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On Bloom-type characterizations of the higher-order commutators of Marcinkiewicz integrals

UDC 517.9 Let $\Omega$ be homogeneous of degree zero, have mean value zero, and integrable on the unit sphere.  For $m\in \mathbb N,$ let $b\in  L_{\rm loc}^1(\mathbb R^n)$ and let the higher-order commutator of the  Marcinkiewicz integral $\mu_{\O...

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Date:2024
Main Authors: Chen, Yanping, Tian, Tian
Format: Article
Language:English
Published: Institute of Mathematics, NAS of Ukraine 2024
Online Access:https://umj.imath.kiev.ua/index.php/umj/article/view/7466
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Journal Title:Ukrains’kyi Matematychnyi Zhurnal
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Ukrains’kyi Matematychnyi Zhurnal
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author Chen, Yanping
Tian, Tian
Chen, Yanping
Tian, Tian
author_facet Chen, Yanping
Tian, Tian
Chen, Yanping
Tian, Tian
author_sort Chen, Yanping
baseUrl_str https://umj.imath.kiev.ua/index.php/umj/oai
collection OJS
datestamp_date 2024-07-15T03:05:02Z
description UDC 517.9 Let $\Omega$ be homogeneous of degree zero, have mean value zero, and integrable on the unit sphere.  For $m\in \mathbb N,$ let $b\in  L_{\rm loc}^1(\mathbb R^n)$ and let the higher-order commutator of the  Marcinkiewicz integral $\mu_{\Omega,b}^m$ be defined by \begin{gather*}\mu_{\Omega,b}^{m}(f)(x)=\left(\,\int\limits_{0}^{\infty}\,\left|\,\,\int\limits_{|x-y| \leq t} \frac{\Omega(x-y)}{|x-y|^{n-1}}[b(x)-b(y)]^{m}f(y) dy\right|^{2} \frac{d t}{t^{3}}\right)^{\frac{1}{2}}.\end{gather*} We establish a sparse domination of $\mu_{\Omega,b}^{m}$ for $\Omega\in {\rm Lip}(\mathbb{S}^{n-1})$. Moreover, we also give Bloom-type characterizations of the two-weighted boundedness of the higher-order commutators $\mu_{\Omega,b}^{m}, \mu_{\Omega,\alpha,b}^{*,m}$, and $\mu_{\Omega,S,b}^{m}$, where  the higher-order commutators $ \mu_{\Omega,\alpha,b}^{*,m}$ and $\mu_{\Omega,S,b}^{m}$ are defined, respectively, by \begin{gather*}\mu_{\Omega,\alpha,b}^{*,m}(f)(x)=\left(\,\,\iint\limits_{\mathbb{R}_{+}^{n+1}}\left(\frac{t}{t+|x-y|}\right)^{n\alpha}\left|\,\int\limits_{|y-z| \leq t} \frac{\Omega(y-z)}{|y-z|^{n-1}}[b(x)-b(z)]^{m}f(z) dz\right|^{2}\frac{dydt}{t^{n+3}}\right)^{\frac{1}{2}},\quad \alpha>1,\end{gather*} and \begin{gather*}\mu_{\Omega,S,b}^{m}(f)(x)=\left(\,\,\,\iint\limits_{|x-y|<t}\left|\,\int\limits_{|y-z| \leq t} \frac{\Omega(y-z)}{|y-z|^{n-1}}[b(x)-b(z)]^{m}f(z) dz\right|^{2}\frac{dydt}{t^{n+3}}\right)^{\frac{1}{2}}.\end{gather*}
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spelling umjimathkievua-article-74662024-07-15T03:05:02Z On Bloom-type characterizations of the higher-order commutators of Marcinkiewicz integrals On Bloom-type characterizations of the higher-order commutators of Marcinkiewicz integrals Chen, Yanping Tian, Tian Chen, Yanping Tian, Tian Marcinkiewicz integrals; commutators; two weighted boundedness; $BMO_\eta$ 42B20, 42B25 UDC 517.9 Let $\Omega$ be homogeneous of degree zero, have mean value zero, and integrable on the unit sphere.  For $m\in \mathbb N,$ let $b\in  L_{\rm loc}^1(\mathbb R^n)$ and let the higher-order commutator of the  Marcinkiewicz integral $\mu_{\Omega,b}^m$ be defined by \begin{gather*}\mu_{\Omega,b}^{m}(f)(x)=\left(\,\int\limits_{0}^{\infty}\,\left|\,\,\int\limits_{|x-y| \leq t} \frac{\Omega(x-y)}{|x-y|^{n-1}}[b(x)-b(y)]^{m}f(y) dy\right|^{2} \frac{d t}{t^{3}}\right)^{\frac{1}{2}}.\end{gather*} We establish a sparse domination of $\mu_{\Omega,b}^{m}$ for $\Omega\in {\rm Lip}(\mathbb{S}^{n-1})$. Moreover, we also give Bloom-type characterizations of the two-weighted boundedness of the higher-order commutators $\mu_{\Omega,b}^{m}, \mu_{\Omega,\alpha,b}^{*,m}$, and $\mu_{\Omega,S,b}^{m}$, where  the higher-order commutators $ \mu_{\Omega,\alpha,b}^{*,m}$ and $\mu_{\Omega,S,b}^{m}$ are defined, respectively, by \begin{gather*}\mu_{\Omega,\alpha,b}^{*,m}(f)(x)=\left(\,\,\iint\limits_{\mathbb{R}_{+}^{n+1}}\left(\frac{t}{t+|x-y|}\right)^{n\alpha}\left|\,\int\limits_{|y-z| \leq t} \frac{\Omega(y-z)}{|y-z|^{n-1}}[b(x)-b(z)]^{m}f(z) dz\right|^{2}\frac{dydt}{t^{n+3}}\right)^{\frac{1}{2}},\quad \alpha>1,\end{gather*} and \begin{gather*}\mu_{\Omega,S,b}^{m}(f)(x)=\left(\,\,\,\iint\limits_{|x-y|<t}\left|\,\int\limits_{|y-z| \leq t} \frac{\Omega(y-z)}{|y-z|^{n-1}}[b(x)-b(z)]^{m}f(z) dz\right|^{2}\frac{dydt}{t^{n+3}}\right)^{\frac{1}{2}}.\end{gather*} УДК 517.9 Про характеристики типу Блума комутаторів вищого порядку інтегралів Марцинкевича Нехай $\Omega$ є однорідною нульового степеня, має нульове середнє значення та інтегровна на одиничній сфері.  Для $m\in \Bbb N$ нехай $b\in L_{\rm loc}^1(\Bbb R^n),$ а комутатор вищого порядку інтеграла Марцинкевича $\mu_{\Omega,b} ^m$ визначається тaким чином: \begin{gather*}\mu_{\Omega,b}^{m}(f)(x)=\left(\,\int\limits_{0}^{\infty}\,\left|\,\,\int\limits_{|x-y| \leq t} \frac{\Omega(x-y)}{|x-y|^{n-1}}[b(x)-b(y)]^{m}f(y) dy\right|^{2} \frac{d t}{t^{3}}\right)^{\frac{1}{2}}. \end{gather*} Отримано розріджене домінування $\mu_{\Omega,b}^{m}$ при умові, що $\Omega\in {\rm Lip}(\mathbb{S}^{n-1})$.  Крім того, наведено характеристики типу Блума для двовагової обмеженості  комутаторів вищого порядку $\mu_{\Omega,b}^{m}, \mu_{\Omega,\alpha,b}^{*,m }, \mu_{\Omega,S,b}^{m}$, де комутатори вищого порядку $ \mu_{\Omega,\alpha,b}^{*,m}$ і $\mu_{\Omega,S, b}^{m}$ визначаються, відповідно, таким чином: \begin{gather*}\mu_{\Omega,\alpha,b}^{*,m}(f)(x)=\left(\,\,\iint\limits_{\mathbb{R}_{+}^{n+1}}\left(\frac{t}{t+|x-y|}\right)^{n\alpha}\left|\,\int\limits_{|y-z| \leq t} \frac{\Omega(y-z)}{|y-z|^{n-1}}[b(x)-b(z)]^{m}f(z) dz\right|^{2}\frac{dydt}{t^{n+3}}\right)^{\frac{1}{2}},\quad \alpha>1,\end{gather*} і \begin{gather*}\mu_{\Omega,S,b}^{m}(f)(x)=\left(\,\,\,\iint\limits_{|x-y|<t}\left|\,\int\limits_{|y-z| \leq t} \frac{\Omega(y-z)}{|y-z|^{n-1}}[b(x)-b(z)]^{m}f(z) dz\right|^{2}\frac{dydt}{t^{n+3}}\right)^{\frac{1}{2}}.\end{gather*} Institute of Mathematics, NAS of Ukraine 2024-07-03 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/7466 10.3842/umzh.v76i5.7466 Ukrains’kyi Matematychnyi Zhurnal; Vol. 76 No. 6 (2024); 931–948 Український математичний журнал; Том 76 № 6 (2024); 931–948 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7466/10038 Copyright (c) 2024 Yanping Chen, Tian Tian
spellingShingle Chen, Yanping
Tian, Tian
Chen, Yanping
Tian, Tian
On Bloom-type characterizations of the higher-order commutators of Marcinkiewicz integrals
title On Bloom-type characterizations of the higher-order commutators of Marcinkiewicz integrals
title_alt On Bloom-type characterizations of the higher-order commutators of Marcinkiewicz integrals
title_full On Bloom-type characterizations of the higher-order commutators of Marcinkiewicz integrals
title_fullStr On Bloom-type characterizations of the higher-order commutators of Marcinkiewicz integrals
title_full_unstemmed On Bloom-type characterizations of the higher-order commutators of Marcinkiewicz integrals
title_short On Bloom-type characterizations of the higher-order commutators of Marcinkiewicz integrals
title_sort on bloom-type characterizations of the higher-order commutators of marcinkiewicz integrals
topic_facet Marcinkiewicz integrals
commutators
two weighted boundedness
$BMO_\eta$
42B20
42B25
url https://umj.imath.kiev.ua/index.php/umj/article/view/7466
work_keys_str_mv AT chenyanping onbloomtypecharacterizationsofthehigherordercommutatorsofmarcinkiewiczintegrals
AT tiantian onbloomtypecharacterizationsofthehigherordercommutatorsofmarcinkiewiczintegrals
AT chenyanping onbloomtypecharacterizationsofthehigherordercommutatorsofmarcinkiewiczintegrals
AT tiantian onbloomtypecharacterizationsofthehigherordercommutatorsofmarcinkiewiczintegrals

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