Two-parameter maximal functions associated with homogeneous surfaces in $ℝ^n$
Volume 130 / 1998
Studia Mathematica 130 (1998), 53-65
DOI: 10.4064/sm-130-1-53-65
Abstract
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.
