Open Access
2018 Weighted square function inequalities
Adam Osȩkowski
Publ. Mat. 62(1): 75-94 (2018). DOI: 10.5565/PUBLMAT6211804


For an integrable function $f$ on $[0,1)^d$, let $S(f)$ and $Mf$ denote the corresponding dyadic square function and the dyadic maximal function of $f$, respectively. The paper contains the proofs of the following statements.

(i) If $w$ is a dyadic $A_1$ weight on $[0,1)^d$, then $$ ||S(f)||_{L^1(w)}\leq \sqrt{5}[w]_{A_1}^{1/2}||Mf||_{L^1(w)}. $$ The exponent $1/2$ is shown to be the best possible.

(ii) For any $p>1$, there are no constants $c_p$, $\alpha_p$ depending only on $p$ such that for all dyadic $A_p$ weights $w$ on $[0,1)^d$, $$ ||S(f)||_{L^1(w)}\leq c_p[w]_{A_p}^{\alpha_p}||Mf||_{L^1(w)}. $$


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Adam Osȩkowski. "Weighted square function inequalities." Publ. Mat. 62 (1) 75 - 94, 2018.


Received: 29 March 2016; Revised: 4 November 2016; Published: 2018
First available in Project Euclid: 16 December 2017

zbMATH: 1362.42049
MathSciNet: MR3738183
Digital Object Identifier: 10.5565/PUBLMAT6211804

Primary: 42B25
Secondary: 46E30 , 60G42

Keywords: Bellman function , dyadic , Maximal operator , square function , weight

Rights: Copyright © 2018 Universitat Autònoma de Barcelona, Departament de Matemàtiques

Vol.62 • No. 1 • 2018
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