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Sharp asymptotic estimates for a class of Littlewood–Paley operators

Volume 260 / 2021

Odysseas Bakas Studia Mathematica 260 (2021), 195-206 MSC: Primary 42B25; Secondary 42A45. DOI: 10.4064/sm200514-6-10 Published online: 4 March 2021

Abstract

It is well-known that Littlewood–Paley operators formed with respect to lacunary sets of finite order are bounded on $L^p (\mathbb {R})$ for all $1 \lt p \lt \infty $. In this note it is shown that $$ \| S_{\mathcal {I}_{E_2}} \|_{L^p (\mathbb {R}) \rightarrow L^p (\mathbb {R})} \sim (p-1)^{-2} \quad \ (p \rightarrow 1^+) ,$$ where $S_{\mathcal {I}_{E_2}}$ denotes the classical Littlewood–Paley operator formed with respect to the second order lacunary set $ E_2 = \{ \pm ( 2^k - 2^l ) : k,l \in \mathbb {Z} \text { with } k \gt l \} $. Variants in the periodic setting and for certain lacunary sets of order $N$ are also presented.

Authors

  • Odysseas BakasCentre for Mathematical Sciences
    Lund University
    221 00 Lund, Sweden
    e-mail

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