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

• Autorzy: Gianfranco Marletta , Fulvio Ricci
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 42B25: Maximal functions, Littlewood-Paley theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1998
• Tom: 130
• Numer: 1
• Strony: 53-65

Daty

wydano

1998

otrzymano

1996-12-09

Twórcy

autor

Gianfranco Marletta

• Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

autor

Fulvio Ricci

• Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

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