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We investigate the boundedness for a class of parametric Marcinkiewicz integral operators associated to submanifolds and a class of related maximal operators under the $L(log L)^{α}(𝕊^{n-1})$ condition on the kernel functions. Our results improve and extend some known results.
We prove the boundedness of the Marcinkiewicz integral operators on under the condition that $Ω ∈ L(log L)^{k/2}(𝕊^{n₁-1}× ⋯ ×𝕊^{n_{k}-1})$. The exponent k/2 is the best possible. This answers an open question posed by Y. Ding.
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