Weighted Sobolev spaces of radially symmetric functions
The paper under review is devoted to the investigation of the following inequalities \[ \int\limits_{{\mathbb R}^n} |x|^{\alpha}|\nabla^ku|^p~\!dx\geq c\int\limits_{{\mathbb R}^n} |x|^{\alpha-kp}|u|^p~\!dx, \] \[ \int_{{\mathbb R}^n} |x|^{\alpha}|\Delta^m u|^p dx\geq c \int_{{\mathbb R}^n} |x|^{\alpha-p}|\nabla(\Delta^{m-1}u)|^{p }dx \] on the spaces of radially symmetric functions. The author discusses the existence of extremals and in some cases computes the best constants.
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