Weak Type (1,1) Bounds for Parametric Marcinkiewicz Integrals with Rough Kernels on Block Spaces

Let $\rho>0$, $\Omega\in L^1(S^{n-1})$ $(n\geq 2)$ be a homogeneous function of degree zero and have mean value zero. The parametric Marcinkiewicz integral of higher dimension is defined by $\mu_\rho(f)(x)=\left(\int_0^\infty\left|\int_{|y|\leq t}\frac{\Omega(y)}{|y|^{n-\rho}}f(x-y)\,dy\right|^2\frac{dt}{t^{1+2\rho}}\right)^{1/2}$. In this paper, we show that the parametric Marcinkiewicz integral is of weak type $(1,1)$ with rough kernels belonging to certain block spaces $B_q^{0,0}(S^{n-1})$ for any $q>1$.