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Let $f\in C^\infty(\mathbb R^{d_1}\times\mathbb R^{d_2})\,\cap\, W^{\infty,p}(\mathbb R^{d_1}\times\mathbb R^{d_2})$ for all $1<p<p_0$. Consider the functions $$ F(x_2) \,=\, \int_{\mathbb R^{d_1}} f(x_1,x_2) \,dx_1$$ $$ G(x_2) \,=\, \int_{\mathbb R^{d_1}} \frac{\partial f}{\partial x_2}(x_1,x_2) \,dx_1$$

Are extra conditions really needed in order to have $F(x_2),\,G(x_2)$ well defined and existence of $$\frac{\partial F}{\partial x_2}(x_2)=G(x_2)$$ for all $x_2\in\mathbb R^{d_2}$? Is it simpler if I require only absolute continuity of $F$ and the previous equality to hold only for almost every $x_2$?

tituf
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  • Widely known as the Leibniz rule. Since you need Fubini in the proof, absolute integrability of $f$ is necessary. – vitamin d Jul 23 '21 at 09:51
  • @vitamind is it sufficient? – tituf Jul 23 '21 at 09:55
  • @vitamind moreover isn't it implied by the other hypothesis? – tituf Jul 23 '21 at 10:06
  • I don't really understand your question(s). Do you want to find the most general conditions on $f$ so that it works? – vitamin d Jul 23 '21 at 10:06
  • @vitamind I'd like to find the most general hypothesis on $f$ so that it works. It is unsatisfying to have to look for a uniform (local) control on the derivative in order to apply dominated convergence. I edited – tituf Jul 23 '21 at 10:09
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    Is the derivative in $G$ taken wrt $x_1$ or $x_2$? – vitamin d Jul 23 '21 at 10:25
  • @vitamind $x_2$ thanks, I edit – tituf Jul 23 '21 at 10:27
  • I think you need first one or two conditions on $f$ so that everything exists etc. and then a dominating function, which is not neglible. I imagine somthing like this: For a $g\in\mathcal L^1(\mathbb R^{d_1})$, $\lvert\partial_{x_2} f(x_1,x_2)\rvert\leqslant g(x_1)$. – vitamin d Jul 23 '21 at 10:45
  • @vitamind that is the classical hypothesis, yes, I think one could ask less – tituf Jul 23 '21 at 11:12
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    Right. In this special case it suffices that $\partial_{x_1} f(x_1,\cdot)$ is Riemann integrable over $\mathbb R^{d_1}$. The exchange of the integrals can be justified with Fubini-Tonelli. – vitamin d Jul 23 '21 at 11:37
  • @vitamind this is what I was looking for, would you write an answer? – tituf Jul 23 '21 at 11:55
  • @vitamind with Fubini-Tonelli approach the results holds true only for almost every $x_2$? Or is it possible to conclude for every $x_2$? – tituf Jul 23 '21 at 14:03
  • I found a bunch of related links when I posted this related answer https://math.stackexchange.com/questions/4147220/moving-differential-out-of-integral-of-fourier-transform/4147231#4147231 – Calvin Khor Jul 23 '21 at 14:22

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