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

Sjögren, Peter

doi:10.1090/S0002-9947-99-02222-9

An estimate for a first-order Riesz operator on the affine group
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Trans. Amer. Math. Soc. 351 (1999), 3301-3314 Request permission

Abstract:

On the affine group of the line, which is a solvable Lie group of exponential growth, we consider a right-invariant Laplacian $\Delta$. For a certain right-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 $L^p, \; 1<p\leq 2$. 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.

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