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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