# Choosing the weight in a particular definition of Besov spaces

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?

## Edit:

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?