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Let $\Omega$ be a bounded, non-empty, regular open domain in $\mathbb{R}^d$. Let $1\le p,q\le \infty$ and $s>0$. Let $\mathcal{B}_{p,q}^s(\Omega)$ be the Besov space on $\Omega$ corresponding to these parameters, see eg Triebel's book [1].

Let $PW(\Omega)$ denote the set of functions from $\Omega$ to $\mathbb{R}$, which are the restriction of piecewise linear functions on $\mathbb{R}^d$ (with finitely many pieces).

My question is, what values of $p,q,s$ are such that $PW(\Omega)\subset \mathcal{B}_{p,q}^s(\Omega)$? Clearly, when $p=q=\infty$ and $s=1$ then $\mathcal{B}_{p,q}^s(\Omega)$ is the set of Lipschitz functions on $\Omega$, so $PW(\Omega)\subset \mathcal{B}_{\infty,\infty}^1(\Omega)$; however, what inclusions are known to hold in general?

[1] Triebel, H. (2008). Function spaces and wavelets on domains (No. 7). European Mathematical Society.

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    $\begingroup$ I assume you piece-wise linear functions are at least continuous, sicne you seem to assume they are Lipschitz. Therefore, it should belong to all Besov spaces $\mathrm{B}^{s}_{p,q}(\Omega)$, $p,q\in[1,+\infty]$, $s<1+1/p$. The idea is that piece-wise constant functions, which are derivatives of your considered functions, belong to Besov spaces for $s<1/p$. $\endgroup$
    – ToGle
    Commented Aug 19 at 1:44
  • $\begingroup$ @ToGle Do you have a reference to this? $\endgroup$
    – LittleQuestionBoy
    Commented Aug 19 at 18:37
  • $\begingroup$ This follows from the fundamental multiplier theorem saying that the characteristic function of the half-plane yields a bounded multiplier on Besov spaces ${\mathrm{B}}^s_{p,q}(\mathbb{R}^n)$ provided $p,q\in[1,+\infty]$ and $s\in(-1+\tfrac{1}{p},\tfrac{1}{p})$. $\endgroup$
    – ToGle
    Commented Aug 20 at 3:35
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    $\begingroup$ Can be found for example in the first by book by Triebel Theory of Function spaces $\endgroup$
    – ToGle
    Commented Aug 20 at 3:36
  • $\begingroup$ @ToGle Amazing; thanks a lot :) $\endgroup$
    – LittleQuestionBoy
    Commented Aug 20 at 15:59

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