  In this paper we study Coifman type estimates and weighted norm inequalities
for singular integral operators $T$ and its commutators, given by the
convolution with a vector valued kernel $K$. We define a weaker H\"ormander
type condition associated with Young functions for the vector valued kernels.
With this general framework we obtain as an example the result for the square
operator 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-sided
averages }, Journal of Mathematical Analysis and Applications, Vol 336, Issue
1,\ (2007) 577-592.]
