Singular integrals associated to the Laplacian on the affine group ax+b

Abstract

We consider singular integral operators of the form (a)Z₁L⁻¹Z₂, (b)Z₁Z₂L⁻¹, and (c)L⁻¹Z₁Z₂, where Z₁ and Z₂ are nonzero right-invariant vector fields, and L is the L²-closure of a canonical Laplacian. The operators (a) are shown to be bounded on L^p for all p∈(1, ∞) and of weak type (1, 1), whereas all of the operators in (b) and (c) are not of weak type (p, p) for any p∈[1, ∞).