Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-26T07:02:04.733Z Has data issue: false hasContentIssue false
  • English
  • Français

Weighted Lacunary Maximal Functions on Curves

Published online by Cambridge University Press:  20 November 2018

Jong-Guk Bak
Jong-Guk Bak*
Affiliation:
Department of Mathematics, Florida State University, Tallahassee, Florida 32306, U.S.A. e-mail:bak@math.fsu.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let γ(t) = (t, t2,..., tn) + a be a curve in Rn, where n ≥ 2 and a ∊ Rn. We prove LP-Lq estimates for the weighted lacunary maximal function, related to this curve, defined by

If n = 2 or 3 our results are (nearly) sharp.

Keywords

42B25
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 38 , Issue 3 , 01 September 1995 , pp. 271 - 277
Copyright
Copyright © Canadian Mathematical Society 1995

References

[BP] Benedek, A. and Panzone, R., The spaces LP with mixed norms, Duke Math. J. 28(1961), 301–324.Google Scholar
[B] Bourgain, J., Averages in the plane over convex curves and maximal operators, J. Analyse Math. 47(1986), 69–85.Google Scholar
[C] Calderόn, C. P., Lacunary spherical means, Illinois J. Math. 23(1979), 476–484.Google Scholar
[Ch1] Christ, M., Weak type (1,1) bounds for rough operators, Ann. of Math. 128(1988), 19–42.Google Scholar
[Ch2] Christ, M., Endpoint bounds for singular fractional integral operators, preprint (1988).Google Scholar
[DR] Duoandikoetxea, J. and Rubio, J. L. de Francia, Maximal and singular integral operators via Fourier transform estimates, Invent. Math. 84(1986), 541–561.Google Scholar
[GS] Gelfand, I. M. and Shilov, G. E., Generalized Functions, vol. 1, Academic Press, 1964.Google Scholar
[M] McMichael, D., Damping oscillatory integrals with polynomial phase, Math. Scand. 73(1993), 215–228.Google Scholar
[MSS] Mockenhaupt, G., Seegerand, A. Sogge, C., Wave front sets, local smoothing and Bourgain s circular maximal theorem, Ann. of Math. 136(1992), 207–218.Google Scholar
[NSW] Nagel, A., Stein, E. M. and Wainger, S., Differentiation in lacunary directions, Proc. Nat. Acad. Sci. U.S.A. 75(1978), 1060–1062.Google Scholar
[O1] Oberlin, D. M., Convolution estimates for some measures on curves, Proc. Amer. Math. Soc. 99(1987), 56–60.Google Scholar
[O2] Oberlin, D. M., Operators interpolating between Riesz potentials and maximal operators, Illinois J. Math. 33 (1989), 143-152.Google Scholar
[O3] Oberlin, D. M., Oscillatory integrals with polynomial phase, Math. Scand. 69(1991), 45–56.Google Scholar
[So] Sogge, C., Fourier integrals in classical analysis, Cambridge Tracts in Math. 105, Cambridge Univ. Press, 1993.Google Scholar
[SI] Stein, E. M., Singular integrals and differentiability properties of functions, Princeton Univ. Press, 1970.Google Scholar
[S2] Stein, E. M., Oscillatory integrals in Fourier analysis. In: Beijing lectures in harmonic analysis, Ann. of Math. Studies 112, Princeton Univ. Press, 1986.Google Scholar
[S3] Stein, E. M., Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Univ. Press, 1993.Google Scholar