1) Is it true that for every $f \in L^p (\mathbb{T^n})$ (where $\mathbb{T^n}$ is the n-dimensional torus) there exists a unique $u \in W^{2,p} (\mathbb{T^n})$ :
$$\Delta u = f$$
such that $u$ is zero mean.
2)Moreover is it true that:
$$\| u \|_{W^{2,p}(\mathbb{T^n})} \leq C \| f \|_{L^p(\mathbb{T}^n)}.$$
Hint: I did it when $p \geq 2$. We can solve $ \Delta \tilde{u} = f \chi $ in $\mathbb{R}^n$ where $\chi$ is a cutoff with a compact support in a ball and $\chi \equiv 1$ in a smaller ball $B$ that "contains torus" and using Calderon-Zygmund in $\mathbb{R}^n$ we get $ \| \tilde{u} \|_{W^{2,p}(\mathbb{R}^n)} \leq C_1 \| f \chi \|_{L^p(\mathbb{R}^n)} \leq C_2 \|f \|_{L^p (\mathbb{T}^n)} $. Moreover we can extend $u$ to an $\mathbb{R}^n$ function and observing that $u - \tilde{u}$ is harmonic in $B$ from harmonic properties (HERE IS IMPORTANT TO USE PARSERVAL AND THE ZERO MEAN OF $u$ AND GET $\| u \|_{L^2(\mathbb{T}^n)} \lesssim \| f \|_{L^p (\mathbb{T}^n)}$) we estimate $ \| u - \tilde{u} \|_{W^{2,p}(B)} \lesssim \|f \chi\|_{L^1( \mathbb{T}^n)} \lesssim \|f \chi\|_{L^2( \mathbb{T}^n)}$. Thesis follows from:
$$\| u \|_{W^{2,p}(\mathbb{T}^n)} \leq \| u - \tilde{u} \|_{W^{2,p}(B)} + \| \tilde{u} \|_{W^{2,p}(\mathbb{R}^n)} \lesssim \|f \chi\|_{L^2( \mathbb{T}^n)}.$$
The problem in $p <2$ is that i don't know how to get that $ \|u \|_{L^p (\mathbb{T}^n)} \leq C \|f \|_{L^p(\mathbb{T}^n)}$ to use the harmonic property and conclude the same proof.