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I have a question about the usual definition of a Sobolev spaces.

Especially, let $f \in W^{1,2} (\Bbb R^n)$. I have thought that this notation means that $ \int_{\Bbb R^n} |f|^2 + | \partial_1 f|^2 + \cdots + | \partial_n f|^2 < \infty $ where $\partial_i$ means that $i$th weak partial derivative. But I am wondering that this notation may include the product of derivatives, that is, $ \int_{\Bbb R^n} |f|^2 + | \partial_1 f|^2 + \cdots + |\partial_n f |^2 + |\partial_1 f | | \partial_2 f | + \cdots + | \partial_{n-1} f | | \partial_n f| < \infty.$

Which one is true in general?

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    @DavideGiraudo Really thank you! you are right, they are not mixed derivatives. :)2012-10-08

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We have $\begin{aligned} \int |\partial_1f|^2 + \cdots + |\partial_n f|^2 & \leq \int |\partial_1 f|^2 + \cdots + |\partial_n f|^2 + |\partial_1 f \partial_2 f| + \cdots + |\partial_{n-1} f \partial_n f| \\ &\leq \frac{n+1}{2} \int |\partial_1 f|^2 + \cdots + \partial_n f|^2 \end{aligned} $

By the arithmetic-mean-geometric-mean inequality $ |ab| \leq \frac{|a| + |b|}{2} $ applied to the ${n \choose 2}$ "cross terms". So the two definitions are equivalent.