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On an extension of Calderón-Zygmund operators. (English) Zbl 1044.42010
Let $$m,n\in\mathbb{N}$$, $$m\leq n-1$$, and $${\mathcal M}$$ be a compact, smooth, $$m$$-dimensional manifold in $$\mathbb{R}^n$$. Suppose that $${\mathcal M}\cap\{rv: r> 0\}$$ contains at most one point for any unit vector $$v$$. Let $${\mathcal C}(M)$$ denote the cone $$\{r\theta:r> 0,\,\theta\in{\mathcal M}\}$$ equipped with the measure $$ds(r\theta)= r^mdr\,d\sigma(\theta)$$, where $$d\sigma$$ is the induced Lebesgue measure on $${\mathcal M}$$. For a locally integrable function in $${\mathcal C}({\mathcal M})$$ of the form $$K(r\theta)= r^{-m-1} h(r)\Omega(\theta)$$, where $$\Omega$$ satisfies $$\int_{{\mathcal M}}\Omega\,d\sigma= 0$$ and $$h$$ satisfies $$\sup_{R>0} R^{-1} \int^T_0| h(r)|^2\,dr< \infty$$, one defines the singular integral operator $$T_{{\mathcal M},\Omega}$$ by $(T_{{\mathcal M},\Omega} f)(x)= \text{p.v. }\int_{{\mathcal C}({\mathcal M})}f(x- y)\,K(y)\,ds(y).$ The authors prove that if $$\Omega$$ is a function in the Hardy space $$H^1({\mathcal M})$$ and $${\mathfrak M}$$ has a contact of finite order one with every hyperplane, then $$T_{{\mathcal M},\Omega}$$ extends to a bounded operator on the $$L^p(\mathbb{R}^n)$$ spaces for all $$1< p<\infty$$. The result improves an earlier result by Duoandikoetxea and Rubio de Francia where they assumed $$\Omega\in L^q({\mathcal M})$$ for some $$q> 1$$.

##### MSC:
 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B30 $$H^p$$-spaces
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