Singular integral operators of near-product-type on the Heisenberg group

In this paper we establish Lp-boundedness (1<p<∞) for a class of singular convolution operators on the Heisenberg group whose kernels satisfy regularity and cancellation conditions adapted to the implicit (n+1)-parameter structure. The polyradial kernels of this type arose in [A.J. Fraser, An (n+1)-fold Marcinkiewicz multiplier theorem on the Heisenberg group, Bull. Austral. Math. Soc. 63 (2001) 35-58; A.J. Fraser, Convolution kernels of (n+1)-fold Marcinkiewicz multipliers on the Heisenberg group, Bull. Austral. Math. Soc. 64 (2001) 353-376] as the convolution kernels of (n+1)-fold Marcinkiewicz-type spectral multipliers m(L₁,…,Ln,iT) of the n-partial sub-Laplacians and the central derivative on the Heisenberg group. Thus they are in a natural way analogous to product-type Calderón-Zygmund convolution kernels on Rn. Here, as in [A.J. Fraser, An (n+1)-fold Marcinkiewicz multiplier theorem on the Heisenberg group, Bull. Austral. Math. Soc. 63 (2001) 35-58; A.J. Fraser, Convolution kernels of (n+1)-fold Marcinkiewicz multipliers on the Heisenberg group, Bull. Austral. Math. Soc. 64 (2001) 353-376], we extend to the (n+1)-parameter setting the methods and results of Müller, Ricci, and Stein in [D. Müller, F. Ricci, E.M. Stein, Marcinkiewicz multipliers and two-parameter structures on Heisenberg groups I, Invent. Math. 119 (1995) 199-233] for the two-parameter setting and multipliers m(L,iT) of the sub-Laplacian and the central derivative.