Let $\Omega\subseteq \mathbb{R}^d$ a bounded domain with sufficient smooth boundary. We consider the Poisson Equation $$\left\{\begin{array} --\Delta u = f, & \Omega\\ u=0, &\partial\Omega\end{array}\right.$$
Let $\{\lambda_n,\phi_n\}$ to be eigenvalues and eigenfunctions of Dirichlet Laplacian. Let $\mu_n=\frac{1}{\lambda_n}$. Assume $f\in C^\infty(\Omega)$ and $f=\sum_n^\infty a_n\phi_n$, the decay of $a_n$ is sufficiently fast. The solution must be $u=\sum_n^\infty \mu_n a_n \phi_n$.
My question is: can we obtain the $L^\infty$ estimate: $\|u\|_\infty\leq C\|f\|_\infty$ , only using the above eigenfunction expansion?( If we can prove the uniform bound for sufficient smooth function, then by density argument the estimate would hold for all $f\in C(\Omega)$.
At first I tried the case when $\Omega=(0,1)^2$, since the eigenfunction expansion is just Fourier expansion. But it's still hard to do the estimate. I can show that $a_n$ is bounded by $\|f\|_\infty$, but $\mu_n=\frac{1}{4\pi^2|\vec{n}|^2}$, and $\sum_n \mu_n$ diverges, thus we cannot simply estimate $a_n$ as $\|f\|_\infty$ in the summation. Can anyone help me with this problem, or simply the case of $\Omega=(0,1)^2$? Thanks!