Geodesic
The composition operator in Sobolev Morrey spaces
Eurasian mathematical journal, Tome 7 (2016) no. 2, pp. 50-67

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In this paper we prove sufficent conditions on a map $f$ from the real line to itself in order that the composite map $f \circ g$ belongs to a Sobolev Morrey space of real valued functions on a domain of the $n$-dimensional space for all functions $g$ in such a space. Then we prove sufficient conditions on f in order that the composition operator $T_f$ defined by $T_f [g] \equiv f\circ g$ for all functions $g$ in the Sobolev Morrey space is continuous, Lipschitz continuous and differentiable in the Sobolev Morrey space. We confine the attention to Sobolev Morrey spaces of order up to one. 
@article{EMJ_2016_7_2_a3,
     author = {N. Kydyrmina and M. Lanza de Cristoforis},
     title = {The composition operator in {Sobolev} {Morrey} spaces},
     journal = {Eurasian mathematical journal},
     pages = {50--67},
     publisher = {mathdoc},
     volume = {7},
     number = {2},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EMJ_2016_7_2_a3/}
}
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N. Kydyrmina; M. Lanza de Cristoforis. The composition operator in Sobolev Morrey spaces. Eurasian mathematical journal, Tome 7 (2016) no. 2, pp. 50-67. http://geodesic.mathdoc.fr/item/EMJ_2016_7_2_a3/