Abstract
In this paper, we present a weighted Hardy identity related to the Baouendi-Grushin vector fields and its applications in the context of differential inequalities. By selecting appropriate parameters, the Hardy identity related to the Baouendi-Grushin operator implies numerous sharp remainder formulae for Hardy type inequalities. In the commutative case, we obtain improved weighted Hardy inequalities in the setting of the Euclidean space. For example, in a special case, by dropping non-negative remainder terms, related to the Baouendi-Grushin operator, and choosing suitable parameters our identity allows us to derive an improved critical Hardy inequality for the radial derivative operator with a sharp constant that does not depend on the topological dimension. We employ the method of factorizing differential expressions, as used by Gesztesy and Littlejohn in [1]. In this paper, we demonstrate the application of the factorization method in the noncommutative Baouendi-Grushin setting. As an application of the Hardy identity related to the Baouendi-Grushin vector fields, we establish a Hardy inequality for the generalized Landau Hamiltonian (or the twisted Laplacian) with remainder terms.