# Sharp asymptotic behavior of the metric entropy for the unit ball in Besov space

For $$s>0$$ and $$1 \leq p,q \leq \infty$$ let $$B^s_{p,q}$$ be the Besov space defined on $$[0,1]^d$$, and assume $$s > d( \frac 1 p - \frac 1 2)_+$$, such that $$B^s_{p,q}$$ is compactly embedded in $$L^2([0,1]^d)$$. Let $$N_\varepsilon$$ be the minimal covering number of the unit ball in $$B^s_{p,q}$$ with respect to $$\varepsilon$$-Balls in $$L^2([0,1]^d)$$, and $$H_\varepsilon=\log_2 N_\varepsilon$$ its metric entropy. A special instance of the Birman-Solomyak Theorem states that there exist positive constants $$c,C$$ such that

$$c \leq \varepsilon^{d/s} H_\varepsilon \leq C$$ for all $$\varepsilon >0$$.

V. M. Tikhomirov, states, in the Selected Works of A.N. Kolmogorov. Volume III, p. 236, that it is "easy to show" that the limit

$$\lim_{\varepsilon \to 0+} \varepsilon^{d/s} H_\varepsilon$$

exists.

Can an expert show me the "easy" argument (or a link to the relevant literature)? I am not interested in the value of the limit (this seems to be difficult and known only in special cases), but only in an argument in favour of its existence.