Geodesic
$\mathrm A_1$-regularity and boundedness of Riesz transforms in Banach lattices of measurable functions
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 44, Tome 447 (2016), pp. 113-122
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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 $Y$. 
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     author = {D. V. Rutsky},
     title = {$\mathrm A_1$-regularity and boundedness of {Riesz} transforms in {Banach} lattices of measurable functions},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {113--122},
     year = {2016},
     volume = {447},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2016_447_a7/}
}
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D. V. Rutsky. $\mathrm A_1$-regularity and boundedness of Riesz transforms in Banach lattices of measurable functions. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 44, Tome 447 (2016), pp. 113-122. http://geodesic.mathdoc.fr/item/ZNSL_2016_447_a7/

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