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 $


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.


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