Abstract The main purpose of this paper is to establish several general Caffarelli–Kohn–Nirenberg (CKN) inequalities on Carnot groups G (also known as stratified groups). These CKN inequalities are sharp for certain parameter values. In case G is an Iwasawa group, it is shown here that the L 2 {L^{2}} -CKN inequalities are sharp for all parameter values except one exceptional case. To show this, generalized Kelvin transforms K σ {K_{\sigma}} are introduced and shown to be isometries for certain weighted Sobolev spaces. An interesting transformation formula for the sub-Laplacian with respect to K σ {K_{\sigma}} is also derived. Lastly, these techniques are shown to be valid for establishing CKN-type inequalities with monomial and horizontal norm weights.

中文翻译：

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卡诺群上的 Sharp Caffarelli-Kohn-Nirenberg 型不等式

摘要 本文的主要目的是在卡诺群 G（也称为分层群）上建立几个一般的 Caffarelli-Kohn-Nirenberg (CKN) 不等式。对于某些参数值，这些 CKN 不等式很明显。如果 G 是 Iwasawa 群，这里表明 L 2 {L^{2}} -CKN 不等式对于所有参数值都是尖锐的，除了一个例外情况。为了说明这一点，引入了广义开尔文变换 K σ {K_{\sigma}} 并显示为某些加权 Sobolev 空间的等距。还导出了关于 K σ {K_{\sigma}} 的子拉普拉斯算子的有趣变换公式。最后，这些技术被证明对于建立具有单项式和水平范数权重的 CKN 型不等式是有效的。