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feynman456

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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

[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 [url=http://math.stackexchange.com/questions/47951/fourier-transform-of-function-in-l4-3]here[\url].

[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 [url=http://math.stackexchange.com/questions/47951/fourier-transform-of-function-in-l4-3]here[\url].

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