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On a Singular Integral of Christ–Journé Type with Homogeneous Kernel

Published online by Cambridge University Press:  20 November 2018

Yong Ding
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems (BNU), Ministry of Education, Beijing, 100875, People’s Republic of China, e-mail: dingy@bnu.edu.cn
Xudong Lai
Affiliation:
Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin 150001, People’s Republic of China and School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems (BNU), Ministry of Education, Beijing, 100875, People’s Republic of China, e-mail: xudonglai@mail.bnu.edu.cn
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Abstract

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In this paper, we prove that the singular integral defined by

${{T}_{\Omega ,a}}f(x)=\text{p}\text{.}\text{v}\text{.}{{\int }_{{{\mathbb{R}}^{d}}}}\frac{\Omega (x-y)}{|x-y{{|}^{d}}}\cdot {{m}_{x,y}}a\cdot f(y)dy$is bounded on ${{L}^{p}}({{\mathbb{R}}^{d}})$ for $1\,<\,p\,<\,\infty $ and is of weak type (1,1), where $\Omega \,\in L\text{lo}{{\text{g}}^{+}}L({{S}^{d-1}})$ and ${{m}_{x,y}}a\,=:\,\int{_{0}^{1}}\,a(sx\,+\,(1\,-\,s)y)ds$, with $a\,\in \,{{L}^{\infty }}({{\mathbb{R}}^{d}})\,$ satisfying some restricted conditions.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2018

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