Abstract
Given a vector field \({\mathfrak{a}}\) on \({\mathbb{R}^3}\) , we consider a mapping \({x\mapsto \Pi_{\mathfrak{a}}(x)}\) that assigns to each \({x\in\mathbb{R}^3}\) , a plane \({\Pi_{\mathfrak{a}}(x)}\) containing x, whose normal vector is \({\mathfrak{a}(x)}\) . Associated with this mapping, we define a maximal operator \({\mathcal{M}^{\mathfrak{a}}_N}\) on \({L^1_{loc}(\mathbb{R}^3)}\) for each \({N\gg 1}\) by
where the supremum is taken over all 1/N × 1/N × 1 tubes τ whose axis is embedded in the plane \({\Pi_\mathfrak{a}(x)}\) . We study the behavior of \({\mathcal{M}^{\mathfrak{a}}_N}\) according to various vector fields \({\mathfrak{a}}\) . In particular, we classify the operator norms of \({\mathcal{M}^{\mathfrak{a}}_N}\) on \({L^2(\mathbb{R}^3)}\) when \({\mathfrak{a}(x)}\) is the linear function of the form (a 11 x 1 + a 21 x 2, a 12 x 1 + a 22 x 2, 1). The operator norm of \({\mathcal{M}^\mathfrak{a}_N}\) on \({L^2(\mathbb{R}^3)}\) is related with the number given by
Similar content being viewed by others
References
Bourgain J.: Besicovitch type maximal operators and applications to Fourier analysi. Geom. Funct. Anal. 1, 147–187 (1991)
Cordoba A.: The Kakeya maximal function and the spherical summation multipliers. Am. J. Math. 99, 1–22 (1977)
Hörmander L.: Fourier integral operators I. Acta Math. 127, 79–183 (1971)
Greenleaf A., Seeger A.: Fourier integral operators with fold sigularities. J. Reine Angew. Math. 455, 35–56 (1994)
Kim J.: Two versions of Nikodym maximal functions on the Heisenberg group. J. Funct. Anal. 257, 1493–1518 (2009)
Melrose R., Taylor M.: Near peak scattering and the correct Kirchhoff approximation for a convex obstacle. Adv. Math. 55, 242–315 (1985)
Sogge C.: Concerning Nikodym-type sets in 3-dimensional curved spaces. J. Am. Math. Soc. 12(1), 1–31 (1999)
Stein E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Press, Princeton (1993)
Tao T.: From rotating needles to stability of waves: emerging connections between combinatorics, analysis, and PDE. Not. Am. Math. Soc. 48, 294–303 (2001)
Wainger S.: Applications of Fourier transform to averages over lower dimensional sets. Proc. Symp. Pure Math. 35(1), 85–94 (1979)
Wolff T.: An imporved bound for Kakeya type maximal function. Revista. Math. Iberoamericana 11, 651–674 (1995)
Wolff, T.: Recent work connected with the Kakeya problem. In: Prospects in Mathematics (Princeton, NJ, 1996), pp. 129–162. Amer. Math. Soc., Providence (1999)
Author information
Authors and Affiliations
Corresponding author
Additional information
The author was supported by the Korea Research Foundation(KRF) grant funded by the Korea government (MEST) (No.2011-0002587).
Rights and permissions
About this article
Cite this article
Kim, J. Nikodym Maximal Functions Associated with Variable Planes in \({\mathbb{R}^{3}}\) . Integr. Equ. Oper. Theory 73, 455–480 (2012). https://doi.org/10.1007/s00020-012-1985-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00020-012-1985-5