Abstract
Abstract This paper is concerned with establishing Lp estimates for a class of maximal operators associated to surfaces of revolution with kernels in Lq(Sn−1 × Sm−1), q > 1. These estimates are used in extrapolation to obtain the Lp boundedness of the maximal operators and the related singular integral operators when their kernels are in the L(logL)κ(Sn−1 × Sm−1) or in the block space $\begin{array}{} B^{0,\kappa-1}_ q \end{array}$(Sn−1 × Sm−1). Our results substantially improve and extend some known results.
Highlights
Introduction and main resultsLet n, m ≥, and let RN (N = n or m) be the N-dimensional Euclidean space
Let KΩ,h(x, y) = Ω(x, y )|x|−n|y|−m h(|x|, |y|), where h is a measurable function on R+ × R+ and Ω is an integrable function on Sn− × Sm− that satis es
When φ(t) = ψ(t) = t, TΩ,h,φ,ψ is just the classical singular integral operator introduced by Fe erman in [1] in which he
Summary
Let SN− be the unit sphere in RN equipped with the normalized Lebesgue surface measure dσ = dσ(·). Let KΩ,h(x, y) = Ω(x , y )|x|−n|y|−m h(|x|, |y|), where h is a measurable function on R+ × R+ and Ω is an integrable function on Sn− × Sm− that satis es. Ψ : R+ → R, consider the singular integral operator TΩP ,h,P,φ,ψ de ned, initially for C∞ functions on Rn+ × Rm+ , by TΩP ,h,P,φ,ψ(f )(x, y) = p.v eiP (u)+iP (v). When P (u) = and P (v) = , we denote TΩP ,h,P,φ,ψ by TΩ,h,φ,ψ. When φ(t) = ψ(t) = t, TΩ,h,φ,ψ (denoted by TΩ,h) is just the classical singular integral operator introduced by Fe erman in [1] in which he
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