Boundedness of generalized higher commutators of Marcinkiewicz integrals

Let be a finite family of locally integrable functions. Then, we introduce generalized higher commutator of Marcinkiwicz integral as follows: where When , and ω is homogeneous of degree zero and satisfies the cancelation condition, we prove that is bounded from Lp(n) to Ls(n) where 1 < p < n/β and 1/s = 1/p - β/n. Moreover, if Ω also satisfies some Lq-Dini condition, then is bounded from Lp(n) to (n) and on certain Hardy spaces. The article extends some known results.