A new approach to sharp Moser-Trudinger and Adams type inequalities: a rearrangement-free argument
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 \(0 < \gamma < n\) in the form of the first theorem with \(\alpha = 0\), where \[ \sup_{u \in W^{\gamma, p}(\mathbb R^n), \;\|(\tau I - \Delta)^{\gamma/2} u \|_{p} \leq 1} \int_{\mathbb R^n} \phi(\beta_0(n, \gamma)|u|^{p^{\prime}})\, dx < \infty, \] \[ \phi(t) = e^t - \sum_{j = 0}^{j_p - 2} \frac{ t^j}{j!} ,\,\, j_{p} = \min \{ j \in \mathbb N: j \geq p \} \geq p, \] and the result is sharp in the same sense. The introduction has a summary of the historical development of Moser-Trudinger and Adams-type inequalities. They conclude with a version applicable to Carnot groups.
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