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Abstract.We prove the L p boundedness of the Marcinkiewicz integral operators 1. Introduction. Marcinkiewicz integrals have been studied by many authors, dating back to the investigations of such operators by Zygmund on the circle and by Stein on R n .We shall be primarily concerned with Marcinkiewicz integrals on the product space R n × R m , since the more general setting of R n 1 × · · · × R n k can be handled similarly (see Section 4).For n, m ≥ 2, x ∈ R n \{0}, y ∈ R m \{0}, we let x = x/|x| and y = y/|y|. Let Ω ∈ L 1 (S n−1 × S m−1 ) be a function satisfying the following cancellation conditions:

Abstract. We obtain certain sharp L p bounds for the generalized parametric Marcinkiewicz integrals M (λ ) Ω,h,ρ . The singular kernels are allowed to be rough on the unit sphere as well as in the radial direction. By the virtue of these estimates along with an extrapolation argument we obtain some new and improved results on generalized parametric Marcinkiewicz integrals. Our conditions on Ω and h are known to be the weakest conditions in their respective classes. One of our main results answers a question posed by Fan and Wu.Mathematics subject classification (2010): Primary 42B20, Secondary 42B25, 42B35, 42B99.

Abstract:We prove the boundedness of several classes of rough integral operators on Triebel-Lizorkin spaces. Our results represent improvements as well as natural extensions of many previously known results.2010 Mathematics Subject Classification: Primary: 42B20; Secondary: 42B15, 42B25.

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