RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Zap. Nauchn. Sem. POMI: Year: Volume: Issue: Page: Find

 Zap. Nauchn. Sem. POMI, 2016, Volume 447, Pages 113–122 (Mi znsl6297)

$\mathrm A_1$-regularity and boundedness of Riesz transforms in Banach lattices of measurable functions

D. V. Rutsky

St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, St. Petersburg, Russia

Abstract: Suppose that $X$ is a Banach lattice of measurable functions on $\mathbb R^n\times\Omega$ having the Fatou property. We show that the boundedness of all Riesz transforms $R_j$ in $X$ is equivalent to the boundedness of the Hardy–Littlewood maximal operator $M$ in both $X$ and $X'$, and thus to the boundedness of all Calderón–Zygmund operators in $X$. We also prove a result for the case of operators between lattices: if $Y\supset X$ is a Banach lattice with the Fatou property such that the maximal operator is bounded in $Y'$, then the boundedness of all Riesz transforms from $X$ to $Y$ is equivalent to the boundedness of the maximal operator from $X$ to $Y$, and thus to the boundedness of all Calderón–Zygmund operators from $X$ to $Y$.

Key words and phrases: $\mathrm A_1$-regularity, Muckenhoupt weights, reverse Hölder inequality, Hardy–Littlewood maximal operator, Riesz transforms, Calderón–Zygmund operators.

Full text: PDF file (214 kB)
References: PDF file   HTML file

English version:
Journal of Mathematical Sciences (New York), 2018, 229:5, 561–567

Bibliographic databases:

UDC: 517.5

Citation: D. V. Rutsky, “$\mathrm A_1$-regularity and boundedness of Riesz transforms in Banach lattices of measurable functions”, Investigations on linear operators and function theory. Part 44, Zap. Nauchn. Sem. POMI, 447, POMI, St. Petersburg, 2016, 113–122; J. Math. Sci. (N. Y.), 229:5 (2018), 561–567

Citation in format AMSBIB
\Bibitem{Rut16} \by D.~V.~Rutsky \paper $\mathrm A_1$-regularity and boundedness of Riesz transforms in Banach lattices of measurable functions \inbook Investigations on linear operators and function theory. Part~44 \serial Zap. Nauchn. Sem. POMI \yr 2016 \vol 447 \pages 113--122 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl6297} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3580165} \transl \jour J. Math. Sci. (N. Y.) \yr 2018 \vol 229 \issue 5 \pages 561--567 \crossref{https://doi.org/10.1007/s10958-018-3698-z} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85041545253} 

• http://mi.mathnet.ru/eng/znsl6297
• http://mi.mathnet.ru/eng/znsl/v447/p113

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. D. V. Rutsky, “Vector-valued boundedness of harmonic analysis operators”, St. Petersburg Math. J., 28:6 (2017), 789–805
2. Rutsky D.V., ““a(1)-Regularity and Boundedness of Calderon-Zygmund Operators” With Some Remarks (Vol 221, Pg 231, 2014)”, Studia Math., 248:3 (2019), 217–231
•  Number of views: This page: 98 Full text: 19 References: 21