About Riesz transforms on the Heisenberg groups
Let \(X_1,\dots,X_{2n}\), \(Z\) be the left invariant vector fields on the Heisenberg group \(H^n\) given by \(X_{2j-1}=\partial_{x_j}+2y_j\partial_z\), \(X_{2j}=\partial_{y_j}-2x_j\partial_z\) and \(Z=\partial_z\). Set \(\Delta=-\sum^{2n}_{i=1}X^2_i\) and consider the Riesz transforms \(R_i=X_i\Delta^{-1/2}\) for \(i\in\{1,\dots,2n\}\). Using an explicit formula for the heat kernel associated with the sublaplacian \(\Delta\) the authors prove that for all \(p\in]1,\infty[\) there exists a \(C_p>1\) such that \[ C^{-1}_p|f|_{L^p(H^n)}\leq \left|\left(\sum^{2n}_{i=1}|R_if|^2\right)^{1/2} \right|_{L^p(H^n)} \leq C_p|f|_{L^p(H^n)} \] uniformly for all \(n\in \mathbb{N}^*\) and \(f\in C^\infty_0(H^n)\). Note that \(C_p\) is independent of \(n\).
scientific article; zbMATH DE number 931963
scientific article