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The $L^p$ boundedness(1 < p < ∞) of Littlewood-Paley’s g-function, Lusin’s S function, Littlewood-Paley’s $g{*}_{\lambda }$-functions, and the Marcinkiewicz function is well known. In a sense, one can regard the Marcinkiewicz function as a variant of Littlewood-Paley’s g-function. In this note, we treat counterparts ${\mu }_S^{\varrho }$ and ${\mu }_{\lambda }^{*,\varrho }$ to S and $g{*}_{\lambda }$. The definition of ${\mu }_S^{\varrho }\left(f\right)$ is as follows: ${\mu }_S^{\varrho }\left(f\right)\left(x\right)=({ʃ}_{|y-x|<t}|1/t^{\varrho }{ʃ}_{\left|z\right|\le t}\Omega \left(z\right)/{\left(\right|z|}^{n-\varrho }{\left)f(y-z)dz\right|}^2\left(dydt\right)/\left(t^{n+1}\right){)}^{1/2}$, where Ω(x) is a homogeneous function of degree 0 and Lipschitz continuous of order β (0 < β ≤ 1) on the unit sphere $S^{n-1}$, and ${ʃ}_{S^{n-1}}\Omega \left(x^{\text{'}}\right)d\sigma \left(x^{\text{'}}\right)=0$. We show that...

Grafakos-Kalton [Collect. Math. 52 (2001)] discussed the boundedness of multilinear Calderón-Zygmund operators on the product of Hardy spaces. Then Lerner et al. [Adv. Math. 220 (2009)] defined $A_{p⃗}$ weights and built a theory of weights adapted to multilinear Calderón-Zygmund operators. In this paper, we combine the above results and obtain some estimates for multilinear Calderón-Zygmund operators on weighted Hardy spaces and also obtain a weighted multilinear version of an inequality for multilinear...