  In this paper we obtain a pointwise sparse domination for generalized
H\"ormander operators and also for iterated commutators with those operators.
As a particular case of our result we obtain a extension of the sparse
domination for commutators obtained by A. Lerner, S. Ombrosi and the second
author to iterated commutators. Relying upon that sparse domination a number of
quantitative estimates such as Coifman-Fefferman estimates, strong type
estimates, and endpoint estimates that improve and complete results in papers
by M. Lorente, J.M. Martell, C. P\'erez, M.S. Riveros and A. de la Torre are
obtained. We also provide a new local decay estimate and we also extend results
in a paper due to J.M. Martell, C. Perez and R. Trujillo-Gonzalez to kernels
satisfying generalized H\"ormander conditions. Among other applications, as a
particular case of our result for endpoint estimates, we extend the endpoint
estimate obtained in a work by A. Lerner, S. Ombrosi and the second author to
iterated commutators.
