A Note on Generalized Hardy-Sobolev Inequalities. (8th January 2013)

• Record Type: Journal Article
• Authors: Anoop, T. V.
• Other Names: Ozawa Tohru Academic Editor.
• Is Part Of: International journal of analysis. Volume 2013(2013)
• Publication Date: 2013-01-08
• Subjects: Mathematical analysis -- Periodicals; 515.05
• DOI: 10.1155/2013/784398
• Languages: English
• ISSNs: 2314-498X

Abstract

Abstract : We are concerned with finding a class of weight functions g so that the following generalized Hardy-Sobolev inequality holds: ∫ Ω g u 2 ≤ C ∫ Ω | ∇ u | 2, u ∈ H 0 1 ( Ω ), for some C > 0, where Ω is a bounded domain in ℝ 2 . By making use of Muckenhoupt condition for the one-dimensional weighted Hardy inequalities, we identify a rearrangement invariant Banach function space so that the previous integral inequality holds for all weight functions in it. For weights in a subspace of this space, we show that the best constant in the previous inequality is attained. Our method gives an alternate way of proving the Moser-Trudinger embedding and its refinement due to Hansson.