Hardy and Rellich type inequalities with two weight functions (Q2830472)

The authors prove several sharp two-weight Hardy, Hardy-Poincaré, and Rellich type inequalities on the sub-Riemannian manifold \(\mathbb R^{2n+1}=\mathbb R^n\times\mathbb R^n\times\mathbb R\) defined by the vector fields: NEWLINE\[NEWLINE X_j=\frac{\partial}{\partial x_j} + 2ky_j |z|^{2k-2}\frac{\partial}{\partial l}, \qquad Y_j=\frac{\partial}{\partial y_j} - 2kx_j |z|^{2k-2}\frac{\partial}{\partial l} NEWLINE\]NEWLINE with \(j=1,\ldots,n\). Here \((z,y)=(x,y,l)\in \mathbb R^{2n+1},\) \(|z|=(|x|^2+|y|^2)^{1/2}\) and \(k\geq 1\).