Geodesic
On substitutions with a weight in the space of operator Lipschitz functions
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 50, Tome 512 (2022), pp. 5-14

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Operators of the form $f\mapsto x^\beta f(x^\alpha)$ are treated. Among other things, it is proved that such an operator acts on the class of operator Lipschitz functions on $(0,+\infty)$ if and only if $\alpha+\beta=1$. 
@article{ZNSL_2022_512_a0,
     author = {A. B. Aleksandrov},
     title = {On substitutions with a weight in the space of operator {Lipschitz} functions},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {5--14},
     publisher = {mathdoc},
     volume = {512},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2022_512_a0/}
}
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A. B. Aleksandrov. On substitutions with a weight in the space of operator Lipschitz functions. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 50, Tome 512 (2022), pp. 5-14. http://geodesic.mathdoc.fr/item/ZNSL_2022_512_a0/