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Averages of functions over hypersurfaces in \(\mathbb R^n\). (English) Zbl 0626.42009
Given a \(C^\infty\) hypersurface \(S\) in \(\mathbb R^n\), \(n\geq 3\), the ‘mean-value’ operators \(M_t\) are defined by \[ M_t(f)(x)=\int_{S}f(x-ty)\psi (y)\,d\sigma (y),\quad t>0, \] where \(\psi\) is a fixed function in \(C_0^\infty(S)\) and \(d\sigma\) is the induced Lebesgue measure on \(S\). The authors study \(L^p\) inequalities for the maximal operator \(\mathcal M\) defined by \((\mathcal Mf)(x)=\sup_{t>0}| M_t(f)(x)|.\) It is shown that if the Gaussian curvature of \(S\) does not vanish of infinite order at each point of \(S\), then \(\mathcal M\) is bounded in \(L^p\) in a certain range \(p>p_0(S)\).
Reviewer: B.P.Duggal

MSC:
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42B25 Maximal functions, Littlewood-Paley theory
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