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I'm just asking if there is a name for the space of functions on $\mathbb R^n$ whose norm is defined by

$$ \|f\|=\|\hat f\|_{L^p} $$

for $p\in [1,\infty]$. I find it handy to give it a name when discussing the success/failure of Young's inequality on the Fourier transform, among other things.

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  • $\begingroup$ @Giovanni I'm not asking about whether we identify functions equal a.e. (I'm always doing that). I'm asking whether there is a short term for the $L^p$ norm of the Fourier transform. $\endgroup$
    – Fan Zheng
    Apr 12, 2015 at 4:42
  • $\begingroup$ For $p=1,$ it is related to the algebra of Fourier transforms $\mathcal{F}L^{1}(\mathbb R^{n}).$ Some authors denotes as $A(\mathbb R^{n})$ as well. $\endgroup$
    – Inquisitive
    Apr 12, 2015 at 6:35
  • $\begingroup$ It follows for the function itself that $f(x)\in L_q(\mathbb{R}^n)$, not so? $\endgroup$
    – Sergei
    Apr 12, 2015 at 12:27
  • $\begingroup$ I guess this is true only when $p\in [1,2]$. $\endgroup$
    – Fan Zheng
    Apr 12, 2015 at 19:38
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    $\begingroup$ Sergei's comment is incorrect as FanZheng points out. Hausdorff-Young is not a characterization. I have never seen the space of functions whose FTs are p-integrable given any special name, so I would suggest just inventing some ad hoc notation within your paper and sticking to it. $\endgroup$
    – Yemon Choi
    Apr 19, 2015 at 18:44

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In the literature you can sometimes see them called Fourier-Lebesgue spaces, with notation $\mathcal{F} L^p(\mathbb{R}^n)$, consisting of the set of all tempered distributions whose norm (as you wrote) is finite.

See, e.g., http://arxiv.org/abs/0804.1730 and http://arxiv.org/abs/0801.1444

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