Boundedness of Littlewood-Paley operators relative to non-isotropic dilations

summary:We consider Littlewood-Paley functions associated with a non-isotropic dilation group on $\Bbb R^n$. We prove that certain Littlewood-Paley functions defined by kernels with no regularity concerning smoothness are bounded on weighted $L^p$ spaces, $1<p<\infty $, with weights of the Muckenhoupt class. This, in particular, generalizes a result of N. Rivière (1971).\looseness -