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Proceedings of the American Mathematical Society

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

 

 

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 $.


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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