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:
$\parallel f \parallel^m \; = \; \int_{\mathbb{R}^n} \; \sum_{\alpha_i \: + \: \cdots \: + \: \alpha_n \: = \: m } {m \choose {\alpha_i! \; \cdots \; \alpha_n!}} \left| \frac{\partial^m f(x)}{\partial x_1^{\alpha_1} \cdots \partial x_n^{\alpha_n} } \right|^2 dx$
Best Answer
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 $0\le I\le m$.