Sharp weighted Young's inequalities and Moser-Trudinger inequalities on Heisenberg type groups and Grushin spaces

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 ||K_*W L||_r,W ≤ ||K|| _p,W||L||_q,W, 1 + 1/r = 1/p + 1/q, for first-layer radial weights W on a general Carnot group script G sign and functions K, L:script G sign → ℝ with L first-layer symmetric. The proofs use some sharp estimates for hypergeometric functions.