Geodesic
Sobolev-type integral representations for functions defined on Carnot groups
Matematičeskie trudy, Tome 11 (2008) no. 1, pp. 113-131

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We obtain some integral representations of the form $f(x)=P(f)+K(\nabla f)$ on the Carnot groups, where $P(f)$ is a polynomial and $K$ is an integral operator with a specific singularity. These representations are employed to prove the weak Poincaré inequality. 
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     author = {E. A. Plotnikova},
     title = {Sobolev-type integral representations for functions defined on {Carnot} groups},
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     url = {http://geodesic.mathdoc.fr/item/MT_2008_11_1_a5/}
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E. A. Plotnikova. Sobolev-type integral representations for functions defined on Carnot groups. Matematičeskie trudy, Tome 11 (2008) no. 1, pp. 113-131. http://geodesic.mathdoc.fr/item/MT_2008_11_1_a5/