Maximal averages over hypersurfaces and the Newton polyhedron. (English) Zbl 1243.42026
In this interesting paper the author studies the maximal averages over hypersurfaces and the Newton polyhedron. The author obtains \(L^p\) estimates for the maximal average over hypersurfaces in \(\mathbb R^n\) for \(n > 2\) for the case of \(p > 2\). These estimates are sharp in various cases, including the convex hypersurfaces of finite type considered by several authors. The author also gives a generalization of the result of C. D. Sogge and E. M. Stein [Invent. Math. 82, 543–556 (1985; Zbl 0626.42009)] that for some finite \(p\) that maximal operator corresponding to a hypersurface whose Gaussian curvature does not vanish to infinite order is bounded on \(L^p\) for some finite \(p\). Analogous estimates are proven for Fourier transforms of surface measures, and these are sharp for the same hypersurfaces as in the case of maximal operators.
Reviewer: Lijing Sun (Milwaukee)
MSC:
| 42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
| 35S30 | Fourier integral operators applied to PDEs |
Citations:
Zbl 0626.42009References:
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