Geodesic
Singular symmetric functionals
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 30, Tome 290 (2002), pp. 42-71

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This is a continuation of the study started in [3]. A linear functional $f$ on a rearrangement invariant space $E$ on $(0, \infty)$ is said to be symmetric if for $x, y\in E$ the condition $$ \int\limits^t_0x^*(s)sd\le\int\limits^t_0y^*(s)ds,\quad t>0, $$ implies that $f(x)\le f(y)$. A new construction of singular symmetric functionals on the Marcinkiewicz space $M(\psi)$ is presented and studied in detail. A necessary and sufficient condition in terms of $\psi$ is obtained for the seminorms equal to distance to $M(\psi)\cap L_1$ and $M(\psi)\cap L_{\infty}$ to be recoverable in terms of the symmetric singular functionals on $M(\psi)$. 
@article{ZNSL_2002_290_a3,
     author = {P. G. Dodds and B. De Pagter and A. A. Sedaev and E. M. Semenov and F. A. Sukochev},
     title = {Singular symmetric functionals},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {42--71},
     publisher = {mathdoc},
     volume = {290},
     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2002_290_a3/}
}
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P. G. Dodds; B. De Pagter; A. A. Sedaev; E. M. Semenov; F. A. Sukochev. Singular symmetric functionals. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 30, Tome 290 (2002), pp. 42-71. http://geodesic.mathdoc.fr/item/ZNSL_2002_290_a3/