Sobolev norm and Beppo-Levi norm

Question

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?

Answer 1

You should maybe clarify your question; what are the general definitions you are trying to compare, and what kind of result are you interested in?

The first quantity in your question, which you call Sobolev, when $1 < p < \infty $, is equivalent to the following: $$ \|f\| _{L^p} ^p + \|\Delta f\| _{L^p}^p. $$

The second quantity in your question, which you call Beppo Levi, is equivalent to $$ \| \Delta f \|_{L^2}^2.$$

So for $p=2$ the first quantity controls the second one (after taking appropriate $p$-th and square roots). For $p\neq2$ the situation is more complicated but in general they are not comparable.

Answer 2

Perhaps there is a simple connection that I have overlooked, but I don't think this was the sort of answer you intended. I'll work in two variables so that the notation is easier to understand. Let the Beppo-Levi semi-norm of order $m$ be

$\parallel f \parallel_{BL_m(\mathbb{R}^2)}^2 \; = \; \int_{\mathbb{R}^2}\sum_{i=0}^m \binom{m}{i}\left( \; \left| \frac{\partial^mf}{\partial x_1^i\partial x_2^{m-i}}\right|^2 \right) \; dx_1 \; dx_2 $

Then

$\parallel f \parallel_{\mathcal{W}^2(\mathbb{R}^2)}^2 \; = \; \parallel f \parallel_{BL_0(\mathbb{R}^2)}^2+\parallel f \parallel_{BL_1(\mathbb{R}^2)}^2+\parallel f \parallel_{BL_2(\mathbb{R}^2)}^2$

For $p\neq 2$ I don't think there is any connection. Is this of any use?

Answer 3

To attempt an answer to your first question:

For $p=2$, the Sobolev space you've defined assigns an infinite norm to the function $f(x_1,x_2)=1$, while the Beppo-Levi SEMI-norm assigns a quantity of zero to this function (as alext87 already noted in his comment). In the way you've defined them, this means that you cannot say that $\|f\|_{W^2} \leq \|f\|_{BL}$. On the other hand, the Sobolev norm contains more terms than the BL semi-norm, and so you find that $\|f\|_{BP} \leq \|f\|_{W^2}$.

If you'd like to show this inequality: $\|f\|_{W^2} \leq \|f\|_{BL}$, then for the above reasons, you will need to make some extra assumptions on $f$. I will now try to explain a sufficient assumption on $f$ which allows an inequality like this to exist.

Let $B=B(0,1)$ be the ball (in $\mathbb{R}^2$) centered at the origin and of radius 1. For convenience, we will assume that $f:B \rightarrow \mathbb{R}$, and that the Sobolev and BL norm/semi-norm you wrote have been generalized for such an $f$ in the following way:

$\|f\|^2_{W^2(B)} = \int_B \sum_{0 \leq i_1,i_2 \leq 2, i_1 + i_2 \leq 2} (|\frac{\partial^{i_1 + i_2}f}{\partial^{i_1}x_1 \partial^{i_2} x_2}|)^2 dx_1 dx_2$

and similarly for $\|f\|_{BL(B)}$ (only integrate over $B$).

Then, the Poincare inequality (see http://en.wikipedia.org/wiki/Poincare_inequality), states that as long as $f$, $\partial_{x_1}f$, and $\partial_{x_2}f$ have mean zero, i.e., $\int_B f(x_1, x_2) dx_1 dx_2 = \int_B \partial_{x_1}f(x_1, x_2) dx_1 dx_2 = \int_B \partial_{x_2}f(x_1 x_2) dx_1 dx_2 = 0$, then we have

$\|f\|_{W^2(B)} \leq C \|f\|_{BL(B)}$

For some constant $C$, and for all functions $f$. Here, $C$ is independent of $f$. To summarize, there exists a constant $C$, so that for all $f$ with $f$, $\partial_{x_1}f$, and $\partial_{x_2}f$ of mean zero, we have $\|f\|_{BL(B)} \leq \|f\|_{W^2(B)} \leq C\|f\|_{BL(B)}$

For unbounded domains, as in how your above question was asked for $\mathbb{R}^2$, the above answer would need to be modified. Generally, to understand these things, it helps to look at the case where the domain is the unit ball, since there the relationships are clearer.

For the second question:

I think your definition for the Sobolev space in two dimensions in part 1 is the standard one, except for exact placement of the constants (for instance you removed the 2, which was your choice, no-one told you to put the 2 there!). In many instances, no-one cares what constants you put where, as long as you have all the derivatives present with a positive constant for each one. If you want to write out the Sobolev norm for 3 derivatives, just make sure to also include all of the partial derivatives of order 3 in your sum above.

Regarding your Beppo-Levi semi-norm, I've never heard it called by this name, but in some circles it seems to be. I've seen it referred to as the homogeneous Sobolev semi-norm in my own limited circle, but I'm certain this isn't universal (a quick search on google shows both terms being used). The homogeneous Sobolev semi-norm would (to me) usually be denoted by $\|f\|_{\dot{W}^2}$, with the dot standing for homogeneous.