Mapping properties of the Marcinkiewicz integrals on homogeneous groups

Abstract

Under the cancellation property and the Lipschitz condition on kernels,we prove that the Marcinkiewicz integrals defined on a homogeneous group mathbb{H}are bounded from H^{1}left(mathbb{H}right) to L^{1}left(mathbb{H}right),from L_{c}^inftyleft(mathbb{H}right) to BMOleft(mathbb{H}right), andfrom L^{p}left(mathbb{H}right) to L^{p}left(mathbb{H}right)for 1<p<infty assuming the L^{q}-boundedness for some q > 1.