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My question concerts the following equality in the photo below in the proof of the Gagliardo-Nirenberg-Sobolev inequality. Take $u$ to be compactly supported and in $C^1_c(\mathbb{R}^n)$, then

enter image description here

It looks like Fubini's is being used, but I'm having trouble seeing this explicitly. For simplicity, let's take $n=2$ and simplify notation by setting $|Du|=f$ to get

$$\int_{-\infty}^\infty\left(\int_{-\infty}^\infty f(y_1,x_2)dy_1 \cdot \int_{-\infty}^\infty f(x_1,y_2)dy_2\right)dx_1=\int_{-\infty}^\infty f(y_1,x_2)dy_1\cdot \int_{-\infty}^\infty\int_{-\infty}^\infty f(x_1,y_2) dy_2dx_1.$$

I would appreciate if someone could hold my hand through the $n=2$ case.

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1 Answer 1

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Not Fubini's. A friend pointed out this is just pulling out the $\int f(y_1,x_2)dy_1$ since this term is completely independent of $x_1$.

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