Monotonicity in Banach function spaces. (Q2846675)

The paper gives a survey of results achieved by the author in the area of weighted norm inequalities and related topics. First, the author defines the monotone envelopes for different partial orders given on sets of non-negative functions on \(\mathbb{R}\). In such a way he obtains, e.g., the least decreasing majorant \(f^{\downarrow }\) of a given function \(f\), or, by changing the partial order, the level function \(f^{o}\) of \(f\) originally introduced by \textit{I. Halperin} [Canad. J. Math. 5, 273--288 (1953; Zbl 0052.11303)]. The similarity between the monotone envelopes with respect to the different partial orders are demonstrated and applications to weighted norm inequalities for positive operators are given. Second, for a Banach function space \(X\), the author defines the down space \(D(X)\) of \(X\). Relations between monotone envelopes and norms of the down space and its dual are studied. For example, it is proved that if \(X\) is a Banach function space and \(g \in D(X)'\), then \(\| g\|_{D(X)'}=\| g^{\downarrow}\|_{X'}\); another result states that \(\| f\|_{D(X)}\leq \| f^{o}\|_X\) provided that \(f^{o} \in X\) (the last inequality can be replaced by the equality if \(X\) is a~universally rearrangement invariant space with the Fatou property). Moreover, two equivalent norms on the down space are given and results are applied to factorize Hardy's inequalities and to characterize embeddings of certain classes of quasiconcave functions between weighted Lebesgue spaces. Such embeddings are then used to find a~new characterization of the dual of the Lorentz space \(\Gamma ^p(v)\) and to establish conditions which are necessary and sufficient for the boundedness of the Fourier transform as a map between Lorentz spaces. In this connection a new Lorentz-type space \(\Theta^p(u)\) (satisfying \(\Gamma ^p(u) \subset \Theta ^p(u) \subset \Lambda ^p(u)\)) is introduced. In the last section of the paper it is shown that \((D(L^1_{\lambda }(\mathbb{R})), D(L^{\infty }_{\lambda })(\mathbb{R}))\) is a Calderón couple of spaces (here \(\lambda \) stands for a measure on \(\mathbb{R}\) satisfying \(\lambda (-\infty , x]<\infty \) for all \(x\in \mathbb{R}\)) and that this implies that the down spaces of universally rearrangement invariant spaces are precisely the exact interpolation spaces between \(D(L^1_{\lambda }(\mathbb {R}))\) and \(D(L^{\infty }_{\lambda })(\mathbb{R})\).NEWLINENEWLINEFor the entire collection see [Zbl 1180.46003].