We establish the $L^p$ boundedness for a class of singular integral operators under the $H^1$ kernel condition. These operators were introduced by Duoandikoetxea and Rubio de Francia as an extension of the classical Calderón-Zygmund operators.
"On an extension of Calderón-Zygmund operators." Illinois J. Math. 46 (4) 1079 - 1088, Winter 2002. https://doi.org/10.1215/ijm/1258138467