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1998 | 130 | 1 | 53-65
Tytuł artykułu

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

Treść / Zawartość
Warianty tytułu
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.
Słowa kluczowe
Czasopismo
Rocznik
Tom
130
Numer
1
Strony
53-65
Opis fizyczny
Daty
wydano
1998
otrzymano
1996-12-09
Twórcy
  • Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
autor
  • Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy, fricci@polito.it
Bibliografia
  • [B] J. Bourgain, Averages in the plane over convex curves and maximal operators, J. Anal. Math. 47 (1968), 69-85.
  • [CRW] A. Carbery, F. Riccci, and J. Wright, Rev. Mat. Iberoamericana, to appear.
  • [C] Y.-K. Cho, Multiparameter maximal operators and square functions on product spaces, Indiana Univ. Math. J.43 (1994), 459-491.
  • [CDMM] M. Cowling, S. Disney, G. Mauceri, and D. Müller, Damping oscillatory integrals, Invent. Math. 101 (1990), 237-260.
  • [dG] M. de Guzmán, Differentiation of Integrals in $ℝ^n$, Lecture Notes in Math. 481, Springer, 1975.
  • [I] A. Iosevich, Maximal operators associated to families of flat curves in the plane, Duke Math. J. 76 (1995), 631-644.
  • [IS] A. Iosevich and E. Sawyer, Maximal averages over mixed homogenous surfaces, preprint.
  • [M] G. Marletta, Maximal functions with mitigating factors in the plane, J. London Math. Soc., to appear.
  • [MRZ] G. Marletta, F. Ricci, and J. Zienkiewicz, Two-parameter maximal functions associated with degenerate homogeneous surfaces in $ℝ^3$, this issue, 67-75.
  • [MSS] G. Mockenhaupt, A. Seeger, and C. Sogge, Wave front sets, local smoothing, and Bourgain's circular maximal theorem, Ann. of Math. 136 (1992), 207-218.
  • [RS] F. Ricci and E. M. Stein, Multiparameter singular integrals and maximal functions, Ann. Inst. Fourier (Grenoble) 42 (1992), 637-670.
  • [RdF] J. L. Rubio de Francia, Maximal functions and Fourier transforms, Duke Math. J. 53 (1986), 395-404.
  • [S] E. M. Stein, Maximal functions: spherical means, Proc. Nat. Acad. Sci. 73 (1976), 2174-2175.
  • [SW] E. M. Stein and S. Wainger, Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc. 84 (1978), 1239-1295.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-smv130i1p53bwm
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