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We study the Marcinkiewicz integral operator M f(x) = ( ∫∞ −∞ | ∫ |y|≤2t f (x − (y))(Ω(y)/|y|n−1)dy|2dt/22t)1/2, where is a polynomial mapping from Rn into Rd and Ω is a homogeneous function of degree zero on Rn with mean value zero over the unit sphere Sn−1. We prove an Lp boundedness result of M for rough Ω. 2000 Mathematics Subject Classification. 42B20,… (More)
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.
where, p.v. denotes the principal value. It is known that if Φ is of finite type at 0 (see Definition 2.2) and Ω ∈ 1(Sn−1), then TΦ,Ω is bounded on Lp for 1<p <∞ . Moreover, it is known that TΦ,Ω may fail to be bounded on Lp for any p if the finite-type condition is removed. In , Fan et al. showed that the Lp boundedness of the operator TΦ,Ω still… (More)
In this paper we study the boundedness properties of certain oscillatory integrals with polynomial phase. We obtain sharp estimates for these oscillatory integrals. By the virtue of these estimates and extrapolation we obtain L boundedness for these oscillatory integrals under rather weak size conditions on the kernel function. Keywords—Fourier transform,… (More)
In this paper, we obtain sharp L estimates of two classes of maximal operators related to rough singular integrals and Marcinkiewicz integrals. These estimates will be used to obtain similar estimates for the related singular integrals and Marcinkiewicz integrals. By the virtue of these estimates and extrapolation we obtain the L boundedness of all the… (More)
We establish the L boundedness of singular integrals on product domains with rough kernels in L(logL) and are supported by subvarieties.
In this paper, we study the L boundedness of a class of parametric Marcinkiewicz integral operators with rough kernels in L(log L)(Sn−1). Our result in this paper solves an open problem left by the authors of ().