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

Abstract

In this paper we establish L p -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 ( L 1 , … , L n , i T ) 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 R n . 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 , i T ) of the sub-Laplacian and the central derivative.