Hardy–Sobolev–Maz’ya inequalities for polyharmonic operators

Abstract

Let K be affine, that is, $K=\{z=(x,y)\in {\mathbb {R}}^{n+m}: y_{1}=\cdots =y_{m}=0\}$. We compute the sharp constant of Hardy inequality related to the distance d(z, K) for polyharmonic operator. Moreover, we show that there exists a constant $C>0$ such that for each $u\in C^{\infty }_{0}({\mathbb {R}}^{n+m}\setminus K)$, there holds

$$\begin{aligned} \int _{{\mathbb {R}}^{n+m}}|\nabla ^{k} u|^{2}\mathrm{d}x\mathrm{d}y-c_{m,k}\int _{{\mathbb {R}}^{n+m}}\frac{u^{2}}{|y|^{2k}}\mathrm{d}x\mathrm{d}y\ge C\left( \int _{{\mathbb {R}}^{n+m}}|y|^{\gamma }|u|^{p}\mathrm{d}x\mathrm{d}y\right) ^{\frac{n+m-2k}{n+m}}, \end{aligned}$$

where $2\le k<\frac{m+n}{2}$, $2<p\le \frac{2(n+m)}{n+m-2k}$, $\gamma =\frac{(n+m-2k)p}{2}-n-m$ and $c_{m,k}$ is the sharp Hardy constant. These inequalities generalize the result of Maz’ya (case $k=1$) and Lu and the second author (case $m=1$ for polyharmonic operators). In order to prove the main result, we establish some Poincaré–Sobolev inequalities on hyperbolic space which is of independent interest.