Sharp Weighted Young's Inequalities and Moser–Trudinger Inequalities on Heisenberg Type Groups and Grushin Spaces

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• 作者：Jeremy T. Tyson
• 关键词：Carnot group ; Grushin plane ; nonlinear potential theory ; Moser– ; Trudinger inequality ; Young's inequality
• 刊名：Potential Analysis
• 出版年：2006
• 出版时间：June 2006
• 年：2006
• 卷：24
• 期：4
• 页码：357-384
• 全文大小：474 KB

文摘

We obtain sharp weighted Moser–Trudinger inequalities for first-layer symmetric functions on groups of Heisenberg type, and for xx-symmetric functions on the Grushin plane. To this end, we establish weighted Young's inequalities in the form || K_*W L ||_r,W \leqslant || K ||_p,W || L ||_q,W , 1 + 1 \mathord/ \vphantom 1 r r = 1 \mathord/ \vphantom 1 p p + 1 \mathord/ \vphantom 1 q q{\left\| {K_{{*W}} L} \right\|}_{{r,W}} \leqslant {\left\| K \right\|}_{{p,W}} {\left\| L \right\|}_{{q,W}} ,\,1 + 1 \mathord{\left/ {\vphantom {1 r}} \right. \kern-\nulldelimiterspace} r = 1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p + 1 \mathord{\left/ {\vphantom {1 q}} \right. \kern-\nulldelimiterspace} q, for first-layer radial weights WW on a general Carnot group \mathbbG\mathbb{G} and functions K, L:\mathbbG ? \mathbbRK,\,L:\mathbb{G} \to \mathbb{R} with LL first-layer symmetric. The proofs use some sharp estimates for hypergeometric functions.