Semi-almost periodic Fourier multipliers on rearrangement-invariant spaces with suitable Muckenhoupt weights
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ORIGINAL ARTICLE
Semi-almost periodic Fourier multipliers on rearrangementinvariant spaces with suitable Muckenhoupt weights C. A. Fernandes1 • A. Yu. Karlovich1 Received: 4 November 2019 / Accepted: 7 January 2020 Ó Sociedad Matemática Mexicana 2020
Abstract Let XðRÞ be a separable rearrangement-invariant space and w be a suitable Muckenhoupt weight. We show that for any semi-almost periodic Fourier multiplier a on XðR; wÞ ¼ ff : fw 2 XðRÞg there exist uniquely determined almost periodic Fourier multipliers al ; ar on XðR; wÞ, such that a ¼ ð1 uÞal þ uar þ a0 ; for some monotonically increasing function u with uð1Þ ¼ 0, uðþ1Þ ¼ 1 and some continuous and vanishing at infinity Fourier multiplier a0 on XðR; wÞ. This result extends previous results by Sarason (Duke Math J 44:357–364, 1977) for L2 ðRÞ and by Karlovich and Loreto Herna´ndez (Integral Equ Oper Theor 62:85– 128, 2008) for weighted Lebesgue spaces Lp ðR; wÞ with weights in a suitable subclass of the Muckenhoupt class Ap ðRÞ. Keywords Rearrangement-invariant Banach function space Boyd indices Muckenhoupt weight Almost periodic function Semi-almost periodic function Fourier multiplier
Mathematics Subject Classification Primary 42A45 Secondary 46E30
Dedicated to Professor Yuri I. Karlovich on the occasion of his 70th birthday. This work was partially supported by the Fundac¸a˜o para a Cieˆncia e a Tecnologia (Portuguese Foundation for Science and Technology) through the projects UID/MAT/00297/2019 (Centro de Matema´tica e Aplicac¸o˜es). & A. Yu. Karlovich [email protected] C. A. Fernandes [email protected] 1
Centro de Matema´tica e Aplicac¸o˜es, Departamento de Matema´tica, Faculdade de Cieˆncias e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, 2829–516 Caparica, Portugal
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C. A. Fernandes, A. Yu. Karlovich
1 Introduction Let CðRÞ be the C -algebra of all continuous functions on the two-point compactification of the real line R ¼ ½1; þ1 and _ ¼ ff 2 CðRÞ : f ð1Þ ¼ f ðþ1Þg; CðRÞ where R_ ¼ R [ f1g is the one-point compactification of the real line. Let APP denote the set of all almost periodic polynomials, that is, finite sums of the form P c ek , where k k2K ek ðxÞ :¼ eikx ;
x 2 R;
ck 2 C and K R is a finite subset of R. The smallest closed subalgebra of L1 ðRÞ that contains APP is denoted by AP and called the algebra of (uniformly) almost periodic functions. Sarason [36] introduced the algebra of semi-almost periodic functions as the smallest closed subalgebra of L1 ðRÞ that contains AP and CðRÞ: SAP :¼ algL1 ðRÞ fAP; CðRÞg: It is not difficult to see that AP and SAP are C -subalgebras of L1 ðRÞ. Theorem 1.1 (Sarason [36], see also [10, Theorem 1.21]) Let u 2 CðRÞ be any function for which uð1Þ ¼ 0 and uðþ1Þ ¼ 1. If a 2 SAP, then there exist _ such that a0 ð1Þ ¼ 0 and al ; ar 2 AP and a0 2 CðRÞ a ¼ ð1 uÞal þ uar þ a0 :
ð1:1Þ
The functions al ; ar are uniquely determined by a and independent of the particular choice of u. The maps a7!al and a7!ar are C -algebra homomorphisms of SAP onto AP. The uniquely determined
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