A Note on the Boundedness of Doob Maximal Operators on a Filtered Measure Space

Let <i>M</i> be the Doob maximal operator on a filtered measure space and let <i>v</i> be an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>A</mi><mi>p</mi...

Descripción completa

Guardado en:

Detalles Bibliográficos
Autores principales:	Wei Chen, Jingya Cui
Formato:	article
Lenguaje:	EN
Publicado:	MDPI AG 2021
Materias:	filtered measure space Doob maximal operator weighted inequality principal set Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/0f1400cc7d4b42d583ed52d030becab4
Etiquetas:	Agregar Etiqueta Sin Etiquetas, Sea el primero en etiquetar este registro!

id	oai:doaj.org-article:0f1400cc7d4b42d583ed52d030becab4
record_format	dspace
spelling	oai:doaj.org-article:0f1400cc7d4b42d583ed52d030becab42021-11-25T18:17:30ZA Note on the Boundedness of Doob Maximal Operators on a Filtered Measure Space10.3390/math92229532227-7390https://doaj.org/article/0f1400cc7d4b42d583ed52d030becab42021-11-01T00:00:00Zhttps://www.mdpi.com/2227-7390/9/22/2953https://doaj.org/toc/2227-7390Let <i>M</i> be the Doob maximal operator on a filtered measure space and let <i>v</i> be an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>A</mi><mi>p</mi></msub></semantics></math></inline-formula> weight with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mo>+</mo><mo>∞</mo></mrow></semantics></math></inline-formula>. We try proving that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mo>∥</mo><mi>M</mi><mi>f</mi><mo>∥</mo></mrow><mrow><msup><mi>L</mi><mi>p</mi></msup><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></msub><mo>≤</mo><msup><mi>p</mi><mo>′</mo></msup><msubsup><mrow><mo>[</mo><mi>v</mi><mo>]</mo></mrow><msub><mi>A</mi><mi>p</mi></msub><mfrac><mn>1</mn><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></mfrac></msubsup><msub><mrow><mo>∥</mo><mi>f</mi><mo>∥</mo></mrow><mrow><msup><mi>L</mi><mi>p</mi></msup><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></msub><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>/</mo><msup><mi>p</mi><mo>′</mo></msup><mo>=</mo><mn>1</mn><mo>.</mo></mrow></semantics></math></inline-formula> Although we do not find an approach which gives the constant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>p</mi><mo>′</mo></msup><mo>,</mo></mrow></semantics></math></inline-formula> we obtain that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mo>∥</mo><mi>M</mi><mi>f</mi><mo>∥</mo></mrow><mrow><msup><mi>L</mi><mi>p</mi></msup><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></msub><mo>≤</mo><msup><mi>p</mi><mfrac><mn>1</mn><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></mfrac></msup><msup><mi>p</mi><mo>′</mo></msup><msubsup><mrow><mo>[</mo><mi>v</mi><mo>]</mo></mrow><msub><mi>A</mi><mi>p</mi></msub><mfrac><mn>1</mn><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></mfrac></msubsup><msub><mrow><mo>∥</mo><mi>f</mi><mo>∥</mo></mrow><mrow><msup><mi>L</mi><mi>p</mi></msup><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></msub><mo>,</mo></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><munder><mo movablelimits="false" form="prefix">lim</mo><mrow><mi>p</mi><mo>→</mo><mo>+</mo><mo>∞</mo></mrow></munder><msup><mi>p</mi><mfrac><mn>1</mn><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></mfrac></msup><mo>=</mo><mn>1</mn><mo>.</mo></mrow></semantics></math></inline-formula>Wei ChenJingya CuiMDPI AGarticlefiltered measure spaceDoob maximal operatorweighted inequalityprincipal setMathematicsQA1-939ENMathematics, Vol 9, Iss 2953, p 2953 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	filtered measure space Doob maximal operator weighted inequality principal set Mathematics QA1-939
spellingShingle	filtered measure space Doob maximal operator weighted inequality principal set Mathematics QA1-939 Wei Chen Jingya Cui A Note on the Boundedness of Doob Maximal Operators on a Filtered Measure Space
description	Let <i>M</i> be the Doob maximal operator on a filtered measure space and let <i>v</i> be an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>A</mi><mi>p</mi></msub></semantics></math></inline-formula> weight with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mo>+</mo><mo>∞</mo></mrow></semantics></math></inline-formula>. We try proving that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mo>∥</mo><mi>M</mi><mi>f</mi><mo>∥</mo></mrow><mrow><msup><mi>L</mi><mi>p</mi></msup><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></msub><mo>≤</mo><msup><mi>p</mi><mo>′</mo></msup><msubsup><mrow><mo>[</mo><mi>v</mi><mo>]</mo></mrow><msub><mi>A</mi><mi>p</mi></msub><mfrac><mn>1</mn><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></mfrac></msubsup><msub><mrow><mo>∥</mo><mi>f</mi><mo>∥</mo></mrow><mrow><msup><mi>L</mi><mi>p</mi></msup><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></msub><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>/</mo><msup><mi>p</mi><mo>′</mo></msup><mo>=</mo><mn>1</mn><mo>.</mo></mrow></semantics></math></inline-formula> Although we do not find an approach which gives the constant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>p</mi><mo>′</mo></msup><mo>,</mo></mrow></semantics></math></inline-formula> we obtain that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mo>∥</mo><mi>M</mi><mi>f</mi><mo>∥</mo></mrow><mrow><msup><mi>L</mi><mi>p</mi></msup><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></msub><mo>≤</mo><msup><mi>p</mi><mfrac><mn>1</mn><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></mfrac></msup><msup><mi>p</mi><mo>′</mo></msup><msubsup><mrow><mo>[</mo><mi>v</mi><mo>]</mo></mrow><msub><mi>A</mi><mi>p</mi></msub><mfrac><mn>1</mn><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></mfrac></msubsup><msub><mrow><mo>∥</mo><mi>f</mi><mo>∥</mo></mrow><mrow><msup><mi>L</mi><mi>p</mi></msup><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></msub><mo>,</mo></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><munder><mo movablelimits="false" form="prefix">lim</mo><mrow><mi>p</mi><mo>→</mo><mo>+</mo><mo>∞</mo></mrow></munder><msup><mi>p</mi><mfrac><mn>1</mn><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></mfrac></msup><mo>=</mo><mn>1</mn><mo>.</mo></mrow></semantics></math></inline-formula>
format	article
author	Wei Chen Jingya Cui
author_facet	Wei Chen Jingya Cui
author_sort	Wei Chen
title	A Note on the Boundedness of Doob Maximal Operators on a Filtered Measure Space
title_short	A Note on the Boundedness of Doob Maximal Operators on a Filtered Measure Space
title_full	A Note on the Boundedness of Doob Maximal Operators on a Filtered Measure Space
title_fullStr	A Note on the Boundedness of Doob Maximal Operators on a Filtered Measure Space
title_full_unstemmed	A Note on the Boundedness of Doob Maximal Operators on a Filtered Measure Space
title_sort	note on the boundedness of doob maximal operators on a filtered measure space
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/0f1400cc7d4b42d583ed52d030becab4
work_keys_str_mv	AT weichen anoteontheboundednessofdoobmaximaloperatorsonafilteredmeasurespace AT jingyacui anoteontheboundednessofdoobmaximaloperatorsonafilteredmeasurespace AT weichen noteontheboundednessofdoobmaximaloperatorsonafilteredmeasurespace AT jingyacui noteontheboundednessofdoobmaximaloperatorsonafilteredmeasurespace
_version_	1718411363423879168

A Note on the Boundedness of Doob Maximal Operators on a Filtered Measure Space

Ejemplares similares