A sharpened form of Adams-type inequalities on higher-order Sobolev spaces $W^{m,\frac {n}{m}}(\mathbb {R}^n)$: a simple approach

Chen, Lu ; Lu, Guozhen ; Zhu, Maocun

Canadian mathematical bulletin, Tome 65 (2022) no. 4, pp. 895-905

Résumé

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, $$ \begin{align*}\sup_{\|\nabla^mu\|_{\frac{n}{m}}^{\frac{n}{m}}+\|u\|_{\frac{n}{m}}^{\frac{n}{m}}\leq1}\int_{\mathbb{R}^n}\Phi\Big(\beta_{n,m} (1+\alpha \|u\|_{\frac{n}{m}}^{\frac{n}{m}} )^{\frac{m}{n-m}}|u|^{\frac{n}{n-m}}\Big)dx<\infty, \end{align*} $$for any $0<\alpha <1$. 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.

Détail

DOI : 10.4153/S0008439521001028

Mots-clés : Adams inequalities, best constants, rearrangement-free argument, Moser-Trudinger inequalities