A multiplier theorem for Fourier transforms

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 ${L_q}(0,\infty )$.