A multiplier theorem for Fourier transforms
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- by James D. McCall PDF
- Trans. Amer. Math. Soc. 189 (1974), 359-369 Request permission
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 $\smallint _{ - \infty }^\infty |f(x + iy){|^p}dx \leq C < \infty$ 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
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Additional Information
- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 189 (1974), 359-369
- MSC: Primary 30A78; Secondary 42A68
- DOI: https://doi.org/10.1090/S0002-9947-1974-0409829-6
- MathSciNet review: 0409829