Abstract. On the affine group of the line, which is a solvable Lie group of exponential growth, we consider a right-invariant Laplacian ∆. For a certain right-invariant vector field X, we prove that the first-order Riesz operator X ∆ −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≤2. Locally, the operator is a standard singular integral. The main part of the proof therefore concerns the behaviour of the kernel of the operator at infinity and involves cancellation. 1. Introduction an
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