Proceedings of the Japan Academy, Series A, Mathematical Sciences

Weighted inequalities for spherical maximal operator

Ramesh Manna

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Given a set $E=(0, \infty)$, the spherical maximal operator $\mathcal{M}$ associated to the parameter set $E$ is defined as the supremum of the spherical means of a function when the radii of the spheres are in $E$. The aim of this paper is to study the following inequality \begin{equation} ∫_{\mathbf{R}^{n}} (\mathcal{M}f(x))^{p} φ(x) dx ≤ B_{p} ∫_{\mathbf{R}^{n}} |f(x)|^{p} φ(x) dx, \label{Lb1} \end{equation} holds for $p > \frac{2n}{n-1}$ with the continuous spherical maximal operator $\mathcal{M}$ and where the nonnegative function $\phi$ is in some weights obtained from the $A_{p}$ classes. As an application, we will get the boundedness of vector-valued extension of the spherical means.

Article information

Proc. Japan Acad. Ser. A Math. Sci., Volume 91, Number 9 (2015), 135-140.

First available in Project Euclid: 29 October 2015

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Zentralblatt MATH identifier

Primary: 42B25: Maximal functions, Littlewood-Paley theory
Secondary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)

Spherical maximal operator oscillatory integrals $A_{p}$ weights


Manna, Ramesh. Weighted inequalities for spherical maximal operator. Proc. Japan Acad. Ser. A Math. Sci. 91 (2015), no. 9, 135--140. doi:10.3792/pjaa.91.135.

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  • J. Duoandikoetxea and L. Vega, Spherical means and weighted inequalities, J. London Math. Soc. (2) 53 (1996), no. 2, 343–353.
  • J. Bourgain, Averages in the plane over convex curves and maximal operators, J. Analyse Math. 47 (1986), 69–85.
  • E. M. Stein, Maximal functions. I. Spherical means, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), no. 7, 2174–2175.
  • G. Mockenhaupt, A. Seeger and C. D. Sogge, Wave front sets, local smoothing and Bourgain's circular maximal theorem, Ann. of Math. (2) 136 (1992), no. 1, 207–218.
  • C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107–115.
  • D. S. Kurtz, Littlewood-Paley and multiplier theorems on weighted $L^{p}$ spaces, Trans. Amer. Math. Soc. 259 (1980), no. 1, 235–254.
  • J. García-Cuerva and J. L. Rubio de Francia, Weighted norm ineqalities and related topics, North-Holland Mathematics Studies, 116, North-Holland, Amsterdam, 1985.
  • E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean spaces, Princeton Mathematical Series, 32, Princeton Univ. Press, Princeton, NJ, 1971.
  • E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, 43, Princeton Univ. Press, Princeton, NJ, 1993.
  • E. M. Stein and S. Wainger, Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1239–1295.