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I've been reading about Besov spaces (my reference thus far has been "Mathematical foundations of infinite-dimensional statistical models" (Nickl & Gine), and I've been struggling a bit with the interpretation of the parameters given when describing a Besov space. I normally see the spaces written as $B_{pq}^s$. I understand that the $s$ represents something akin to Holder continuity / level of differentiability, but getting a concrete hold on what each of $p,q,s$ ($q$ in particular) has been something of a tricky task.

In particular, I'm looking for a description of what each of $p,q,s$ tells us about the space in question. I can look up inclusions/equivalences to e.g. Holder/Sobolev spaces on my own. I'm interested in the slightly more qualitative side of matters.

Edit: Thanks to Ian's helpful comment, I feel relatively at peace with my understanding of $s$ and $p$ - right now, my focus is on getting a qualitative understanding of how $q$ affects the type of functions lying in a given Besov space. I current have it in my head as some control over the tail decay of the wavelet coefficients, but this is still quite unsatisfying; it doesn't tell me as much about the function as I'd like.

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    $\begingroup$ Well the first thing to understand is that $B^s_{p,q}$ embeds into $W^{\lfloor s \rfloor,p}$. The remainder of the definition says that a certain function involving $s-\lfloor s \rfloor$ and $p$ is in $L^q$, which provides a further refinement of the precise regularity of $f$. $\endgroup$
    – Ian
    Mar 7 '17 at 14:04
  • $\begingroup$ Hi @Ian - thanks for this. This second ("remainder of...") statement is what I'm most interested in. Which definition do you have in mind when interpreting this? $\endgroup$
    – πr8
    Mar 7 '17 at 15:06
  • $\begingroup$ Just the one from Wikipedia (I'm no expert in this corner of analysis...) $\endgroup$
    – Ian
    Mar 7 '17 at 15:22
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    $\begingroup$ I see - I'll think on this for a bit. I'm open to the idea that $q$ might not readily admit a simpler interpretation, but am still hopeful that there's something out there. Thank you in any case. $\endgroup$
    – πr8
    Mar 7 '17 at 15:29
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The way I understand the Besov spaces is by looking at the embedding $$ B^{s+\varepsilon}_{p,q} ⊆ B^s_{p,1} ⊆ B^s_{p,q_0} ⊆ B^s_{p,q_1} \subseteq B^s_{p,\infty} \subseteq B^{s-\varepsilon}_{p,q} $$ if $q_0 ≤ q_1$. So the third parameter is an additional refinement of the regularity scale (but a very subtle "log correction"). Then, you have the links with the Sobolev spaces $$ \begin{align*} H^s &= B^s_{2,2} \\ C^s &= B^s_{\infty,\infty} &(s\notin\mathbb N) \\ B^0_{p,2} &\subseteq L^p ⊆ B^0_{p,p} & (p\geq 2) \end{align*} $$ And with $\mathcal M$ the spaces of radon measures, $$ L^1 \subseteq \mathcal M ⊆ B^0_{1,\infty} $$

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