Geodesic
Vector-valued boundedness of harmonic analysis operators
Algebra i analiz, Tome 28 (2016) no. 6, pp. 91-117.

Voir la notice de l'article provenant de la source Math-Net.Ru

Let $S$ be a space of homogeneous type, $X$ a Banach lattice of measurable functions on $S \times \Omega$ with the Fatou property and nontrivial convexity, and $Y$ some Banach lattice of measurable functions with the Fatou property. Under the assumption that the Hardy–Littlewood maximal operator $M$ is bounded both in $X$ and $X’$, it is proved that the boundedness of $M$ in $X (Y)$ is equivalent to its boundedness in $\mathrm L_{s}(Y)$ for some (equivalently, for all) $1 s \infty$. With $S = \mathbb R^n$, the last condition is known as the Hardy–Littlewood property of $Y$ and is related to the $\mathrm {UMD}$ property. For lattices $X$ with nontrivial convexity and concavity, the UMD property implies the boundedness of all Calderón–Zygmund operators in $X (Y)$ and is equivalent to the boundedness of a single nondegenerate Calderón–Zygmund operator. The $\mathrm {UMD}$ property of $Y$ is characterized in terms of the $\mathrm A_{p}$-regularity of both $\mathrm L_{\infty } (Y)$ and $\mathrm L_{\infty } (Y’)$. The arguments are based on an improved version of the divisibility property for $\mathrm A_{p}$-regularity.
Keywords: $A_p$-regularity, BMO-regularity, Hardy-Littlewood maximal operator, Calderón–Zygmund operators. 
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D. V. Rutsky. Vector-valued boundedness of harmonic analysis operators. Algebra i analiz, Tome 28 (2016) no. 6, pp. 91-117. http://geodesic.mathdoc.fr/item/AA_2016_28_6_a4/

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