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.


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