Simple proof that Fourier transform is an isomorphism between $L^p$ spaces for $p \neq 2$?

It is known that the Fourier transform $\mathcal F$ maps $L^2 \to L^2$ as an (isometric) isomorphism and $L^1 \to L^\infty$ as bounded operator. Via Riesz-Thorin this result can be extended to give that $\mathcal F$ also maps $L^p \to L^{p'}$ where $1 = \frac{1}{p} + \frac{1}{p'}$ as a bounded operator, i.e. the Hausdorff-Young equality holds: $$ \|\mathcal F(u) \|_{p'} \leq \|u\|_p $$ for all $u \in L^p(\mathbb{R}^n)$, $p \in (1,2)$.

How do we know that the Fourier transform is only an isomorphism from $L^2 \to L^2$? One could argue that no $L^p$ space is isomorphic to an $L^q$ space for $p \neq q$ using the invariants type and cotype as suggested here but, considering that I only want to show that $\mathcal F$ is not an isomorphism from $L^p \to L^{p'}$, I suppose there is a more simple approach directly related to the Fourier transform. I am thinking of showing the fourier transform from $L^p \to L^{p'}$ is simply not surjective.

Can you think of an easier approach to this problem?

Solution 1:

Here is a (modified) argument which I used repeatedly in a paper of mine. Possibly, this is some form of type and cotype argument (I am not too familiar with these notions), but maybe it is elementary enough for you:

I will make extensive use of the Khintchine inequality, which states that if $\varepsilon_1, \dots, \varepsilon_N$ are independent and uniformly distributed in $\{1, -1\}$, then we have $$ \mathbb{E} \left|\sum_{n=1}^N \varepsilon_n x_n\right|^p \asymp \left(\sum_{n=1}^N |x_n|^2\right)^p , $$ for arbitrary $x_1,\dots, x_N \in \Bbb{C}$. Here, $\Bbb{E}$ denotes the expected value with respect to $\varepsilon = (\varepsilon_1,\dots, \varepsilon_N)$. It is crucial that the implied constants above are independent of $x_1,\dots, x_N$ and of $N$; they only depend on the choice of $p \in (0,\infty)$.

Now, assume that $\mathcal{F} : L^p \to L^q$ is an isomorphism. Let us first rule out the case $q = \infty$: If we had $q=\infty$, then there would be some $f \in L^p$ with $\widehat{f} \equiv 1$, where I write $\widehat{f} = \mathcal{F}f$. But we also have $\widehat{\delta_0} \equiv 1$, where we are using the formalism of tempered distributions, and where $\delta_0$ is the Dirac measure at the origin. But this implies $f = \delta_0$ as tempered distributions, which is absurd.

Hence $q \in (0,\infty)$. Now, let $\varepsilon_1,\dots, \varepsilon_N$ as above. For $n \in \Bbb{N}$, let $y_n := (n,0,\dots,0) \in \Bbb{R}^d$, and set $f_n := T_{y_n} 1_{(0,1)^d}$, where $(T_y f)(x) = f(x-y)$. Note that $\widehat{f_n} = M_{-y_n} \widehat{f}$ for $f := 1_{(0,1)}$, where $M_\xi f(x) = e^{2\pi i \langle x, \xi \rangle} f(x)$ denotes the modulation of $f$. Finally, observe that the supports of $f_n, f_m$ are disjoint for $n \neq m$. All in all, these properties imply \begin{align*} N^{q/p} = \Bbb{E} \, \left\| \sum_{n=1}^N \varepsilon_n \cdot 1_{(n,n+1)} \right\|_{L^p}^q & \asymp \Bbb{E} \, \left\| \sum_{n=1}^N \varepsilon_n \cdot \mathcal{F}[ 1_{(n,n+1)} ] \right\|_{L^q}^q \\ & = \Bbb{E} \int \left| \sum_{n=1}^N \varepsilon_n e^{2\pi i \langle y_n, \xi\rangle} \right|^q \cdot |\widehat{f}(\xi)|^q \, d\xi \\ (\text{Fubini's theorem}) & = \int \Bbb{E} \left| \sum_{n=1}^N \varepsilon_n e^{2\pi i \langle y_n, \xi\rangle} \right|^q \cdot |\widehat{f}(\xi)|^q \, d\xi \\ (\text{Khintchine inequality}) & \asymp N^{q/2} \cdot \| \widehat{f} \|_{L^q}^q \\ & \asymp N^{q/2} \cdot \|f\|_{L^p}^q = N^{q/2}, \end{align*} where the implied constants are independent of $N \in \Bbb{N}$. This can only hold for $p = 2$.

Now, let us show that also $q = 2$. To see this, note for arbitrary $f \in L^p = L^2$ by Plancherel's theorem that $$ \|\widehat{f} \|_{L^2} = \| f \|_{L^2} = \| f \|_{L^p} \asymp \|\widehat{f}\|_{L^q} . $$ Thus, we get for arbitrary $L^2$ functions $g \in L^2$ that $\|g\|_{L^2} \asymp \| g \|_{L^q}$, which easily implies $q = 2$, by considering e.g. $g = \sum_{n=1}^N f_n$ with $f_n$ as above.