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Multiparameter maximal functions along dilation-invariant hypersurfaces. (English) Zbl 0578.42018

Summary: Consider the hypersurface \(x_{n+1}=\prod^{n}_{1}x_ i^{\alpha_ i}\) in \(R^{n+1}\). The associated maximal function operator is defined as the supremum of means taken over those parts of the surface lying above the rectangles \(\{0\leq x_ i\leq h_ i,i=1,...,n\}\). We prove that this operator is bounded on \(L^ p\) for \(p>1\). An analogous result is proved for a quadratic surface in \(R^ 3\).

MSC:

42B25 Maximal functions, Littlewood-Paley theory
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