Hardy-Sobolev-type inequalities with monomial weights
In this paper, the author gives some conditions on \(A,B \in \mathbb{R}^N\) and \(p \geq 1\) for the existence of \(p^*\) and a constant \(C>0\) such that the Hardy-Sobolev-type inequality \[ \Big(\int_{\mathbb{R}^N}| x^B u(x)|^{p^{*}}dx\Big)^{\frac{1}{p^*}} \leq \Big(\int_{\mathbb{R}^N}| x^A \nabla u(x)|^{p}dx\Big)^{\frac{1}{p}} \] holds for all \(u\in C^{\infty}_{C} (\mathbb{R}^N)\) and \(N \geq 1\). In addition, the author gives an alternative technique to prove the above Hardy-Sobolev-type inequality with monomial weight using the elementary calculus tools instead of the Alexandroff-Bakelman-Pucci (ABP) method, the theory of Muckhenhoupt weights or other general techniques in the literature on weighted Sobolev inequalities. Furthermore, it is shown that this elementary method in comparison with the ABP method works not only for domains of the form \(\mathbb{R}^{N}_{A}\), that is, a domain of the form \[ \mathbb{R}^{N}_{I,+}=\{(x_1, x_2,\dots,x_N)\in \mathbb{R}^{N}:x_i \geq 0 \text{ for } i\in I\}, \] or \[ \mathbb{R}^{N}_{I,-}=\{(x_1, x_2,\dots,x_N)\in \mathbb{R}^{N}:x_i \leq 0 \text{ for } i\in I\}, \] where \(I\) is any subset of \(\{1,2,\dots,N\}\). Finally, a weighted trace inequality related to the fractional Laplacian was derived and proved.
scientific article; zbMATH DE number 6705698
scientific article