We present here a new proof of the theorem of Birman and Solomyak on the metric entropy of the unit ball of a Besov space $B_{\pi ,q}^s$ on a regular domain of ${ℝ}^d.$ The result is: if s - d(1/π - 1/p)+> 0, then the Kolmogorov metric entropy satisfies H(ε) ~ ε-d/s. This proof takes advantage of the representation of such spaces on wavelet type bases and extends the result to more general spaces. The lower bound is a consequence of very simple probabilistic exponential inequalities. To prove the upper bound,...

We present here a new proof of the theorem of Birman and Solomyak on the metric entropy of the unit ball of a Besov space $B_{\pi ,q}^s$ on a regular domain of ${ℝ}^d.$ The result is: if $s-d{(1/\pi -1/p)}_{+}$$\>0,$ then the Kolmogorov metric entropy satisfies $H\left(ϵ\right)\sim {ϵ}^{-d/s}$. This...