A note on singular integrals with angular integrability

Abstract

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<p˜≤p<p˜n/(n−1) $1<\tilde{p}\leq p<\tilde{p}n/(n-1)$ or p˜n/(p˜+n−1)<p≤p˜<∞ $\tilde{p}n/(\tilde{p}+n-1)< p\leq \tilde{p}<\infty $. The same conclusions hold for the well-known Riesz transforms and directional Hilbert transforms. It should be pointed out that our proof is based on the Calderón–Zygmund’s rotation method.

Keywords

• Singular integrals
• Rough kernels
• Directional Hilbert transforms
• Riesz transforms
• Mixed radial-angular spaces