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Two-parameter maximal functions associated with degenerate homogeneous surfaces in \(\mathbb{R}^3\). (English) Zbl 0922.42014

This paper develops the work of G. Marletta and F. Ricci [Stud. Math. 130, No. 1, 53-65 (1998)]. Suppose that \(\Gamma :{\mathbb R}^2\to{\mathbb R}\) is homogeneous of degree \(d\) in \({\mathbb R}^+\), and consider the maximal operator \({\mathcal M}\), given by \[ {\mathcal M}f(x)=\sup \Biggl\{\int_B | f(x-(as,b\Gamma(s)))| ds: a,b\in{\mathbb R}^+\Biggr\}, \] where \(B\) stands for the unit ball in \({\mathbb R}^2\). The authors show that \({\mathcal M}\) is bounded on \(L^p({\mathbb R}^3)\) for \(p\) in the range \((p_\Gamma,\infty)\), where \(p_\Gamma\) is determined in terms of the geometry of the graph of \(\Gamma\). In this paper, an extra principal curvature of the graph of \(\Gamma\) is allowed to vanish, but some control of the vanishing is imposed.

MSC:

42B25 Maximal functions, Littlewood-Paley theory
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