Estimates for entropy numbers of sets of smooth functions on complex spheres

Abstract

In this paper we investigate the asymptotic behavior of entropy numbers of multiplier operators ${\Lambda }_{\ast }$ and $\Lambda$, defined for functions on the complex sphere ${\Omega }_d$ of ${ℂ}^d$, associated with sequences of multipliers of the type ${\left\{{\lambda }_{m,n}^{\ast }\right\}}_{m,n\in N}$, ${\lambda }_{m,n}^{\ast }=\lambda (m+n)$ and ${\left\{{\lambda }_{m,n}\right\}}_{m,n\in N}$, ${\lambda }_{m,n}=\lambda (max\{m,n\})$, respectively, for a bounded function $\lambda$ defined on $[0,\infty )$. If the operators ${\Lambda }_{\ast }$ and $\Lambda$ are bounded from $L^p\left({\Omega }_d\right)$ into $L^q\left({\Omega }_d\right)$, $1\le p,q\le \infty$, and $U_p$ is the closed unit ball of $L^p\left({\Omega }_d\right)$, we study lower and upper estimates for the entropy numbers of the sets ${\Lambda }_{\ast }{U}_p$ and $\Lambda U_p$ in $L^q\left({\Omega }_d\right)$. As application we obtain, in particular, estimates for the entropy numbers of classes of Sobolev, of finitely differentiable, infinitely differentiable and analytic functions on the complex sphere, in $L^q\left({\Omega }_d\right)$, which are order sharp in several important situations.