John-Lowner ellipsoids and entropy of multiplier operators on rank $1$ compact homogeneous manifolds

We present a new method of the evaluation of entropy, which is based on volume estimates for John-Löwner ellipsoids induced by the eigenfunctions of Laplace-Beltrami operator on compact homogeneous manifolds $\mathbb{M}^{d}$ of rank $1$. This approach gives the sharp orders of entropy in the situations where the known methods meet difficulties of fundamental nature. In particular, we calculate the sharp orders of the entropy of the Sobolev classes $W_{p}^{\gamma }(\mathbb{M}^{d})$, $\gamma>0$, in $L_{q}(\mathbb{M}^{d})$, $1 \leq q \leq p \leq \infty$. Bibliography: 35 titles.