Optimal inequalities on quasinormed function spaces (Q2712747)

For a domain \(\Omega\subset\mathbb{R}^n\) let \(M(\Omega)\) denote the real-valued measurable functions on \(\Omega\) and \(M_+(\Omega)\) the nonnegative ones. Let \(T\) be a linear operator and \(\rho_R\), \(\rho_D\) quasinorms on \(M_+(\Omega)\) satisfying \(\rho_R(|Tf|)\leq C\rho_D(|Tf|)\) for \(f\in M(\Omega)\). The author poses the problem of determining whether these quasinorms are optimal in an appropriate class. Two cases are considered. The first involves weighted versions of the classical Hardy operator \((Tf)(x)= \int^x_0 f(y) dy\) or its dual and the quasinorms are rearrangement invariant. The second involves more general Hardy type operators on weighted Orlicz and Orlicz-Lorentz spaces, the quasinorms being determined by weights. Proofs are not provided.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00028].