Weighted Hardy-Adams inequality on unit ball of any even dimension

Abstract

In this study, we obtain the weighted Hardy-Adams inequality of any even dimension n≥4n\ge 4. Namely, for u∈C0∞(Bn)u\in {C}_{0}^{\infty }\left({{\mathbb{B}}}^{n}) with ∫Bn∣∇n2u∣2dx−∏k=1n⁄2(2k−1)2∫Bnu2(1−∣x∣2)ndx≤1,\mathop{\int }\limits_{{{\mathbb{B}}}^{n}}{| {\nabla }^{\frac{n}{2}}u| }^{2}{\rm{d}}x-\mathop{\prod }\limits_{k=1}^{n/2}{\left(2k-1)}^{2}\mathop{\int }\limits_{{{\mathbb{B}}}^{n}}\frac{{u}^{2}}{{\left(1-{| x| }^{2})}^{n}}{\rm{d}}x\le 1, then the following Hardy-Adams inequalities hold: ∫Bnexp22−n−ϑβ0n2,n(uφϑ)2(1−∣x∣2)2ϑ+n−4dx≤Cn,ϑ,ϑ∈3−n2,0\begin{array}{r}\mathop{\displaystyle \int }\limits_{{{\mathbb{B}}}^{n}}\exp \left({2}^{2-n-{\vartheta }}{\beta }_{0}\left(\frac{n}{2},n\right){\left(u{\varphi }_{{\vartheta }})}^{2}\right){(1-{| x| }^{2})}^{2{\vartheta }+n-4}{\rm{d}}x\le {C}_{n,{\vartheta }},\hspace{0.33em}\hspace{0.33em}{\vartheta }\in \left(\frac{3-n}{2},0\right]\end{array} and ∫BnΦ222−n−ϑβ0n2,n(uφϑ)2(1−∣x∣2)2ϑ+n−4dx≤Cn,ϑ,ϑ∈(2−n,3−n2],\mathop{\int }\limits_{{{\mathbb{B}}}^{n}}{\Phi }_{2}\left({2}^{2-n-{\vartheta }}{\beta }_{0}\left(\frac{n}{2},n\right){\left(u{\varphi }_{{\vartheta }})}^{2}\right){(1-{| x| }^{2})}^{2{\vartheta }+n-4}{\rm{d}}x\le {C}_{n,{\vartheta }},\hspace{0.33em}\hspace{0.33em}{\vartheta }\in \left(2-n,\frac{3-n}{2}\right], where Φ2(t)=et−∑1j=0tjj!{\Phi }_{2}\left(t)={e}^{t}-\mathop{\sum ^{1}}\limits_{j=0}\frac{{t}^{j}}{j\!}, φϑ(x)=21−∣x∣2n+ϑ−22{\varphi }_{{\vartheta }}\left(x)={\left(,\frac{2}{1-{| x| }^{2}}\right)}^{\tfrac{n+{\vartheta }-2}{2}} and Cn,ϑ{C}_{n,{\vartheta }} is a positive constant independent of uu. The crucial method in this article is borrowed from Lu and Yang [Adv. Math. 319 (2017)] and Li, Lu and Yang [Adv. Math. 333 (2018)]. We also apply set level method which was first developed by Lam and Lu [J. Diff. Equa. 255 (2013)]. The new ingredient here is to utilize the relationship between ΔH{\Delta }_{{\mathbb{H}}} and the Laplace-Beltrami operator Δϑ{\Delta }_{{\vartheta }}, where Δϑ{\Delta }_{{\vartheta }} is developed by Liu and Peng [Indiana Univ. Math. J. 3 (2009), 1457–1492]. Furthermore, the Hardy-Adams inequality in this article is the weighted version of the result of Li et al. [Adv. Math. 333 (2018), 350–385].

Keywords

• hardy-adams inequality
• poincaré-sobolev inequality
• laplace-beltrami operators δϑ
• primary 35j20
• 46e35