I am studying functional analysis with a book of Stein and Shakarchi. Now I have trouble in solving the question for Hausdorff-Young inequality on torus.
The problem is to show that when $f$ is in $L^p$-space on $d$-dimensional torus and $a_n$ is Fourier coefficient w.r.t. $e^{2πi nx}$ for $n \in \mathbb{Z}^d$, $\|a_n\|_q \leqslant \|f\|_p$ for $\dfrac{1}{p}+\dfrac{1}{q} \leqslant 1$.
I solved some part of this problem. In fact, I showed the estimate for $(1, 0)$, $(2, 2)$, $(0, 0)$ on Riesz diagram. With Interpolation theorem, it is only left for me to show an estimate on $(0, 1)$.
I am trying to show every partial l1 sum of an is bounded by the supremum norm of $f$ using Cauchy-Schwartz inequality but it failed because of the $n$ term. Is there any other way to get estimate for $\|a_n\|_1 \leqslant \|f\|_∞$ or is there a counter example for this inequality?
I got a counterexample. Take $f(x) = x$ on $[0, 1].$ Then $a_n = -1/(2\pi in)$ whose sum diverges.