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

Let ρ>0, Ω∈ L¹(Sⁿ⁻¹) (n≥ 2) be a homogeneous function of degree zero and have mean value zero. The parametric Marcinkiewicz integral of higher dimension is defined by μ_ρ(f)(x)=(∫₀^∞|∫|y|≤ tΩ(y)|y|n-ρf(x-y)\,dy|²dtt1+2ρ)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 Bq0,0(Sⁿ⁻¹) for any $q>1$.