Entropy numbers of Besov classes of generalized smoothness on the sphere

We investigate the asymptotic behavior of the entropy numbers of Besov classes $BB_{p,\theta }^\Omega \left( {\mathbb{S}^{d - 1} } \right)$ of generalized smoothness on the sphere in $L_q \left( {\mathbb{S}^{d - 1} } \right)$ for 1 ≤ p, q, θ ≤ ∞, and get their asymptotic orders. We also obtain the exact orders of entropy numbers of Sobolev classes $BW_p^r \left( {\mathbb{S}^{d - 1} } \right)$ in $L_q \left( {\mathbb{S}^{d - 1} } \right)$ when p and/or q is equal to 1 or ∞. This provides the last piece as far as exact orders of entropy numbers of $BW_p^r \left( {\mathbb{S}^{d - 1} } \right)$ in $L_q \left( {\mathbb{S}^{d - 1} } \right)$ are concerned.