I've asked this question on math.stackexchange.com but I'm not satisfied by the answers I got, so I've decided to ask here instead. As always I apologize if my notation is not precise enough. I am a computer scientist and not a mathematician.

I would like to understand the difference between a Sobolev norm and a Beppo-Levi norm. Because notation can be an issue, I'll give examples, based on what I've read, of what I suspect either norm to be in hope that someone would be kind enough to confirm or correct my suspicions.

Example of Sobolev norm (involves partial derivatives of all powers up to $m = 2$)

$\parallel f \parallel_{\mathcal{W}^2}^p \; = \; \int_{\mathbb{R}^2} \left( \; \mid f \mid^p \; + \; \left| \frac{\partial f}{\partial x_1}\right|^p \; + \left| \frac{\partial f}{\partial x_2}\right|^p \; + \; \left| \frac{\partial^2 f}{\partial x_1^2}\right|^p \; + \; \left| \frac{\partial^2 f}{\partial x_1 \partial x_2}\right|^p \; + \; \left| \frac{\partial^2 f}{\partial x_2^2}\right|^p \; \right) \; dx_1 \: dx_2$

Edit: I changed $2 \; \left| \frac{\partial^2 f}{\partial x_1 \partial x_2}\right|^p$ to $\left| \frac{\partial^2 f}{\partial x_1 \partial x_2}\right|^p$ above.

Example of Beppo-Levi norm (partial derivatives are always in powers of two)

$\parallel f \parallel_{BL}^2 \; = \; \int_{\mathbb{R}^2} \left( \; \left| \frac{\partial^2f}{\partial x_1^2}\right|^2 + 2 \left|\frac{\partial^2f}{\partial x_1 \partial x_2}\right|^2 + \left|\frac{\partial^2f}{\partial x_2^2}\right|^2 \; \right) \; dx_1 \; dx_2 $

Edit: The Sobolev norm is often written as

$\parallel f \parallel \; = \; \left( \int_{\mathbb{R}} \; ( \; \mid f(x) \mid^p \; + \; \mid D^1 f(x) \mid^p \; + \; \ldots \; + \; \mid D^\alpha f(x) \mid^p \; ) \; dt \right)^\frac{1}{p}$

or

$\parallel f \parallel \; = \; \left( \int_{\mathbb{R}} \sum_{\alpha = 0 }^d \mid D^\alpha f(x) \mid^p dt \right)^\frac{1}{p}$

But what if $f$ is a multivariate function? Surely, I'll need to write more differential terms as I did in my original/first definition of the Sobolev norm. Is this the case? And if so, what is the standard notation?