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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}(ℝ^{n}) embeds into C^{j}_{B}(ℝ^{n}). (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}(ℝ^{n}) 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?

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# Question regarding Sobolev embeddings

Can you offer guidance or do you also need help?

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