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Two-parameter maximal functions associated with homogeneous surfaces in $ℝ^n$

100%

Marletta G., Ricci F.

Studia Mathematica

1998

tom 130

nr 1

53-65

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.

Two-parameter maximal functions associated with degenerate homogeneous surfaces in ℝ³

81%

Marletta G., Ricci F., Zienkiewicz J.

Studia Mathematica

1998

tom 130

nr 1

67-75

We consider the two-parameter maximal operator $Mf(x)= sup_{a,b>0}$ ʃ_{|s| < 1} |f(x-(as,bΓ(s)))|ds$ on a homogeneous surface $x_3 = Γ(x_1,x_2)$ in $ℝ^3$. We assume that the curvature of the level set $Γ(x_1,x_2) = 1$ has a degeneracy of finite order k at a given point. We prove that the operator M is bounded on $L^p$ if and only if $p > max{3/2, 2k/(k+1)}$.

Ograniczanie wyników

2 Studia Mathematica

2 Marletta G.

2 Ricci F.

1 Zienkiewicz J.

2 1998

Wyniki wyszukiwania

Two-parameter maximal functions associated with homogeneous surfaces in $ℝ^n$

Two-parameter maximal functions associated with degenerate homogeneous surfaces in ℝ³