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A sharpened form of Adams-type inequalities on higher-order Sobolev spaces
$W^{m,\frac {n}{m}}(\mathbb {R}^n)$: a simple approach
Published online by Cambridge University Press: 15 December 2021
Abstract
In this paper, we develop an extremely simple method to establish the sharpened Adams-type inequalities on higher-order Sobolev spaces
$W^{m,\frac {n}{m}}(\mathbb {R}^n)$
in the entire space
$\mathbb {R}^n$
, which can be stated as follows: Given
$\Phi \left ( t\right ) =e^{t}-\underset {j=0}{\overset {n-2}{\sum }} \frac {t^{j}}{j!}$
and the Adams sharp constant
$\beta _{n,m}$
. Then,
$$ \begin{align*}\sup_{\|\nabla^mu\|_{\frac{n}{m}}^{\frac{n}{m}}+\|u\|_{\frac{n}{m}}^{\frac{n}{m}}\leq1}\int_{\mathbb{R}^n}\Phi\Big(\beta_{n,m} (1+\alpha \|u\|_{\frac{n}{m}}^{\frac{n}{m}} )^{\frac{m}{n-m}}|u|^{\frac{n}{n-m}}\Big)dx<\infty, \end{align*} $$
$0<\alpha <1$
. Furthermore, we construct a proper test function sequence to derive the sharpness of the exponent
$\alpha $
of the above Adams inequalities. Namely, we will show that if
$\alpha \ge 1$
, then the above supremum is infinite.
Our argument avoids applying the complicated blow-up analysis often used in the literature to deal with such sharpened inequalities.
MSC classification
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- Copyright
- © Canadian Mathematical Society, 2021
Footnotes
The first author was supported partly by the National Natural Science Foundation of China (No. 11901031). The second author was supported partly by the Simons Foundation. The third author was supported partly by the National Natural Science Foundation of China (No. 12071185).
References
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