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I have a very basic question about how to define the convolution product of the fundamental solution $\Phi$ of the Laplace equation (so that $\Delta \Phi = \delta_0$) with a function with compact support $f\in C^2_c(\mathbb{R}^n)$.

In the book I'm studying (Evans, PDEs), $\Phi$ is defined as a function on $\mathbb{R}^n\backslash \{0\}$. I suppose that to make sense of $\Phi * f$, one has to see $\Phi$ as a distribution on $\mathbb{R}^n$ (not $\mathbb{R}^n\backslash \{0\}$) ? The problem that I see, is that it seems that $\Phi$ is not $L^1_{loc}(\mathbb{R}^n)$, so how can it be a distribution on $\mathbb{R}^n$ ? Or am I missing something ?

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Why do you think, $\Phi$ is not locally integrable? Check the growth for $x\to 0$. – Vobo Sep 28 '12 at 15:10

The fundamental solution of the Laplace equation is locally integrable and therefore defines a distribution. The same holds for its partial derivatives of first order (but not of the second order).

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