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I've got two questions:

(1). I would like to know if by definition the Sobolev space norm is defined only by differentials with respect to one variable i.e.

$\parallel f \parallel \; = \; \left[ \int_{\mathbb{R}} \left( \left| f \right|^p \; + \; \left| \frac{\partial f}{\partial x} \right|^p \; + \ldots + \; \left| \frac{\partial^m f}{\partial x^m} \right|^p \right) dx \right]^{\frac{1}{p}} $

What would the corresponding Sobolev space norm be if $f$ were a function of two variables e.g. $f(x,y)$

(2). If the Sobolev space norm exists for multivariate functions, is it correct to say that the Beppo-Levi space norm a special case of the Sobolev space norm? From the little I've read, it appears that the Beppo-Levi space norm is given by

$\parallel f \parallel \; = \; \int_{\mathbb{R}^2} \sum_{i = 0}^m {m \choose i} \left( \frac{\partial^m f}{\partial x^i \partial y^{m - i}} \right)^2 dx dy$

What if the partial derivative is raised to the power of 3. Would this still be a Beppo-Levi space norm?

Edit: based on Robin's answer and the paper "Spline functions and stochastic filtering" by Christine Thomas-Agnan, here's my understanding of the

Sobolev norm:

$\parallel f \parallel^p \; = \; \sum_{ 0 \: \le \: ( \: \mid \alpha \mid \: = \: \alpha_1 \: + \: \cdots \: + \: \alpha_n \: ) \: \le \: p}\int_{\mathbb{R}^n} \; \left| \frac{\partial^{\mid \alpha \mid} f}{\partial x_1^{\alpha_1} \: \cdots \: \partial x_n^{\alpha_n}} \right|^p \: dx$

Beppo-Levi norm:

$\parallel f \parallel^m \; = \; \int_{\mathbb{R}^n} \; \sum_{\alpha_i \: + \: \cdots \: + \: \alpha_n \: = \: m } {m \choose {\alpha_i! \; \cdots \; \alpha_n!}} \left| \frac{\partial^m f(x)}{\partial x_1^{\alpha_1} \cdots \partial x_n^{\alpha_n} } \right|^2 dx$

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The Beppo-Levi norm raises the partial derivative to the power 2 only. – alext87 Sep 13 '10 at 15:25
Got that. Thanks. – Olumide Sep 13 '10 at 15:35
up vote 2 down vote accepted

The Sobolev norm is also defined for regions in $\mathbb{R}^n$. You would typically have $$\|f\|^p=\sum_{(i_1,\ldots,i_n)}\int\left| \frac{\partial^I f}{\partial x_1^{i_1}\cdots\partial x_n^{i_n}} \right|^pd\mathbf{x}$$ where $I=i_1+\cdots+i_n$ and the sum is over all tuples with $0\le I\le m$.

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Thanks. Just to be sure, the Sobolev norm is summed over all $I$th differentials and in the range $0 \le I \le m$, in which case the Beppo-Levi space norm is the Sobolev norm where $I = 2$ – Olumide Sep 13 '10 at 14:44
In which case, for the sake of clarity the Sobolev space can be given as $\|f\|_{\mathcal{H}^m}^p= \sum_{I = 0}^m \sum_{(I = i_1 + \ldots + i_n)}\int\left|\frac{\partial^I f}{\partial x_1^{i_1}\cdots\partial x_n^{i_n}}\right|^pd\mathbf{x}$ ? – Olumide Sep 13 '10 at 14:52
If you like. I'm not familiar with the B-L norm, but the formula you gave only involves $m$-th derivatives, and so isn't equivalent to a Sobolev norm. – Robin Chapman Sep 13 '10 at 15:03
I think it involves $I$th derivatives to the power of $p$, where $0 \le I \le m$. This appears to be consistent with the Sobolev norm in my question. This norm was taken from (page 10 of) the paper "On Different Facets of Regularization Theory" by Zhe Chen and Simon Haykin, and is taken from a classic text "Sobolev Spaces" by Adams R.A (1975) – Olumide Sep 13 '10 at 15:17
My intention is to capture norms like, if $m = 2$, $\parallel f \parallel_{\mathcal{H}^2}^p \; = \; \mid f \mid^p \; + \; \left| \frac{\partial f}{\partial x_1}\right|^p \; + \left| \frac{\partial f}{\partial x_2}\right|^p \; + \; \left| \frac{\partial^2 f}{\partial x_1^2}\right|^p \; + \; 2\, \left| \frac{\partial^2 f}{\partial x_1 \partial x_2}\right|^p \; + \; \left| \frac{\partial f}{\partial x_2}\right|^p$ – Olumide Sep 13 '10 at 15:23

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