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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Author: Peter Sjögren
Journal: Trans. Amer. Math. Soc. 351 (1999), 3301-3314
MSC (1991): Primary 43A80, 42B20; Secondary 22E30
Published electronically: March 29, 1999
MathSciNet review: 1475695
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Abstract | References | Similar Articles | Additional Information

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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Additional Information

Peter Sjögren
Affiliation: Department of Mathematics\ Chalmers University of Technology and Göteborg University\ S-412 96 Göteborg\ Sweden
Email: peters@math.chalmers.se

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02222-9
PII: S 0002-9947(99)02222-9
Received by editor(s): December 15, 1996
Received by editor(s) in revised form: August 15, 1997
Published electronically: March 29, 1999
Article copyright: © Copyright 1999 American Mathematical Society