An extension of Calderón-Zygmund type singular integral with non-smooth kernel (Q1982526)

In this article the authors studied the following singular integral \[Tf(x)=\mathrm{p.v.}\int_{\mathbb{R}^n}\frac{\Omega(y)}{|y|^{n-\beta}}f(x-y)dy,\] where \(f\in L^1(\mathbb{R}^n)\cap L^q(\mathbb{R}^n)\), \(1\leq q<\infty\) and \(0<\beta<n\). This type of singular integral can be viewed as an extension of the classical Calderón-Zygmund type singular integral. Moreover, this kind of singular integral appears in the approximation of the surface quasi-geostrophic equation from the generalized surface quasi-geostrophic equation. The authors established an estimate of the singular integral in the \(L^q\) space for \(1<q<\infty\) and a weak \((1,1)\) type of the singular integral when \(0<\beta<(q-1)n\) and \(\Omega\in L^\infty(\mathrm{S}^{n-1})\). Particularly, the bounds do not depend on \(\beta\). Personally speaking, this paper is very interesting and very well written. This paper involves a large amount of definitions, notation and references, which increases its richness. Meanwhile, these elements also add a lot of difficulty. Overall, this article is a very nice piece of work.