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. © 2006 Elsevier Inc. All rights reserved