Entropy of Multiplier Operators on Compact Globally Symmetric Spaces of Rank 1

Estimates of entropy of multiplier operators $\Lambda = \{\lambda_{k}\}_{k=1}^{\infty}$ on compact globally symmetric spaces of rank 1 (i.e. on $S^{d}$, $P^{d}(\RR)$, $P^{d}(\CC)$, $P^{d}(\HH)$, $P^{16}({\rm Cay})$ ) are established. It is shown that these estimates have sharp orders in different important cases. In particular, we give sharp orders of entropy of standard Sobolev's classes $W^{\gamma}_{p}(M^{d})$ in $L_{q}(M^{d})$ for all $1 < p, q < \infty$.