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real-valued function on with compact support. Then for there is a constant depending only on and such that

:

with .The case is due to Sobolev and the case to Gagliardo and Nirenberg independently. The Gagliardo-Nirenberg-Sobolev inequality implies directly the Sobolev embedding

:

The embeddings in other orders on are then obtained by suitable iteration.

Hardy-Littlewood-Sobolev lemma

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