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Journal of the London Mathematical Society 1987 s2-36(1):137-142; doi:10.1112/jlms/s2-36.1.137
© 1987 by London Mathematical Society
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© Oxford University Press

A Note on a Littlewood-Paley Inequality for Arbitrary Intervals in R2

Fernando Soria

Department of Mathematics, University of Chicago Illinois, USA

Received 28 February 1986.

Let P = {Ri}i≥1 be an arbitrary collection of mutually disjoint intervals in Rn. For every i ≥ 1, let Si denote the multiplier operator of symbol Ri; that is, (Sif)^= {chi}RFormula. Consider now the Littlewood-Paley square function


Formula

It has recently been proved by J. L. Rubio de Francia (when n = 1) and by J.-L. Journé (for general n) that the sublinear operatorf-> {Delta}f is bounded on Lp for 2 ≤ p < {infty}. The purpose of this note is to present a simple proof of the boundedness of {Delta} for n = 2 or, with more generality, for arbitrary intervals in Rn whose sides have no more than two different sizes. The main ingredient of the proof is the use of a covering lemma due to Journé which has been found to have many interesting applications in the setting of product domains.

Current address: Universidad Autónoma de Madrid, Division de Matematicas, 28049 Madrid, Spain


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