An estimate for a first-order Riesz operator on the affine group

On the affine group of the line, which is a solvable Lie group of exponenti al growth, we consider a right-invariant Laplacian Delta. For a certain rig ht-invariant vector field X, we prove that the first-order Riesz operator X Delta (-1/2) is of weak type (1, 1) with respect to the left Haar measure of the group. This operator is therefore also bounded on L-p, 1 < p less th an or equal to 2. Locally, the operator is a standard singular integral. Th e main part of the proof therefore concerns the behaviour of the kernel of the operator at infinity and involves cancellation.