Estimates for operators related to the sub-Laplacian with drift in Heisenberg groups
Let \(\mathbb{H}_n=\mathbb{R}^n\times\mathbb{R}^n\times\mathbb{R}\) be the \((2n+1)\)-dimensional Heisenberg group with its generic point denoted by \(g=(x,y,t)\). Using the left-inariant vector fields \(X_j=\frac{\partial}{\partial x_j}+2y_j\frac{\partial}{\partial t}\) and \(Y_j=\frac{\partial}{\partial y_j}-2x_j\frac{\partial}{\partial t}\) for \(j=1,\dots,n\), one defines the horizontal gradient \(\nabla=(X_1,\dots,X_n,Y_1,\dots,Y_n)\) and the sub-Laplacian \(\Delta=\sum_{j=1}^n(X_j^2+Y_j^2)\). Any vector \(v=(a_1,\dots,a_n,b_1,\dots,b_n)\in\mathbb{R}^n\times\mathbb{R}^n\) gives rise to the sub-Laplacian with a drift \(\Delta_v=\Delta+2\sum_{j=1}^n(a_jX_j+b_jX_j)\) and to the measure on \(\mathbb{H}_n\) given by \(\text{d}\mu_v(g)=\psi_v(g)\text{d}g\), where \(\psi_v(g)=\exp(\sum_{j=1}^n(a_jx_j+b_jy_j))\). Moreover, the Riesz transform of order \(k=1,2,\dots\) is \(\mathcal{R}_k=\nabla^k(-\Delta_v)^{-k/2}\). \par In the above notation, one proves that if \(k\) is a positive integer and \(k\ne 2\), then the Riesz transform \(\mathcal{R}_k\) is of weak type \((1,1)\) for the measure \(\text{d}\mu_v\) if and only if \(k=1\). Moreover, defining the Littlewood-Paley-Stein operator associated with \(\Delta_v\) by \(\mathcal{H}_kf(g)=\sup_{h>0}\vert h^{k/2}\nabla^k \text{e}^{h\Delta_v}f(g)\vert\) for \(k=0,1,2,\dots\), one proves that the operator \(\mathcal{H}_k\) is of weak type \((1,1)\) with respect to the measure \(\text{d}\mu_v\) if and only if \(k\in\{0,1\}\).
scientific article; zbMATH DE number 7457089
scientific article