A note on singular integrals with angular integrability

Journal of Inequalities and Applications

Feng Liu¹ 

[1] College of Mathematics and System Sciences, Shandong University of Science and Technology;

关键词: Singular integrals;    Rough kernels;    Directional Hilbert transforms;    Riesz transforms;    Mixed radial-angular spaces;   

DOI  :  10.1186/s13660-019-2214-4

来源: DOAJ

【 摘 要 】

Abstract In this note we study the rough singular integral TΩf(x)=p.v.∫Rnf(x−y)Ω(y/|y|)|y|ndy, $$ T_{\varOmega }f(x)=\mathrm{p.v.} \int _{\mathbb{R}^{n}}f(x-y)\frac{\varOmega (y/ \vert y \vert )}{ \vert y \vert ^{n}}\,dy, $$ where n≥2 $n\geq 2$ and Ω is a function in LlogL(Sn−1) $L\log L(\mathrm{S} ^{n-1})$ with vanishing integral. We prove that TΩ $T_{\varOmega }$ is bounded on the mixed radial-angular spaces L|x|pLθp˜(Rn) $L_{|x|}^{p}L_{\theta }^{\tilde{p}}( \mathbb{R}^{n})$, on the vector-valued mixed radial-angular spaces L|x|pLθp˜(Rn,ℓp˜) $L_{|x|}^{p}L_{\theta }^{\tilde{p}}(\mathbb{R}^{n},\ell ^{\tilde{p}})$ and on the vector-valued function spaces Lp(Rn,ℓp˜) $L^{p}(\mathbb{R}^{n}, \ell ^{\tilde{p}})$ if 1

【 授权许可】

Unknown