A Note on Generalized Parabolic Marcinkiewicz Integrals with Grafakos–Stefanov Kernels

This paper focuses on studying the generalized Marcinkiewicz integral operators with mixed homogeneity over symmetric spaces. By making an appropriate decomposition of the aforementioned operators and tracking certain estimates, the boundedness of these operators is established from the homogeneous Triebel–Lizorkin space F.p0,τ(Rd) to the Lp(Rd) space for all p,τ∈(2+2α1+2α,2α+2) provided that the kernel functions belong to the Grafakos–Stefanov class. The main results generalize and improve some previously known results on Marcinkiewicz and generalized Marcinkiewicz operators.