Lp estimates for maximal functions along surfaces of revolution on product spaces

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.

On the boundedness of parabolic maximal operators on product domains

“…We should state here that for the case 𝛾 = 2 and 𝛼 1 = ⋯ = 𝛼 𝑚 = 𝛽 1 = ⋯ = 𝛽 𝑛 = 1, Theorem 1.2 extends and improves the results given in [3,4,5,6]. Further, in the proof of Theorem 1.2, we employ the extrapolation method found in [1,13], which is considered a new alternative technique. More novelty, we can utilize the boundedness of the operator  Ω,𝛾 to satisfy the boundedness of the singular integral operator 𝑇 Ω,𝜑 .…”

“…By the conclusions of Theorem 1.1 and applying the same extrapolation argument used in [1,13], we obtain the following result:…”

“…), and then we shall use these obtained estimates in the extrapolation argument that employed in [1] to get some new extended results in the parabolic maximal functions. Moreover, we shall present and prove several applications of our main result.…”

“…In the proof of this theorem, we employ similar arguments used in the proof of [Theorem 1.6, [1]] and [Theorem 1.1, [23]]. By the duality,…”

A boundedness result for Marcinkiewicz integral operator

“…1 . We refer the readers to [1][2][3][4][11][12][13][14] for more background information and related results.…”