Following Giovanni Leoni's excellent book (or the Wikipedia article) one possible way to define the Besov spaces $B^{s,p,\theta}(\mathbb R ^d)$, with $s\in(0,1)$ the fractional "order of derivative" and $\theta,p\in[1,+\infty]$ two exponents, is to require that a function $u\in B^{s,p,\theta}(\mathbb R ^d)$ iff

  1. $u\in L^p(\mathbb R^d)$, and
  2. the Besov semi norm is finite $$ |u|^{B^{s,p,\theta}}:=\sum\limits_{i=1}^d\left(\int_0^\infty\|\Delta_i^h u\| ^\theta_{L^p(\mathbb R^d)}\frac{\mathrm d h}{h^{1+\theta s}}\right)^{\frac{1}{\theta}}<+\infty, $$ where $\Delta^h_i u(x):=u(x+he_i)-u(x)$ is the h-difference quotient in the i-th direction. (if $\theta=+\infty$ just replace the $L^\theta$ norm by the $L^\infty$ one)

Actually, I find it more natural to include the $h^{\theta s}$ factor appearing in the denominator inside the difference quotient, i-e write (completely equivalently) $$ |u|^{B^{s,p,\theta}}=\sum\limits_{i=1}^d\left(\int_0^\infty\|\tilde\Delta_i^{s,h} u\| ^\theta_{L^p(\mathbb R^d)}\frac{\mathrm d h}{h}\right)^{\frac{1}{\theta}}<+\infty, $$ with the more transparent "$\mathcal C^{0,s}$ Hölder"-like difference quotient $$ \tilde\Delta_i^{s,h} u:=\frac{u(x+he_i)-u(x)}{h^s} $$ somehow measuring $s$ derivatives of $u$. Anyway, my question is:

Why is it natural to choose the weight $\frac{\mathrm d h}{h}$ inside the $L^\theta$ norm with respect to $h$ ?

Of course one should use a measure that "charges" $h=0$ (here $\frac{1}{h}$ is not integrable in any neighbourhood of zero) so that the $h$-difference quotient is forced to be small for small $h$, but why $\frac{1}{h}$ specifically? I am surely missing the point here, but I can't see any natural reason for this choice except perhaps for $\frac{\mathrm d h}{h}=\mathrm d (\log h)$ (we want a logarithmic scale? why so?) As a side-question, I suspect that the nonintegrability at infinity does not really play a role, or presumably a very minor minor one compared to the behaviour for small $h$. Is this so? For example, would a truncation $\int_0^H(\dots)\frac{\mathrm d h}{h}$ give an equivalent behaviour/regularity?


This post on Terry Tao's blog indeed mentions at the very end that, in many applications, the number of scales is logarithmic. This suggests that the $\frac{\mathrm d h}{h}$ is here to account for this logarithmic mixture. But the question remains: Is this natural for some reason that escapes me, or is it just because we (I?) didn't find yet any interesting applications where a non-logarithmic weight would be useful or needed?


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