Suppose $f(x)$ is a continuous compactly supported function in $ R^N$ where $N \ge 3$.

Consider the Newtonian potential of $f$ (at least I think this is what it is called)

$$ v(x)=\int_{R^N} \frac{C_N}{|x-y|^{N-2}} f(y) dy$$ where $C_N$ is appropriate positive constant.

My question is that (at least formally) we have $ -\Delta v(x) = f(x)$ in $ R^N$.

My question is there a way to see the equation is satisfied for all $x$.
I know if $ f$ is slightly better than continuous then I think one can do this.

I realize that we can't expect $ v \in C^2$.
thanks Craig

  • 2
    $\begingroup$ It is always true in the sense of generalized functions but then it is trivial because you can pass all the convolutions and differentiations on the $C_0^\infty$ mollifier. If you want it pointwise, you have to tell in exactly what sense you want to understand the Laplacian in this equation. $\endgroup$
    – fedja
    Aug 9, 2014 at 1:02
  • $\begingroup$ Convolving with this kernel is the same as multiplying the Fourier transform by $|k|^{-2}$, which gives a pretty good general idea of what smoothness to expect for what $f$'s (here $\widehat{f}\in L^2$ is what you know about the decay of $\widehat{f}$). $\endgroup$ Aug 9, 2014 at 2:05
  • $\begingroup$ thanks for the comments. @fedja. I am not sure in exactly what sense I want to understand it. But in whatever way we make sense of it I want it to hold pointwise (not sure if I am even asking a well defined question). $\endgroup$
    – Craig
    Aug 9, 2014 at 6:43


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