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Maximal functions and singular integrals associated to polynomial mappings of \(\mathbb R^ n\). (English) Zbl 1036.42013

The authors consider convolution operators on \(\mathbb R^n\) of the form \[ T_Pf(x)=\int_{\mathbb R^m} f(x-P(y)) K(y)\, dy, \] where \(P\) is a polynomial defined on \(\mathbb R^m\) with values in \(\mathbb R^n\) and \(K\) is a smooth Calderón-Zygmund kernel on \(\mathbb R^m\). Their main concern is to study the weak type 1-1 for this kind of operators and other related ones. They prove that under certain conditions on \(P\) and \(K\), the above operator can be written as a classical convolution operator associated to a “rough” Calderón-Zygmund kernel on \(\mathbb R^n\). This approach allows the authors to study the weak type 1-1 estimate and the strong type \(p\). They also study the weak type 1-1 for related maximal operators and what they call super maximal operators, defined by taking the suprema over \(P\) ranging in certain classes of polynomials of bounded degree.

MSC:

42B25 Maximal functions, Littlewood-Paley theory
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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References:

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