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A Note on a Littlewood‐Paley Inequality for Arbitrary Intervals in R 2
Author(s) -
Soria Fernando
Publication year - 1987
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-36.1.137
Subject(s) - mathematics , combinatorics , sublinear function , disjoint sets , bounded function , multiplier (economics) , lemma (botany) , discrete mathematics , product (mathematics) , mathematical analysis , ecology , geometry , poaceae , biology , economics , macroeconomics
Let P = { R i } i ⩾1 be an arbitrary collection of mutually disjoint intervals in R n . For every i ⩾ 1, let S i denote the multiplier operator of symbol R i ; that is, ( S i f )≙ χ R f ^ . Consider now the Littlewood‐Paley square functionΔ f ( x ) =(∑ i| S i f ( x ) |2 )1 2It has recently been proved by J. L. Rubio de Francia (when n = 1) and by J.‐L. Journé (for general n ) that the sublinear operator f → Δ f is bounded on L p for 2 ⩽ p < ∞. The purpose of this note is to present a simple proof of the boundedness of Δ for n = 2 or, with more generality, for arbitrary intervals in R n 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.