Estimates for Restrictions of Monotone Operators on the Cone of Decreasing Functions in Orlicz Space

The restriction of a monotone operator $P$ to the cone $\Omega$ of nonnegative decreasing functions from a weighted Orlicz space $L_{\varphi,v}$ without additional a priori assumptions on the properties of the Orlicz function $\varphi$ and the weight function $v$ is considered. An order-sharp two-sided estimate of the norm of this restriction is established by using a specially constructed discretization procedure. Similar estimates are also obtained for monotone operators over the corresponding Orlicz–Lorentz spaces $\Lambda_{\varphi,v}$. As applications, descriptions of associated spaces for the cone $\Omega$ and the Orlicz–Lorentz space are obtained. These new results are of current interest in the theory of such spaces.