Question regarding Sobolev embeddings

[phibonacci]

Hi,

I'm having a bit of trouble understanding certain Sobolev embedding theorems. For instance, if 1 ≤p<∞, and m,j are non-negative integers and mp>n then W^j+m,p(ℝⁿ) embeds into C^j_B(ℝⁿ). (See for instance Adams and Fournier, Sobolev Spaces, 2nd edition, Theorem 4.12 p. 85). But does this result mean that every function in W^j+m,p(ℝⁿ) is continuously differentiable in the classical sense up to order j (with the function and its classical derivatives up to order j being bounded and continuous), or does it simply mean that the function has weak derivatives up to order j, and that the function and these weak derivatives are continuous and bounded?

 

[jvicens]

Hi there,

Thank you for reaching out about this topic. The Sobolev embedding theorems can be quite complex and can often be confusing to understand. Let me try to clarify the result you mentioned.

In general, the Sobolev embedding theorems state that certain Sobolev spaces (such as Wj+m,p(ℝn)) can be continuously embedded into other function spaces (such as CjB(ℝn)). This means that every function in Wj+m,p(ℝn) can be mapped to a function in CjB(ℝn) in a continuous way. However, this does not necessarily mean that the functions in Wj+m,p(ℝn) are continuously differentiable in the classical sense up to order j.

Instead, what the result means is that the functions in Wj+m,p(ℝn) have weak derivatives up to order j that are also continuous and bounded. This is the main difference between classical derivatives and weak derivatives - weak derivatives only need to exist in a certain sense, while classical derivatives must be continuous.

I hope this helps to clarify the result for you. If you have any further questions, please don't hesitate to ask.