Fourier transform question

1. Jun 27, 2011

feynman456

Suppose $f \in L^{4/3}(\mathbb{R}^2)$ and denote its Fourier transform by $\mathscr{F}(f).$ Is it true that the function $g:\mathbb{R}^2 \rightarrow \mathbb{C}$ defined by
$$g(x)=|x|^{-1}\mathscr{F}(f)(x)$$ is in $L^{4/3}(\mathbb{R}^2)$ also?

Simply appealing to Hausdorff-Young and Hölder's inequality doesn't suffice.

Edit: It turns out that this can be proved using the Marcinkiewicz interpolation theorem[\url], as described here[\url].

Last edited: Jun 27, 2011