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## Studia Mathematica

1998 | 130 | 1 | 53-65
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### Two-parameter maximal functions associated with homogeneous surfaces in $ℝ^n$

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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$only at the origin. Słowa kluczowe Kategorie tematyczne Czasopismo Rocznik Tom Numer Strony 53-65 Opis fizyczny Daty wydano 1998 otrzymano 1996-12-09 Twórcy autor • Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy autor • Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy Bibliografia • [B] J. Bourgain, Averages in the plane over convex curves and maximal operators, J. Anal. Math. 47 (1968), 69-85. • [CRW] A. Carbery, F. Riccci, and J. Wright, Rev. Mat. Iberoamericana, to appear. • [C] Y.-K. Cho, Multiparameter maximal operators and square functions on product spaces, Indiana Univ. Math. J.43 (1994), 459-491. • [CDMM] M. Cowling, S. Disney, G. Mauceri, and D. Müller, Damping oscillatory integrals, Invent. Math. 101 (1990), 237-260. • [dG] M. de Guzmán, Differentiation of Integrals in$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.
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