A new approach to sharp Moser-Trudinger and Adams type inequalities: a rearrangement-free argument
type: Item
instance of: scholarly article
title: A new approach to sharp Moser-Trudinger and Adams type inequalities: a rearrangement-free argument
zbMATH Open document ID: 1294.46034
author: Guozhen Lu
author: Nguyen Lam
published in: Journal of Differential Equations
publication date: 2014-01-30T00:00:00Z
review text: The authors develop a new approach to singular Adams-type inequalities on high-order Sobolev spaces \(W^{m, n/m}(\mathbb R^n)\). \textit{B. Ruf} and \textit{F. Sani} [Trans. Am. Math. Soc. 365, No.~2, 645--670 (2013; Zbl 1280.46024)] proved these results for \(m\) even, and the authors [J. Differ. Equations 253, No.~4, 1143--1171 (2012; Zbl 1259.46029); Methods Appl. Anal. 19, No.~3, 243--266 (2012; Zbl 1319.46027)] extended them to \(m\) odd but for a more restrictive norm. They give a more complete generalization here, in which \(m\) need not be an integer. Their first result is that, if \(m\) is a positive integer less than \(n, \tau >0\) and \(0 \leq \alpha <n\), then \[ \sup_{u \in W^{m, n/m}(\mathbb R^n), \;\|(\tau I - \Delta)^{m/2}u \|_{n/m} \leq 1 } \int_{\mathbb R^n} \frac{ \phi(\beta_{\alpha, n, m} |u|^{\frac{n}{n - m} } )} { |x|^{\alpha} } \, dx < \infty, \] where \[ \phi(t) = e^t - \sum_{j = 0}^{j_{n/m} - 2 } \frac{t^j}{j!}, \] \[ j_{n/m} = \min \{ j \in \mathbb N: j \geq n/m \} \geq n/m, \] \[ \beta_{\alpha, n, m} =\left (1 - \frac{\alpha}{n}\right) \beta_0(n, m), \] \[ \beta_0(n,m) = \frac{n}{\omega_{n -1}} \left[\frac { \pi^{\frac{n}2} 2^m \Gamma(\frac{m}2) } { \Gamma(\frac{n - m}2) } \right]^{\frac{n}{n - m} }. \] The result is sharp in the sense that if one replaces \(\beta_{\alpha, n, m}\) by any \(\beta> \beta_{\alpha, n,m}\), the supremum is infinite. The result can be stated in terms of the standard Sobolev norm if \(n = 2m\), where \(m\) is an integer, and the authors obtain \[ \sup_{u \in W^{m, 2}\left(\mathbb R^{2m}\right),\quad \| u \|_{W^{m,2}} \leq 1} \int_{\mathbb R^{2m}} \frac{ \text{exp}((1 - \frac{\alpha}{2m}) \beta_0(2m, m) |u|^2 ) - 1} { |x|^{\alpha} } \, dx < \infty, \] \noindent where \(\beta_0(2m,m)\) is defined above, and the result is sharp in the sense that if one chooses any \(\beta >(1 - \frac{\alpha}{2m}) \beta_0(2m, m) \), the supremum is infinite. There is also a fractional version for an arbitrary real number \(0 < \gamma…
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 6252414
zbMATH Keywords: Moser-Trudinger inequality
zbMATH Keywords: Adams inequality
zbMATH Keywords: Sobolev embeddings
zbMATH Keywords: high-order Sobolev spaces
reviewed by: Raymond Johnson
MaRDI profile type: Publication
full work available at URL: j.jde.2013.04.005
OpenAlex ID: W2077034061
cites work: Trudinger type inequalities in $\mathbf {R}^N$ and their best exponents
cites work: A sharp inequality of J. Moser for higher order derivatives
cites work: A singular Moser-Trudinger embedding and its applications
cites work: On a version of Trudinger-Moser inequality with Möbius shift invariance
cites work: Q3581209
cites work: Fundamental solution for the \(Q\)-Laplacian and sharp Moser--Trudinger inequality in Carnot groups.
cites work: Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality
cites work: Moser-Trudinger and Beckner-Onofri's inequalities on the CR sphere
cites work: Nontrivial Solution of Semilinear Elliptic Equations with Critical Exponent in R
cites work: Q3756181
cites work: Conformal deformations of metric on \({\mathcal S}\) 2
cites work: Moser--Trudinger trace inequalities
cites work: Moser-Trudinger inequalities without boundary conditions and isoperimetric problems
cites work: Best constants for Moser-Trudinger inequalities on the Heisenberg group
cites work: Sharp constants for Moser‐Trudinger inequalities on spheres in complex space ℂn
cites work: Best constants for Moser-Trudinger inequalities, fundamental solutions and one-parameter representation formulas on groups of Heisenberg type
cites work: On an inequality by N. Trudinger and J. Moser and related elliptic equations
cites work: \(N\)-Laplacian equations in \(\mathbb{R}^N\) with critical growth
cites work: Extremal functions for the Trudinger-Moser inequality in 2 dimensions
cites work: Hardy Spaces on Homogeneous Groups. (MN-28), Volume 28
cites work: Sharp borderline Sobolev inequalities on compact Riemannian manifolds
cites work: Adams inequalities on measure spaces
cites work: Q3531572
cites work: Trudinger-Moser inequality on the whole plane with the exact growth condition
cites work: Existence and nonexistence of maximizers for variational problems associated with Trudinger--Moser type inequalities in \({\mathbb{R}^N}\)
cites work: Q5480316
cites work: Upper bound of the best constant of the Trudinger-Moser inequality and its application to the Gagliardo-Nirenberg inequality
cites work: Sharp Adams type inequalities in Sobolev spaces \(W^{m,\frac{n}{m}}(\mathbb R^n)\) for arbitrary integer \(m\)
cites work: Sharp singular Adams inequalities in high order Sobolev spaces
cites work: Sharp Moser-Trudinger inequality on the Heisenberg group at the critical case and applications
cites work: Q5500028
cites work: On Nonuniformly Subelliptic Equations of Q−sub-Laplacian Type with Critical Growth in the Heisenberg Group
cites work: Sharp affine and improved Moser-Trudinger-Adams type inequalities on unbounded domains in the spirit of Lions
cites work: Moser's inequality on the ball Bn for functions with mean value zero
cites work: Moser-Trudinger inequality on compact Riemannian manifolds of dimension two
cites work: Extremal functions for the Moser-Trudinger inequalities on compact Riemannian manifolds
cites work: Extremal functions for Moser-Trudinger type inequality on compact closed 4-manifolds
cites work: A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbb{R}^n$
cites work: Extremal functions for Moser’s inequality
cites work: The concentration-compactness principle in the calculus of variations. The limit case. I
cites work: The concentration-compactness principle in the calculus of variations. The limit case. II
cites work: Adams' inequalities for bi-Laplacian and extremal functions in dimension four
cites work: A sharpened Moser-Pohozaev-Trudinger inequality with mean value zero in \(\mathbb R^2\)
cites work: MOSER–TRUDINGER INEQUALITY ON CONFORMAL DISCS
cites work: Q5603485
cites work: Q5516356
cites work: A sharp Trudinger-Moser type inequality for unbounded domains in \(\mathbb R^2\)
cites work: Sharp Adams-type inequalities in ℝⁿ
cites work: Q5607484
cites work: Q4108681
cites work: Adams' inequality and limiting Sobolev embeddings into Zygmund spaces
cites work: Moser-Trudinger type inequalities for the Hessian equation
cites work: A symmetrization result for elliptic equations with lower-order terms
cites work: Q5547919
cites work: A Hardy-Moser-Trudinger inequality
cites work: Adams type inequalities and related elliptic partial differential equations in dimension four
cites work: Q5520312
DOI: 10.1016/J.JDE.2013.04.005