Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A weighted inequality for the maximal Bochner-Riesz operator on $ {\bf R}\sp 2$

Author: Anthony Carbery
Journal: Trans. Amer. Math. Soc. 287 (1985), 673-680
MSC: Primary 42B10
MathSciNet review: 768732
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For $ f \in \mathcal{S}({{\mathbf{R}}^2})$, let $ (T_R^\alpha f)\hat \emptyset (\xi ) = (1 - \vert\xi {\vert^2}{R^2})_ + ^\alpha \hat f(\xi )$. It is a well-known theorem of Carleson and Sjölin that $ T_1^\alpha $ defines a bounded operator on $ {L^4}$ if $ \alpha > 0$. In this paper we obtain an explicit weighted inequality of the form

$\displaystyle \int {\mathop {\sup }\limits_{0 < R < \infty } \vert T_R^\alpha f(x){\vert^2}w(x)\;dx \leqslant \int {\vert f{\vert^2}{P_\alpha }w(x)\;dx,} } $

with $ {P_\alpha }$ bounded on $ {L^2}$ if $ \alpha > 0$. This strengthens the above theorem of Carleson and Sjölin. The method gives information on the maximal operator associated to general suitably smooth radial Fourier multipliers of $ {{\mathbf{R}}^2}$.

References [Enhancements On Off] (What's this?)

  • [1] Anthony Carbery, The boundedness of the maximal Bochner-Riesz operator on 𝐿⁴(𝑅²), Duke Math. J. 50 (1983), no. 2, 409–416. MR 705033
  • [2] -, Radial Fourier multipliers and associated maximal functions, Conf. Harmonic Analysis (El Escorial 1983), preprint.
  • [3] Lennart Carleson and Per Sjölin, Oscillatory integrals and a multiplier problem for the disc, Studia Math. 44 (1972), 287–299. (errata insert). Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, III. MR 0361607,
  • [4] A. Córdoba, The multiplier problem for the polygon, Ann. of Math. (2) 105 (1977), no. 3, 581–588. MR 0438022,
  • [5] -, An integral inequality for the disc multiplier, preprint.
  • [6] Charles Fefferman, A note on spherical summation multipliers, Israel J. Math. 15 (1973), 44–52. MR 0320624,
  • [7] José L. Rubio de Francia, Weighted norm inequalities and vector valued inequalities, Harmonic analysis (Minneapolis, Minn., 1981) Lecture Notes in Math., vol. 908, Springer, Berlin-New York, 1982, pp. 86–101. MR 654181
  • [8] E. M. Stein, Some problems in harmonic analysis, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 3–20. MR 545235
  • [9] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 42B10

Retrieve articles in all journals with MSC: 42B10

Additional Information

Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society