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A multiplier theorem for $ H\sp{1}({\bf R}\sp{n})$


Author: Daniel M. Oberlin
Journal: Proc. Amer. Math. Soc. 73 (1979), 83-87
MSC: Primary 42B15
DOI: https://doi.org/10.1090/S0002-9939-1979-0512063-2
MathSciNet review: 512063
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Abstract: We prove the following theorem.

Theorem. Let m be a nonnegative measurable function on $ [0,\infty )$. For $ n \geqslant 2$, the two conditions below are equivalent:

(a) $ \smallint {\;_{{R^n}}}\vert\hat f(x)\vert m(\vert x\vert)\;dx < \infty $ for each $ f \in {H^1}({R^n})$,

(b) $ \sup \{ {2^{(n - 1)k}}\smallint _{{2^k}}^{{2^{k + 1}}}m(r)\;dr: - \infty < k < \infty \} < \infty $.


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

  • [1] R. J. Bagby, Fourier multipliers from $ {H^1}({R^n})$ to $ {L^p}({R^n})$, Rocky Mountain J. Math. 7 (1977), 323-332. MR 0438021 (55:10942)
  • [2] R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645. MR 0447954 (56:6264)
  • [3] R. Johnson, Convoluteurs of $ {H^p}$ spaces, Bull. Amer. Math. Soc. 81 (1975), 711-714. MR 0370054 (51:6283)
  • [4] E. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, N. J., 1971. MR 0304972 (46:4102)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1979-0512063-2
Keywords: $ {H^1}({R^n})$, multiplier, atom
Article copyright: © Copyright 1979 American Mathematical Society

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