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In this note the authors give the L 2 (R") boundedness of a class of parametric Marcinkiewicz integral H P aA with kernel function ft in Z. log + L(5""') and radial function h(\x |) 6 /°°(L«)(R + )forl < q < oo. As its corollary, the Z/(R")(2

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In this paper, it was proved that the commutator H β,b generated by an n-dimensional fractional Hardy operator and a locally integrable function b is bounded from L p 1 (R n ) to L p 2 (R n ) if and only if b is a CṀO(R n ) function, where 1/p1 − 1/p2 = β/n, 1 < p1 < ∞, 0 β < n. Furthermore, the characterization of H β,b on the homogenous Herz spaceK α,p q (R n ) was obtained.