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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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