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

Abstract

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