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Let $\mathbb{T}^d \sim \mathbb{R}^d/\mathbb{Z}^d$. I know that $$W^{k,2}(\mathbb{T})\equiv\{f\in W^{k,2}([0,1]); f^{(i)}(0) = f^{(i)}(1), i = 0, \ldots, k-1\}.$$ Are there similar characterizations for $W^{k,p}(\mathbb{T}^d)$? Or some reference for this?

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  • $\begingroup$ Depends on how similar you want. When $d > 1$ generally the trace to the boundary of $[0,1]^d$ of the $k-1$-th order derivatives is not defined in $L^\infty$. So a formally similar statement will not even make sense. // Can you explain what you are looking for or what you want to use it for? $\endgroup$ Nov 4, 2015 at 14:43
  • $\begingroup$ I just want to know what further conditions are needed (e.g. in terms of traces) to ensure that a function in $W^{k,p}([0,1]^d)$ also belongs to $W^{k,p}(\mathbb{T}^d)$. $\endgroup$
    – Housen
    Nov 4, 2015 at 20:23

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