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International Journal of Analysis
Volume 2013 (2013), Article ID 784398, 9 pages
Research Article

A Note on Generalized Hardy-Sobolev Inequalities

Department of Mathematics, Indian Institute of Science, Bangalore 560012, India

Received 16 September 2012; Accepted 20 November 2012

Academic Editor: Tohru Ozawa

Copyright © 2013 T. V. Anoop. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We are concerned with finding a class of weight functions so that the following generalized Hardy-Sobolev inequality holds: , for some , where is a bounded domain in . 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.