{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,15]],"date-time":"2026-04-15T17:04:53Z","timestamp":1776272693585,"version":"3.50.1"},"reference-count":15,"publisher":"American Mathematical Society (AMS)","issue":"4","license":[{"start":{"date-parts":[[2010,11,13]],"date-time":"2010-11-13T00:00:00Z","timestamp":1289606400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Proc. Amer. Math. Soc."],"abstract":"

\n The main goal of this paper is to prove an optimal limiting Sobolev inequality in two dimensions for H\u00f6lder continuous functions. Additionally, from this inequality we derive the double logarithmic inequality\n \n \\[\n \n \n \n \n \u2016\n \n <\/mml:mo>\n u<\/mml:mi>\n \n \n \u2016\n \n <\/mml:mo>\n \n \n L<\/mml:mi>\n \n \n \u221e\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:msub>\n \n \u2a7d\n \n <\/mml:mo>\n \n \n \n \u2016\n \n <\/mml:mo>\n \n \u2207\n \n <\/mml:mi>\n u<\/mml:mi>\n \n \n \u2016\n \n <\/mml:mo>\n \n \n L<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n \n 2<\/mml:mn>\n \n \u03c0\n \n <\/mml:mi>\n \n \u03b1\n \n <\/mml:mi>\n <\/mml:msqrt>\n <\/mml:mfrac>\n \n ln<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n \n \n (<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mstyle>\n 1<\/mml:mn>\n +<\/mml:mo>\n 6<\/mml:mn>\n \n 2<\/mml:mn>\n \n \u03c0\n \n <\/mml:mi>\n \n \u03b1\n \n <\/mml:mi>\n <\/mml:msqrt>\n \n \n \n \n \u2016\n \n <\/mml:mo>\n u<\/mml:mi>\n \n \n \u2016\n \n <\/mml:mo>\n \n \n \n \n \n \n \n \n C<\/mml:mi>\n \n \u02d9\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n \n <\/mml:mrow>\n \n \n \n \u03b1\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n \n \n \u2016\n \n <\/mml:mo>\n \n \u2207\n \n <\/mml:mi>\n u<\/mml:mi>\n \n \n \u2016\n \n <\/mml:mo>\n \n \n L<\/mml:mi>\n \n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:mfrac>\n <\/mml:mstyle>\n \n ln<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n (<\/mml:mo>\n 1<\/mml:mn>\n +<\/mml:mo>\n \n 2<\/mml:mn>\n \n \u03c0\n \n <\/mml:mi>\n \n \u03b1\n \n <\/mml:mi>\n <\/mml:msqrt>\n \n \n \n \n \u2016\n \n <\/mml:mo>\n u<\/mml:mi>\n \n \n \u2016\n \n <\/mml:mo>\n \n \n \n \n \n \n \n \n C<\/mml:mi>\n \n \u02d9\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n \n <\/mml:mrow>\n \n \n \n \u03b1\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n \n \n \u2016\n \n <\/mml:mo>\n \n \u2207\n \n <\/mml:mi>\n u<\/mml:mi>\n \n \n \u2016\n \n <\/mml:mo>\n \n \n L<\/mml:mi>\n \n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:mfrac>\n <\/mml:mstyle>\n )<\/mml:mo>\n \n <\/mml:msqrt>\n \n \n \n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mstyle>\n <\/mml:msqrt>\n <\/mml:mrow>\n \\|u\\|_{L^{\\infty }} \\leqslant \\frac {\\|\\nabla u\\|_{L^2}}{\\sqrt {2\\pi \\alpha }} \\sqrt {\\ln \\Bigl (1+6\\sqrt {2\\pi \\alpha } \\tfrac {\\|u\\|_{{\\left .\\rm \\! \\dot C\\right .^{\\!\\alpha }}}}{\\|\\nabla u\\|_{L^{2}}} \\sqrt {\\ln (1+\\sqrt {2\\pi \\alpha } \\tfrac {\\|u\\|_{{\\left .\\rm \\! \\dot C\\right .^{\\!\\alpha }}}} {\\|\\nabla u\\|_{L^{2}}} )\\!}\\, \\Bigr )}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n \\]\n <\/disp-formula>\n for functions\n \n \n \n \n u<\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n \n W<\/mml:mi>\n 0<\/mml:mn>\n \n 1<\/mml:mn>\n ,<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msubsup>\n (<\/mml:mo>\n \n B<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n u\\in W^{1,2}_0(B_1)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n on the unit disk\n \n \n \n \n B<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n B_1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n in\n \n \n \n \n \n R<\/mml:mi>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:msup>\n \\mathbb R^2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ,\n \n \n \n \n \n \u03b1\n \n <\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n (<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n ]<\/mml:mo>\n .<\/mml:mo>\n <\/mml:mrow>\n \\alpha \\in (0,1].<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n <\/p>","DOI":"10.1090\/s0002-9939-09-10159-4","type":"journal-article","created":{"date-parts":[[2009,12,21]],"date-time":"2009-12-21T13:43:53Z","timestamp":1261403033000},"page":"1461-1470","source":"Crossref","is-referenced-by-count":1,"special_numbering":"610","title":["An optimal limiting 2\ud835\udc37 Sobolev inequality"],"prefix":"10.1090","volume":"138","author":[{"given":"Andrei","family":"Biryuk","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2009,11,13]]},"reference":[{"key":"1","unstructured":"R. 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