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Consider $\mathbb{R}^d$ and $D_i = \partial /\partial x_i$, in many cases

$$D_i(f*g) = D_if*g,$$ given one of $f,g$ is smooth and the other is $L^p$ integrable.

I am wondering if there is a general approach for proofs?

The major step would be how to interchange $D_i$ with integral of the convoloution

$$D_i (f*g)(x) = \lim_{x_n\to x} \int_{\mathbb{R}^d} \frac{f(x_n-y)-f(x-y)}{x_n-x} g(y) \, \mathrm{d}y \\ = \int_{\mathbb{R}^d} \lim_{x_n\to x}\left[\frac{f(x_n-y)-f(x-y)}{x_n-x}\right] g(y) \, \mathrm{d}y, $$ the second equality requires a certain convergence theorem, but I usually cannot find a good function to should the $L^1$ boundedness.

For example, by taking $f$ is in $C^{\infty}_0$ or $\mathcal{S}$(Schwartz function), when $g\in L^p, p=1\text{ or }\infty$, we can use the method post here.

And I close with two questions:

  1. Is it can be extended to $C^\infty$?
  2. How to modify for $p\in(1,\infty)$. Should I use Hoelder or Young's inequality?
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  • If $f\in C^\infty$ and $g\in L^p$ (any $p$), the integral in the definition of $f*g$ may fail to converge everywhere. Consider $f(x)=\exp(x^2)$ and $g(x)=1/(x^2+1)$ on $\mathbb R$. – ˈjuː.zɚ79365 Jul 01 '13 at 14:46

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