Suppose [itex]f \in L^{4/3}(\mathbb{R}^2)[/itex] and denote its Fourier transform by [itex]\mathscr{F}(f). [/itex] Is it true that the function [itex]g:\mathbb{R}^2 \rightarrow \mathbb{C}[/itex] defined by(adsbygoogle = window.adsbygoogle || []).push({});

[tex]g(x)=|x|^{-1}\mathscr{F}(f)(x)[/tex] is in [itex]L^{4/3}(\mathbb{R}^2) [/itex] 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].

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Fourier transform question

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**