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Let $\Omega$ be a bounded smooth domain in $\mathbb R^n$. The Laplacian $\Delta$ acts on functions on $\Omega$. From elliptic regularity (I haven't worked out all the details), we have that

$$ \Delta : C^{2,\alpha} (\Omega) \to C^{0, \alpha}(\Omega),\ \ \alpha >0$$

and

$$\Delta : W^{k,p}(\Omega) \to W^{k-2, p} (\Omega),\ \ k\ge 2, p \in (1,\infty)$$

are both Fredholm operators of index zero. The Schauder ($L^p$) estimates does not hold for $\alpha = 0$ (or $p=1$). So my question is: Are

$$ \Delta : C^2(\Omega) \to C^0(\Omega), \ \ \ W^{2,1} (\Omega) \to L^1(\Omega)$$

Fredholm operators?

Remark: to be precise, the operator

$$ \Delta : \{ u\in C^{2,\alpha} (\Omega)\cap C^0(\overline\Omega)\ |\ u|_{\partial \Omega} = 0\}\to C^{0,\alpha}(\Omega)$$

is not only Fredholm of index zero, but also bijective. This is found e.g. in Theorem 6.13 in Gilbarg-Trudinger. Assuming better regularity on the boundary, one also has $u\in C^{2,\alpha}(\overline \Omega)$ (Theorem 6.14).

The more relevant case is probably Theorem 6.15, where they show a Fredholm alternatives for the operator $Lu = \Delta u - \lambda u$.

The higher regularity in section 6.4 implies that

$$ \Delta : \{ u\in C^{k+2,\alpha} (\Omega)\cap C^0(\overline\Omega)\ |\ u|_{\partial \Omega} = 0\}\to C^{k,\alpha}(\Omega)$$

is also bijective for all $k\geq 0$.

The relevant $L^p$-theory can be found in Chapter 9 of the same book.

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No, the cokernels are infinite dimensional for $C^2\to C^0$ and $W^{2,1}\to L^1$. It suffices to take one function in $C^0$ or $L^1$ for which the Poisson equation fails to have a solution in $C^2$ or $W^{2,1}$; then translate it around, spanning an infinite-dimensional subspace.

Here is an explicit example for the continuous case. For $L^1$, an example is a function like this, $$\frac{x}{|x|^{n+1}(-\log |x|)^{1+\epsilon}}$$

The underlying reason for the failure of solvability of Poisson's equation in these cases is that Riesz transforms are not bounded on $C^0$ and $L^1$. Riesz transforms are singular integral operators that recover individual second-order partials from the Laplacian.

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