Abstract
In this paper, the authors establish the Lp-mapping properties of a class of singular integral operators along surfaces of revolution with rough kernels. The size condition on the kernels is optimal and much weaker than that for the classical Calderon-Zygmund singular integral operators.
Highlights
Introduction and Main ResultsLet Rn, n 2, be the n-dimensional Euclidean space and S n 1 be the unit sphere in Rn equipped with the normalized Lebesgue measure d d
L1 S n 1 be a homogeneous function of degree zero on Rn and satisfy
We would like to remark that the main ideas in the proofs of our results are taken from [7,9,17]
Summary
Define the singular integral operator T ,h in Rn 1 along by. Inspired by Al-Salman’s work [17], we shall establish the following main result in this paper. Lp Rn 1 , for 1 p , provided that the lower dimensional maximal operator M defined by. 2 p , provided that the maximal operator M in (1.3) is bounded on Lp R2 for 1 p. It is worth pointing out that the size condition is much weaker than that for the classical Calderón-Zygmund singular integral operators. We would like to remark that the main ideas in the proofs of our results are taken from [7,9,17]. Throughout this paper, we always use letter C to denote positive constants that may vary at each occurrence but are independent of the essential variables
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