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

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).

MSC:

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