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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 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 provide a new universal...

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 then the Kolmogorov metric entropy satisfies . 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 provide...