Rough maximal singular integral and maximal operators supported by subvarieties on Triebel–Lizorkin spaces

By Feng Liu, Qingying Xue, K. Yabuta. Nonlinear Analysis, 2018

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Abstract In this paper, our object of investigation is the maximal singular integrals with rough kernels associated to polynomial mapping P as well as the corresponding compound submanifolds, which is defined by T h , Ω , P ∗ f ( x ) : = sup ϵ > 0 | ∫ | y | > ϵ f ( x − P ( y ) ) h ( | y | ) Ω ( y ) | y | n d y | . We show that the operator T h , Ω , P ∗ is bounded on Triebel–Lizorkin spaces and Besov spaces when the rough kernel Ω ∈ L ( log + L ) β ( S n − 1 ) for some β ∈ ( 0 , 1 ] . Similar results can be extended to Hardy–Littlewood maximal operators with rough kernels associated to the mapping P .

Subfields: Applied Mathematics

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