Showing that Sobolev Space $H^m$ is in $L^\infty$

Question

I'm very new to Fourier analysis/Sobolev spaces and am stuck on this exercise. I found proofs of more general embedding theorems for Sobolev spaces and some similar questions on here, but they are too complex for my level of understanding.

What I'm working on:

Let $\Omega := (-\pi, \pi)^d, m>d/2$. Show $H^m(\Omega)\subset L^\infty(\Omega)$ by proving $f_k\in\mathcal{\ell}^1(\mathbb{Z}^d)$ for $f\in H^m(\Omega)$.

($f_k$ are the Fourier coefficients of $f$, and $H^m = W^{m,2}$).

My approach:

I have a theorem that states that $u\in H^m(\Omega)$ is equivalent to $ (|k|^mu_k)_{k\in\mathbb{Z}^d}\in\ell^2(\mathbb{Z}^d)$ for $m,d\in\mathbb{N}$. I'm guessing I'm supposed to use it in this proof, but I simply have to idea where to begin.

Answer 1

Hint: multiply and divide by $1+|k|^m$ and then apply Hölder's inequality.