Hardy-Sobolev-Rellich, Hardy-Littlewood-Sobolev and Caffarelli-Kohn-Nirenberg inequalities on general Lie groups (Q6536753)

Let $G$ be a connected Lie group with identity \(e\) and let \(\{X_1, \dots, X_n\}\) be a family of linearly independent left-invariant vector fields of \(G\) satisfying Hörmander's condition. On \(G,\) fix a right Haar measure \(\rho,\) a continuous positive character \(\chi\) and a measure \( \mu_ \chi\) given by \( \mu_ \chi = \chi \, \rho.\) Consider the differential operator \(\Delta_\chi =-\sum_{j=1}^n (X_j^2+X_j(\chi)(e) X_j).\) The form of the first order drift term in the expression of \(\Delta_\chi\) is both necessary and sufficient to ensure that the differential operator \(\Delta_\chi\) is essentially self-adjoint on \(L^2(\mu_\chi).\) It thus naturally leads to the Sobolev spaces adapted to \(\Delta_\chi\) (defined using functional calculus). \N\NIn the present paper, the authors build on the work of \textit{T.~Bruno} et al. [J. Funct. Anal. 276, No.~10, 3014--3050 (2019; Zbl 1466.46024)] which concerns embedding properties of these Sobolev spaces, estimates for the associated heat kernel etc., to prove classical inequalities such as Hardy-Sobolev-Rellich, Hardy-Littlewood-Sobolev, Caffarelli-Kohn-Nirenberg, Gagliardo-Nirenberg inequalities, uncertainty principles in this new setting. Their results are comparable to the classical ones in case \(G\) is unimodular and \(\chi\) is the trivial character, as then \(\Delta_\chi\) reduces to the usual sub-Laplacian associated with \(\{X_1, X_2, \dots ,X_n\}.\)