Two-parameter maximal functions associated with homogeneous surfaces in $ℝ^n$
Studia Mathematica, Tome 130 (1998) no. 1, pp. 53-65
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Given a hypersurface $x_n = Ꮁ(x_1...,x_{n-1})$ in $ℝ^n$, where Ꮁ is homogeneous of degree d>0, we define the two-parameter maximal operator $$Mf(x) = sup_{a,b>0} ∫_{s∈ℝ^{n-1},|s| 1} |f(x - (as, bᎱ(s)))|ds.$$ We prove that if d ≠ 1 and the hypersurface has non-vanishing Gaussian curvature away from the origin, then M is bounded on $L^p$ if and only if p>n/(n-1). If d = 1, i.e. if the surface is a cone, the same conclusion holds in dimension n ≥ 3 if the surface has n-1 non-vanishing principal curvatures away from the origin and it intersects the hyperplane $x_n = 0$ only at the origin.
@article{10_4064_sm_130_1_53_65,
author = {Gianfranco Marletta and Fulvio Ricci},
title = {Two-parameter maximal functions associated with homogeneous surfaces in $\ensuremath{\mathbb{R}}^n$},
journal = {Studia Mathematica},
pages = {53--65},
publisher = {mathdoc},
volume = {130},
number = {1},
year = {1998},
doi = {10.4064/sm-130-1-53-65},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/sm-130-1-53-65/}
}
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Gianfranco Marletta; Fulvio Ricci. Two-parameter maximal functions associated with homogeneous surfaces in $ℝ^n$. Studia Mathematica, Tome 130 (1998) no. 1, pp. 53-65. doi: 10.4064/sm-130-1-53-65
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