We prove several optimal Moser-Trudinger and logarithmic Hardy-Littlewood-Sobolev inequalities for systems in two dimensions. These include inequalities on the sphere , on a bounded domain and on all of . In some cases we also address the question of existence of minimizers.
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Itai Shafrir, Gershon Wolansky, Moser-Trudinger and logarithmic HLS inequalities for systems. J. Eur. Math. Soc. 7 (2005), no. 4, pp. 413–448DOI 10.4171/JEMS/34