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We prove some weighted Hardy and Rellich inequalities on general Carnot groups with weights associated to the norm constructed by Folland's fundamental solution of the Kohn sub-Laplacian. Mathematics Subject Classification (2000). 26D10, 22E30, 43A80.

In this note we provide simple and short proofs for a class of Hardy-Rellich type inequalities with the best constant, which extends some recent results. MSC: 26D15; 35A23

We present some new matrix spectral problems, based on the real special orthogonal Lie algebra so(4,double-struckR), and construct corresponding soliton hierarchies by means of zero curvature equations associated with these spectral problems. With the aid of symbolic computation by Maple, new soliton hierarchies of Kaup–Newell type, Ablowitz–Kaup–Newell–Segur type and Wadati–Konno–Ichikawa type are obtained to illustrate the use of so(4,double-struckR). Copyright © 2016 John Wiley & Sons, Ltd.