For $p\geqslant 1$ there is some $f\in L^1$ s.t. $\sum_{k\in \Bbb Z }|\hat f(k)|^p=\infty $

Question

Im having trouble with this exercise.

Show that for any chosen $p\geqslant 1$ there is some $f\in L^1((-\pi,\pi])$ such that $\sum_{k\in \Bbb Z }|\hat f(k)|^p=\infty $

There $$ \hat f(k):=\frac1{2\pi}\int_{-\pi }^{\pi }f(x)e^{-ikt}\,\mathrm d t\tag1 $$ is the classical Fourier coefficient.

Let $f(z):=\sum_{k\geqslant 1}\frac1{k^r} z^k$, then $f\in L^2(\partial \Bbb D )$ when $r>1/2$, and because for any space of finite measure we knows that $L^p\subset L^q$ for $p>q$, then $f\in L^1(\partial \Bbb D )$ also and $\sum_{k\in \Bbb Z }|\hat f(k)|^p=\infty $ whenever $rp\leqslant 1$, what happens when $p\in[1,1/r]$, and so we proved the case for all $p\in[1,2)$.

However Im having trouble finding a way to show the statement for $p\geqslant 2$. I tried to find some lower bound for $|\hat f(k)|^p$ (or it sum) using Jensen's inequality and similar ideas but I dont find something useful. Some help will be appreciated, thank you.

Answer 1

If for $0 < \alpha < 1$ you let $f(t) = |t|^{\alpha - 1}$ then a direct calculation shows that for some constant $c$ one has $\hat{f}(k) = c|k|^{-\alpha} + o(|k|^{-\alpha})$ for $k > 0$ and $\hat{f}(k) = \bar{c}|k|^{-\alpha} + o(|k|^{-\alpha})$ for $k < 0$. Thus taking $\alpha$ close enough to $0$ will give an example for any given $p$.

Some details on the calculation: by definition one has $$\hat{f}(k) = {1 \over 2\pi} \int_{-\pi}^{\pi}|t|^{\alpha - 1}e^{-ikt} dt$$ Say $k > 0$. Then changing variables to $u = tk$ this equals $${1 \over 2\pi} k^{-\alpha}\int_{-k\pi}^{k\pi}|u|^{\alpha - 1}e^{-iu} du$$ Since the improper integral $\int_{-\infty}^{\infty}|u|^{\alpha - 1}e^{-iu} du$ converges to some limit $L$ this equals $${L \over 2\pi} k^{-\alpha} - k^{-\alpha}{1 \over 2\pi} \int_{|u| > 2k\pi}|u|^{\alpha - 1}e^{-iu} du$$ Again using that the improper integral converges, the error term is $o(k^{-\alpha})$ as $k \rightarrow \infty$.

Answer 2

By the closed graph theorem, it suffices to prove that there is no $C > 0$ satisfying $\sum_k |\widehat{f}(k)|^p \leq C \cdot \|f\|_{L^1}^p$ for all $f \in L^1$.

To see this, consider the Fejer kernel $F_n$. It satisfies $F_n \geq 0$ and $\|F_n\|_{L^1} = \int F_n \, d x \leq \kappa$, with $\kappa$ independent of $n$. It thus suffices to show that $\sum_k |\widehat{F_n}(k)|^p \to \infty$ as $n\to\infty$. To see this, simply note that $$ \widehat{F_n}(k) = 1 - \frac{|k|}{n} \quad \text{for} \quad |k| \leq n . $$