Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Weighted inequalities for the disc multiplier

Author: Kenneth F. Andersen
Journal: Proc. Amer. Math. Soc. 83 (1981), 269-275
MSC: Primary 42B15
MathSciNet review: 624912
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Charles Fefferman has shown that the disc multiplier is not a bounded operator on $ {L^p}({{\mathbf{R}}^n})$, $ n > 1$, $ p \ne 2$. On the other hand, Carl Herz has shown that when this operator is restricted to radial functions in $ {{\mathbf{R}}^n}$, it is bounded in $ {L^p}({{\mathbf{R}}^n})$ provided $ p$ satisfies $ 2n/(n + 1) < p < 2n/(n - 1)$. In this paper, sufficient conditions on the weight function $ \omega $ are given in order that the disc multiplier restricted to radial functions should be bounded in $ {L^p}({{\mathbf{R}}^n},\omega (\left\vert x \right\vert))$. When applied to power weights $ \omega (r) = {r^\alpha }$ these conditions are also necessary.

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

  • [1] K. F. Andersen and B. Muckenhoupt, Weighted weak type Hardy inequalities with applications to Hilbert transforms and maximal functions, Studia Math. (to appear). MR 665888 (83k:42018)
  • [2] C. Fefferman, The multiplier problem for the ball, Ann. of Math. 94 (1971), 330-336. MR 0296602 (45:5661)
  • [3] C. Herz, On the mean inversion of Fourier and Hankel transforms, Proc. Nat. Acad. Sci. U.S.A. 40 (1954), 996-999. MR 0063477 (16:127b)
  • [4] I. I. Hirschman, Multiplier transformations. II, Duke Math. J. 28 (1961), 45-56. MR 0124693 (23:A2004)
  • [5] C. E. Kenig and P. Tomas, The weak behaviour of spherical means, Proc. Amer. Math. Soc. 78 (1980), 48-50. MR 548082 (80k:42021)
  • [6] B. Muckenhoupt, Hardy's inequalities with weights, Studia Math. 44 (1972), 31-38.
  • [7] -, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226. MR 0293384 (45:2461)
  • [8] E. M. Stein, Some problems in harmonic analysis, Proc. Sympos. Pure Math., vol. 35, part I, Amer. Math. Soc., Providence, R.I., 1979. MR 545235 (80m:42027)
  • [9] E. Stein and G. Weiss, Introduction to Fourier analysis on euclidean spaces, Princeton Univ. Press, Princeton, N.J., 1971. MR 0304972 (46:4102)
  • [10] G. N. Watson, Theory of Bessel functions, Cambridge Univ. Press, London, 1966.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 42B15

Retrieve articles in all journals with MSC: 42B15

Additional Information

Keywords: Disc multiplier, radial functions, spherical harmonics, weighted inequalities
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society