Geodesic
Decompositions of the Sobolev Scale and Gradient--Divergence Scale into the Sum of Solenoidal and Potential Subspaces
Informatics and Automation, Function spaces, approximation theory, and nonlinear analysis, Tome 255 (2006), pp. 136-145

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For the complete Sobolev scale and the gradient–divergence scale, decompositions into direct sums of solenoidal and potential subspaces are found. A smoothing property of solenoidal factorization is proved. Projectors onto the subspaces of solenoidal and potential functions are described. 
@article{TRSPY_2006_255_a9,
     author = {Yu. A. Dubinskii},
     title = {Decompositions of the {Sobolev} {Scale} and {Gradient--Divergence} {Scale} into the {Sum} of {Solenoidal} and {Potential} {Subspaces}},
     journal = {Informatics and Automation},
     pages = {136--145},
     publisher = {mathdoc},
     volume = {255},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2006_255_a9/}
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Yu. A. Dubinskii. Decompositions of the Sobolev Scale and Gradient--Divergence Scale into the Sum of Solenoidal and Potential Subspaces. Informatics and Automation, Function spaces, approximation theory, and nonlinear analysis, Tome 255 (2006), pp. 136-145. http://geodesic.mathdoc.fr/item/TRSPY_2006_255_a9/