Show that an inequality $\|\{a_n\}\|_{L^q}\leq A\|f\|_{L^p}$,for all $f\in L^p$, with $a_n=\frac{1}{2\pi}\int_\theta^{2\pi}f(\theta)e^{-inθ}d\theta$, is possible only if $1/p+1/q \leq 1$
It's an exercise from Stein's functional analysis. The hint says to use the Dirichlet kernel $D_N(\theta)=\sum_{|n|\leq N}e^{in\theta}$ and the fact that $\|D_N\|_{L^p}\approx N^{1-1/p}$ when $p>1$ and $\|D_N\|_{L^1}\approx \log N$. But I can't see what I want to prove is relevant to Dirichlet kernel. What I see is if argue by contradiction,then the pair $(p,q)$ in Riesz diagram is upon the line $x+y=1$ which seems not work.Does anyone know how to prove it？Thanks in advance
Edit: Clearly $D_N(\theta)$ is in $L^p$, then putting this into the inequality yields $(2N+1)^{1/q}\leq A'N^{1-1/p}$, and this only hold when $1/p+1/q \leq 1$ for sufficient large $N$. So,I think I work it out. But thank everyone all the same.