Volume 16, Issue 1 Songting Yin Ahmed A. Abdelhakim Optimal critical exponent Lp inequalities of Hardy type on the sphere via Xiao's method Downloads Abstract PDF Full article PDF DOI { copied = true; setTimeout(() => copied = false, 2000) })" :aria-label="copied ? 'DOI copied' : 'Copy DOI'" :title="copied ? 'Copied' : 'Copy DOI'" > Find all available articles from these authors Songting Yin Ahmed A. Abdelhakim Find more articles with the same keywords Sobolev space Lp Hardy inequalities density agrument compact manifold critical exponent Find more articles with the same subject classes 26D10 35A23 46E35