Surjectivity of the Fourier transform on torus

Question

It's well known that the Fourier transform $\mathcal{F}: L^2(\mathbb{T}) \rightarrow l^2(\mathbb{Z})$ is an isomorphism and $\mathcal{F}: L^1(\mathbb{T}) \rightarrow l^{\infty}(\mathbb{Z})$ is linear and continous (but not surjective even if the image is replaced by $C_0(\mathbb{Z})$).

I would expect by some interpolation argument that:

1) $\mathcal{F}: L^p(\mathbb{T}) \rightarrow l^{p'}(\mathbb{Z})$ is continous for all $1 < p <2$? ($p'$ is the conjugate exponent)

if the first is true:

2) Is it also surjective for all $1< p <2$? (the case $p=2$ is true, but the case $p=1$ is false)

Answer 1

Yes, (1) is true: Hausdorff-Young Inequality .

$\newcommand\FT{\mathcal F}$ But (2) is false. Suppose $1<p<2$ and $\FT:L^\to\ell^{p'}$ is surjective. Since $\FT$ is certainly injective, the Open Mapping Theorem shows that its inverse is bounded: $$||f||_p\le c||\FT f||_{p'}.$$

But let $$f_N(t)=\sum_{k=1}^Ne^{2\pi ikt}.$$Since $f_N$ is "lacunary" it's well known that there exists $c$ with $$||f_N||_p\ge c||f_N||_2=c\sqrt N.$$(I haevn't found a nice reference for this. It "must" be in Kahane "Some Random Series of Functions"; I haven't verified this since the book's miles away...)

Combining two inequalities leads to $$\sqrt N\le c N^{1/p'},$$a contradiction since $1/p'<1/2$.

Answer 2

The trigonometric series $\sum_{n \ge 1} {\frac{\cos 2^nx}{\sqrt n}}$ satisfies $\sum |a_n|^{2+\epsilon} <\infty, \epsilon >0$ so its coefficients are in $l^p(\mathbb z)$ for every $p>2$ but it is not even a Fourier-Stieltjes series (Fourier series of a measure) not to speak a regular Fourier series of an integrable function on the circle

Also by Rademacher series theory for almost all choices of plus/minus $\sum_{n \ge 1} {\pm \frac{\cos x}{\sqrt n}}$ is not a Fourier-Stieltjes series

in both cases, one shows that the series is not summable $C-1$ (or by any "good summability method") almost anywhere and Fourier (Stieltjes) series are summable $C-1$ ae

references - Zygmund, Trigonometric Series Third edition, Chapter 5 (vol I)