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I've currently started to study about Sobolev inequalities and here is Gagliardo-Nirenberg-Sobolev inequality from Evans' book:

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I understand the whole proof except for some details I'm about to ask:

  1. In $\;(12)\;$ the last inequality results from general Holder inequality for $\;(\int_{\mathbb R} \vert Du(x) \vert\;dy_i)^{\frac{1}{n-1}}\;$. However this inequality states that in generall $\;\int_{\mathbb R} \vert u_{x_1}\dots u_{x_n}\vert \;dx \le (\int_{\mathbb R} {\vert u_{x_1} \vert}^{p_1}\;dx)^{1/{p_1}} \dots (\int_{\mathbb R} {\vert u_{x_n} \vert}^{p_n}\;dx)^{1/{p_n}}\;$. Which are the appropriate $\;p_i\;$ here and why I can't see them?
  2. In the first two lines, I don't fully understand why $\;\vert u(x) \vert \le\int_{\mathbb R} \vert Du(x_1,\dots,y_1,\dots,x_n)\;dy_i\;$

Any help would be valuable. Thanks in advance!

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  1. There are $n-1$ terms in the application of generalized Holder, $p_1,\dots,p_{n-1}$ and each $p_i=n-1$ for all $i$.

  2. The first line is the fundamental theorem of calculus. The second line follows from $|u_{x_i}|\leq |Du|$, and increasing the domain of the integral to the whole real line.

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