Maximal functions along twisted surfaces on product domains
Bull. Korean Math. Soc. 2021 Vol. 58, No. 4, 1003-1019
Published online June 25, 2021
Printed July 31, 2021
Ahmad Al-Salman
Yarmouk University
Abstract : In this paper, we introduce a class of maximal functions along twisted surfaces in $\mathbb{R}^{n}\mathbb{\times R}^{m}$ of the form $$\{(\phi (\left\vert v\right\vert )u, \varphi (\left\vert u\right\vert )v):(u,v)\in \mathbb{R}^{n}\mathbb{\times R}^{m}\}.$$ We prove $L^{p}$ bounds when the kernels lie in the space $L^{q}(\mathbb{S}^{n-1}\mathbb{\times S}^{m-1})$. As a consequence, we establish the $L^{p}$ boundedness for such class of operators provided that the kernels are in $L\log L(\mathbb{S}^{n-1}\mathbb{ \times S}^{m-1})$ or in the Block spaces $B_{q}^{0,0}\left( \mathbb{S}^{n-1} \mathbb{\times S}^{m-1}\right) (q>1)$.
Keywords : Maximal functions, singular integrals, product domains, twisted surfaces, block spaces
MSC numbers : Primary 42B20, 42B15,42B25
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