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It is well known that the commutator    of a linear Calderón–Zygmund operator  T  associated with a    function  b  does not satisfy some continuity properties, such as the continuity from the Hardy space  H p   into  L p   for    and weak ( 1 , 1 ). Thus, an alternative result was given by J. Alvarez and C. Pérez, who proved that    is continuous from an atomic Hardy space    into  L p  , where    is a subspace of the Hardy space  H p   for   . In this paper, we study this continuity property in the multilinear setting. More precisely, we study the commutator    of multilinear Calderón–Zygmund operators on product subspaces of Hardy type spaces, where   . The endpoint continuity from the product of sub-Hardy spaces    into  L p   for the commutator    is obtained. Note that    type estimates in linear and multilinear setting were obtained in works of C. Pérez and A. K. Lerner et al., as an alternative result of the type weak ( 1 , 1 ) and multilinear generalizations.

Continuity property for the commutators of multilinear Calderón–Zygmund operators