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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A multiplier theorem for Fourier transforms


Author: James D. McCall
Journal: Trans. Amer. Math. Soc. 189 (1974), 359-369
MSC: Primary 30A78; Secondary 42A68
MathSciNet review: 0409829
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Abstract: A function f analytic in the upper half-plane $ {\Pi ^ + }$ is said to be of class $ {E_p}({\Pi ^ + })(0 < p < \infty )$ if there exists a constant C such that $ \smallint _{ - \infty }^\infty \vert f(x + iy){\vert^p}dx \leq C < \infty $ for all $ y > 0$. These classes are an extension of the $ {H_p}$ spaces of the unit disc U. For f belonging to $ {E_p}({\Pi ^ + })(0 < p \leq 2)$, there exists a Fourier transform f with the property that $ f(z) = 2{(\pi )^{ - 1}}\smallint _0^\infty \hat f(t){e^{izt}}dt$. This makes it possible to give a definition for the multiplication of $ {E_p}({\Pi ^ + })(0 < p \leq 2)$ into $ {L_q}(0,\infty )$ that is analogous to the multiplication of $ {H_p}(U)$ into $ {l_q}$. In this paper, we consider the case $ 0 < p < 1$ and $ p \leq q$ and derive a necessary and sufficient condition for multiplying $ {E_p}({\Pi ^ + })$ into $ {L_q}(0,\infty )$.


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

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0409829-6
PII: S 0002-9947(1974)0409829-6
Keywords: $ {H_p}$ spaces, multipliers
Article copyright: © Copyright 1974 American Mathematical Society