Rough singular integrals with kernels supported by submanifolds of finite type. (English) Zbl 1104.42006

Al-Qassem, Hussain; Al-Salman, Ahmad; Pan, Yibiao

Asian J. Math. 9, No. 1, 19-29 (2005).

Let \(n\geq 2\) and \(S^{n-1}\) be the unit sphere in \(\mathbb{R}^n\) equipped with the normalized Lebesgue measure \(d\sigma\). Suppose that \(\Omega\) is a homogeneous function of degree zero on \(\mathbb{R}^n\) that satisfies \(\Omega\in L^1(S^{n-1})\) and \[ \int_{S^{n-1}} \Omega(x)\,d\sigma= 0. \] Let \(B(0,r)\) be the ball centered at the origin in \(\mathbb{R}^n\) with radius \(r\). For a \(C^\infty\) mapping \(\Phi: B(0,1)\to\mathbb{R}^m\), \(m\in\mathbb{N}\), the authors study the singular integral operator \[ T_\Phi f(x)= \text{p.v.\,}\int_{B(0,1)} f(x- \Phi(y))|y|^{-n} \Omega(y)\,dy \] and its related maximal operators \[ M_\Phi f(x)= \sup_{r> 0}\, r^{-n} \int_{B(0,r)} |f(x- \Phi(y))\Omega(y)|\,dy, \]
\[ T^*_\Phi(x)= \sup_{\varepsilon> 0}\,\Biggl| \int_{\varepsilon<|y|< 1} f(x- \Phi(y))|y|^{-n}\Omega(y)\, dy\Biggr|. \] In the paper, the authors investigate the case that \(\Phi\) is of finite type at \(0\). By assuming \(\Omega\) is in certain block space \(B^{0,0}_q(S^{n-1})\), the authors establish the \(L^p\) boundedness of \(T_*\), \(T^*_\Phi\) and \(M_\Phi\) for \(1<p<\infty\).
It is now known that any \(\Omega\) in \(B^{0, 0}_q(S^{n- 1})\) is a sum of an \(H^1\) function and an \(L(\log L)\) function. This result was recently proved by X. Ye in his Ph.D. Thesis in Zhejiang University, China (2006).

Reviewer: Dashan Fan (Milwaukee)

MSC:

42B20	Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B15	Multipliers for harmonic analysis in several variables
42B25	Maximal functions, Littlewood-Paley theory

Keywords:

rough singular kernel; finite type; block space