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 $ω_{n-1}$ denotes the surface area of unit sphere in $\mathbb R^n$. We also show that this supremum is attained by some function $u^*\in W^{1,n}(Ω)$ such that $\int_Ωu^* dx =0$ and $\|u^*\|_{1,α} =1$. This generalizes a result of Ngo and Nguyen \cite{NN17} in dimension two and a result of Yang \cite{Yang07} for $α=0$, and improves a result of Cianchi \cite{Cianchi05}.
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.
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 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.
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!…
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).
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…
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…
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 …
The existence of an extremal in an exponential Sobolev type inequality, with optimal constant, in Gauss space is established. A key step in the proof is an augmented version of the relevant inequality, which, by contrast, fails for a parallel classical inequality by Moser in the Euclidean space.
