Geodesic
How to recognize constant functions. Connections with Sobolev spaces
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 57 (2002) no. 4, pp. 693-708

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A criterion for a function $f\in L^p$ to belong to $W^{1,p}$ $(p>1)$ or to $BV$ $(p=1)$ is given. Various integral conditions under which a measurable function is constant are discussed. 
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     author = {H. Brezis},
     title = {How to recognize constant functions. {Connections} with {Sobolev} spaces},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
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     publisher = {mathdoc},
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     year = {2002},
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     url = {http://geodesic.mathdoc.fr/item/RM_2002_57_4_a1/}
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H. Brezis. How to recognize constant functions. Connections with Sobolev spaces. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 57 (2002) no. 4, pp. 693-708. http://geodesic.mathdoc.fr/item/RM_2002_57_4_a1/