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In this paper we study Coifman type estimates and weighted norm inequalitiesfor singular integral operators $T$ and its commutators, given by theconvolution with a vector valued kernel $K$. We define a weaker H\"ormandertype condition associated with Young functions for the vector valued kernels.With this general framework we obtain as an example the result for the squareoperator and its commutator given in [ M. Lorente, M.S. Riveros, A. de la Torre\emph{On the Coifman type inequality for the oscillation of the one-sidedaverages }, Journal of Mathematical Analysis and Applications, Vol 336, Issue1,\ (2007) 577-592.]

H\"ormander's conditions for vector-valued kernels of singular integrals and its commutators