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

This article is cited in 2 scientific papers (total in 2 papers)

$\mathrm A_1$-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.

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

Bibliographic databases:

UDC: 517.5
Received: 06.06.2016

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}


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    Citing articles on Google Scholar: Russian citations, English citations
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    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  mathnet  crossref  isi  elib
    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  crossref  isi
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