvements of the Sobolev inequaliies, Poincaré inequalities, and Hardy inequalities.
We show that the Moser-Trudinger inequality holds in a conformal disc if and only if the metric is bounded from above by the Hyperbolic metric. We also find a necessary and sufficient condition for the Moser-Trudinger inequality to hold in an unbounded subset of the two dimensional Euclidean space.
Given a compact closed four dimensional smooth Riemannian manifold, we prove existence of extremal functions for Moser-Trudinger type inequality. The method used is Blow-up analysis combined with capacity techniques.
This paper is devoted to study the sharp Moser-Trudinger type inequalities in whole space $\mathbb R^N$, $N \geq 2$ in more general case. We first compute explicitly the \emph{normalized vanishing limit} and the \emph{normalized concentrating limit} of the Moser-Trudinger type functional associated with our inequalities over all the \emph{normalized vanishing sequences} and the \emph{normalized concentrating sequences}, respectively. Exploiting these limits together with the concentration-compactness principle of Lions type, we give a proof of the exitence of maximizers for these Moser-Truding…
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.
We study a sharp fractional Moser-Trudinger type inequality in dimension 1, its compactness properties and the critical points of a functional associeted to the inequality.
The paper raises a question about the optimal critical nonlinearity for the Sobolev space in two dimensions, connected to loss of compactness, and discusses the pertinent concentration compactness framework. We study properties of the improved version of the Trudinger-Moser inequality on the open unit disk $B\subset\R^2$, recently proved by G. Mancini and K. Sandeep. Unlike the original Trudinger-Moser inequality, this inequality is invariant with respect to Möbius automorphisms of the unit disk, and as such is a closer analogy of the critical nonlinearity $\int |u|^{2^*}$ in the higher dimens…
We consider the problem of finding the optimal exponent in the Moser-Trudinger inequality \[ \sup \left\{\int_Ω\exp{\left(α\,|u|^{\frac{N}{N-s}}\right)}\,\bigg|\,u \in \widetilde{W}^{s,p}_0(Ω),\,[u]_{W^{s,p}(\mathbb{R}^N)}\leq 1 \right\}< + \infty.\] Here $Ω$ is a bounded domain of $\mathbb{R}^N$ ($N\geq 2$), $s \in (0,1)$, $sp = N$, $\widetilde{W}^{s,p}_0(Ω)$ is a Sobolev-Slobodeckij space, and $[\cdot]_{W^{s,p}(\mathbb{R}^N)}$ is the associated Gagliardo seminorm. We exhibit an explicit exponent $α^*_{s,N}>0$, which does not depend on $Ω$, such that the Moser-Trudinger inequality does not ho…
Trudinger-Moser inequality is a substitute to the (forbidden) critical Sobolev embedding, namely the case where the scaling corresponds to $L^\infty$. It is well known that the original form of the inequality with the sharp exponent (proved by Moser) fails on the whole plane, but a few modified versions are available. We prove a precised version of the latter, giving necessary and sufficient conditions for the boundedness, as well as for the compactness, in terms of the growth and decay of the nonlinear function. It is tightly related to the ground state of the nonlinear Schrodinger equation (…
We derive the sharp Moser-Trudinger-Onofri inequalities on the standard $n$-sphere and CR $(2n+1)$- sphere as the limit of the sharp fractional Sobolev inequalities for all $n\ge 1$. On the $2$-sphere and $4$-sphere, this was established recently by S.-Y. Chang and F. Wang. Our proof uses an alternative and elementary argument.
Let $Ω$ be a smooth bounded domain in $\mathbf R^2$ and $λ^{\mathsf N} (Ω)$ the first non-zero Neumann eigenvalue of the operator $-Δ$ on $Ω$. In this paper, for any $γ\in [0, λ^{\mathsf N} (Ω) )$, we establish the following improved Moser-Trudinger inequality \[ \sup_{u} \int_Ω e^{2πu^2} dx < +\infty \] for arbitrary functions $u$ in $H^1(Ω)$ satisfying $\int_Ωu dx =0$ and $\|\nabla u\|_2^2 -α\|u\|_2^2 \leqslant 1$. Furthermore, this supremum is attained by some function $u^*\in H^1(Ω)$. This strengthens the results of Chang and Yang (J. Differential Geom. 27 (1988) 259-296) and of Lu and Yan…
Given a closed Riemann surface $(Σ,g)$ and any positive smooth weight, we use a minmax scheme together with compactness, quantization results and with sharp energy estimates to prove the existence of positive critical points of the functional $$J_{p,β}(u)=\frac{2-p}{2}\left(\frac{p\|u\|_{H^1}^2}{2β} \right)^{\frac{p}{2-p}}-\ln \int_Σ(e^{u_+^p}-1) f dv_g,$$ for every $p\in (1,2)$ and $β>0$, {or} for $p=1$ and $β\in (0,\infty)\setminus 4π\mathbb{N}$. Letting $p\uparrow 2$ we obtain positive critical points of the Moser-Trudinger functional $$F(u):=\int_Σ(e^{u^2}-1)f dv_g$$ constrained to $\mathc…
We establish an improved version of the Moser-Trudinger inequality in the hyperbolic space $\mathbb H^n$, $n\geq 2$. Namely, we prove the following result: for any $0 \leq λ< \left(\frac{n-1}n\right)^n$, then we have $$ \sup_{\substack{u\in C_0^\infty(\mathbb H^n) \int_{\mathbb H^n} |\nabla_g u|_g^n d\text{Vol}_g -λ\int_{\mathbb H^n} |u|^n d\text{ Vol}_g \leq 1}} \int_{\mathbb H^n} Φ_n(α_n |u|^{\frac{n}{n-1}}) d\text{ Vol}_g < \infty, $$ where $α_n = n ω_{n-1}^{\frac1{n-1}}$, $ω_{n-1}$ denotes the surface area of the unit sphere in $\mathbb R^n$ and $Φ_n(t) = e^t -\sum_{j=0}^{n-2}\frac{t^j}{j!…
On the 2-dimensional unit disk $B_1$ we study the Moser-Trudinger functional $$E(u)=\int_{B_1}(e^{u^2}-1)dx, u\in H^1_0(B_1)$$ and its restrictions to $M_Λ:=\{u \in H^1_0(B_1):\|u\|^2_{H^1_0}=Λ\}$ for $Λ>0$. We prove that if a sequence $u_k$ of positive critical points of $E|_{M_{Λ_k}}$ (for some $Λ_k>0$) blows up as $k\to\infty$, then $Λ_k\to 4π$, and $u_k\to 0$ weakly in $H^1_0(B_1)$ and strongly in $C^1_{\loc}(\bar B_1\setminus\{0\})$. Using this we also prove that when $Λ$ is large enough, then $E|_{M_Λ}$ has no positive critical point, complementing previous existence results by Carleson-…
In this paper, using blow-up analysis, we prove a singular Hardy-Morser-Trudinger inequality, and find its extremal functions. Our results extend those of Wang-Ye (Adv. Math. 2012), Yang-Zhu ( Ann. Glob. Anal. Geom. 2016), Csató- Roy (Calc. Var. 2015), and Yang-Zhu (J. Funct. Anal. 2017).
Extending several works, we prove a general Adams-Moser-Trudinger type inequality for the embedding of Bessel-potential spaces $\tilde H^{\frac{n}{p},p}(Ω)$ into Orlicz spaces for an arbitrary domain $Ω\subset \mathbb{R}^n$ with finite measure. In particular we prove $$\sup_{u\in \tilde H^{\frac{n}{p},p}(Ω), \;\|(-Δ)^{\frac{n}{2p}}u\|_{L^{p}(Ω)}\leq 1}\int_Ωe^{α_{n,p} |u|^\frac{p}{p-1}}dx \leq c_{n,p}|Ω|, $$ for a positive constant $α_{n,p}$ whose sharpness we also prove. We further extend this result to the case of Lorentz-spaces (i.e. $(-Δ)^\frac{n}{2p}u\in L^{(p,q)})$. The proofs are simple…
In this article, we study the connection between the fractional Moser-Trudinger inequality and the fractional $\left(\frac{kp}{p-1},p\right)$-Poincaré type inequality for any Euclidean domain and discuss the sharpness of this inequality whose analogous results are well known in the local case. We further provide sufficient conditions on domains for fractional $(q,p)$-Poincaré type inequalities to hold. We also derive Adachi-Tanaka type inequalities in the non-local setting.
The paper is concerned about a sharp form of Anisotropic Moser-Trudinger inequality which involves $L^{n}$ norm. Let \begin{equation*} λ_{1}(Ω) = \inf_{u\in W_0^{1,n}(Ω),u\not\equiv 0} ||F(\nabla u)||_{L^n(Ω)}^n / ||u||_{L^n(Ω)}^n \end{equation*} be the first eigenvalue associated with $n$-Finsler-Laplacian. using blowing up analysis, we obtain that \begin{equation*} \sup_{u\in W_{0}^{1,n}(Ω),||F(\nabla u)||_{L^n(Ω)} = 1} \int_Ωe^{λ_n (1+α||u||_{L^n (Ω)}^n)^{\frac{1}{n-1}} |u|^{\frac{n}{n-1}}}dx \end{equation*} is finite for any $0\leq α<λ_{1}(Ω)$,and the supremum is infinite for any $α\geq λ_…
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-Tanaka-type 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…
Let $Ω$ be a bounded smooth domain in $\mathbb R^n$, $W^{1,n}(Ω)$ be the Sobolev space on $Ω$, and $λ(Ω) = \inf\{\|\nabla u\|_n^n: \int_Ωu dx =0, \|u\|_n =1\}$ be the first nonzero Neumann eigenvalue of the $n-$Laplace operator $-Δ_n$ on $Ω$. For $0 \leq α< λ(Ω)$, let us define $\|u\|_{1,α}^n =\|\nabla u\|_n^n -α\|u\|_n^n$. We prove, in this paper, the following improved Moser--Trudinger inequality on functions with mean value zero on $Ω$, \[ \sup_{u\in W^{1,n}(Ω), \int_Ωu dx =0, \|u\|_{1,α} =1} \int_Ω e^{β_n |u|^{\frac n{n-1}}} dx < \infty, \] where $β_n = n (ω_{n-1}/2)^{1/(n-1)}$, and $ω_{n-…
On a smooth bounded 2-dimensional domain $Ω$ we study the heat flow $u_t=Δu +λ(t)ue^{u^2}$ ($λ(t)$ is such that $d/dt ||u(t,\cdot)||_{H^1_0}=0$) introduced by T. Lamm, F. Robert and M. Struwe to investigate the Moser-Trudinger functional $E(v)=\int_Ω (e^{v^2}-1)dx, v\in H^1_0(Ω).$ We prove that if $u$ blows-up as $t\to\infty$ and if $E(u(t,\cdot))$ remains bounded, then for a sequence $t_k\to\infty$ we have $u(t_k,\cdot)\rightharpoonup 0$ in $H^1_0$ and $\|u(t_k,\cdot)\|_{H^1_0}^2\to 4πL$ for an integer $L\ge 1$. We couple these results with a topological technique to prove that if $Ω$ is not …
We derive a sharp Moser-Trudinger inequality for the borderline Sobolev imbedding of W^{2,n/2}(B_n) into the exponential class, where B_n is the unit ball of R^n. The corresponding sharp results for the spaces W_0^{d,n/d}(Ω) are well known, for general domains Ω, and are due to Moser and Adams. When the zero boundary condition is removed the only known results are for d=1 and are due to Chang-Yang, Cianchi and Leckband. Our proof is based on general abstract results recently obtained by the authors, and on a new integral representation formula for the "canonical" solution of the Poisson equati…
In this paper, we prove a Hardy--Moser--Trudinger inequality in the unit ball $\mathbb B^n$ in $\mathbb R^n$ which improves both the classical singular Moser--Trudinger inequality and the classical Hardy inequality at the same time. More precisely, we show that for any $β\in [0,n)$ there exists a constant $C>0$ depending only on $n$ and $β$ such that \[ \sup_{u\in W^{1,n}_0(\mathbb B^n), \mathcal H(u) \leq 1}\int_{\mathbb B^n} e^{(1-\fracβn)α_n |u|^{\frac n{n-1}}} |x|^{-β} dx \leq C \] where $α_n = n ω_{n-1}^{\frac1{n-1}}$ with $ω_{n-1}$ being the surface area of the unit sphere $S^{n-1} = \pa…
The Trudinger-Moser inequality states that for functions $u \in H_0^{1,n}(Ω)$ ($Ω\subset \mathbb R^n$ a bounded domain) with $\int_Ω|\nabla u|^ndx \le 1$ one has $\int_Ω(e^{α_n|u|^{\frac n{n-1}}}-1)dx \le c |Ω|$, with $c$ independent of $u$. Recently, the second author has shown that for $n = 2$ the bound $c |Ω| $ may be replaced by a uniform constant $d$ independent of $Ω$ if the Dirichlet norm is replaced by the Sobolev norm, i.e. requiring $\int_Ω(|\nabla u|^n + |u|^n)dx \le 1$. We extend here this result to arbitrary dimensions $n > 2$. Also, we prove that for $Ω= \mathbb R^n$ the supremum…
With a view towards Riemannian or sub-Riemannian manifolds, RCD metric spaces and specially fractals, this paper makes a step further in the development of a theory of heat semigroup based $(1,p)$ Sobolev spaces in the general framework of Dirichlet spaces. Under suitable assumptions that are verified in a variety of settings, the tools developed by D. Bakry, T. Coulhon, M. Ledoux and L. Saloff-Coste in the paper "Sobolev inequalities in disguise" allow us to obtain the whole family of Gagliardo-Nirenberg and Trudinger-Moser inequalities with optimal exponents. The latter depend not only on th…
Given $α>0$, we establish the following two supercritical Moser-Trudinger inequalities \[ \sup\limits_{u \in W^{1,n}_{0,{\rm rad}}(B): \int_B |\nabla u|^n dx \leq 1} \int_B \exp\big( (α_n + |x|^α) |u|^{\frac{n}{n-1}} \big) dx < +\infty \] and \[ \sup\limits_{u\in W^{1,n}_{0,{\rm rad}}(B): \int_B |\nabla u|^n dx \leq 1} \int_B \exp\big( α_n |u|^{\frac{n}{n-1} + |x|^α} \big) dx < +\infty, \] where $W^{1,n}_{0,{\rm rad}}(B)$ is the usual Sobolev spaces of radially symmetric functions on $B$ in $\mathbb R^n$ with $n\geq 2$. Without restricting to the class of functions $W^{1,n}_{0,{\rm rad}}(B)$, …
We study the existence and nonexistence of maximizers for variational problem concerning to the Moser--Trudinger inequality of Adimurthi--Druet type in $W^{1,N}(\mathbb R^N)$ \[ MT(N,β, α) =\sup_{u\in W^{1,N}(\mathbb R^N), \|\nabla u\|_N^N + \|u\|_N^N\leq 1} \int_{\mathbb R^N} Φ_N(β(1+α\|u\|_N^N)^{\frac1{N-1}} |u|^{\frac N{N-1}}) dx, \] where $Φ_N(t) =e^{t} -\sum_{k=0}^{N-2} \frac{t^k}{k!}$, $0\leq α< 1$ both in the subcritical case $β< β_N$ and critical case $β=β_N$ with $β_N = N ω_{N-1}^{\frac1{N-1}}$ and $ω_{N-1}$ denotes the surface area of the unit sphere in $\mathbb R^N$. We will show th…
In this paper, suppose $F: \mathbb{R}^{N} \rightarrow [0, +\infty)$ be a convex function of class $C^{2}(\mathbb{R}^{N} \backslash \{0\})$ which is even and positively homogeneous of degree 1. We establish the Lions type concentration-compactness principle of singular Trudinger-Moser Inequalities involving $N$-Finsler--Laplacian operator. Let $Ω\subset \mathbb{R}^{N}(N\geq 2)$ be a smooth bounded domain. $\{u_n\}\subset W_0^{1, N}(Ω)$ be a sequence such that anisotropic Dirichlet norm$\int_ΩF^N (\nabla u_n)dx=1$, $u_n \rightharpoonup u \not \equiv 0$ weakly in $W_0^{1, N}(Ω)$. Then for any $0 …
In this paper we derive a variety of functional inequalities for general homogeneous invariant hypoelliptic differential operators on nilpotent Lie groups. The obtained inequalities include Hardy, Rellich, Hardy-Littllewood-Sobolev, Galiardo-Nirenberg, Caffarelli-Kohn-Nirenberg and Trudinger-Moser inequalities. Some of these estimates have been known in the case of the sub-Laplacians, however, for more general hypoelliptic operators almost all of them appear to be new as no approaches for obtaining such estimates have been available. Moreover, we obtain several versions of local and global wei…
