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Gehring Lemma in dimension 2 retort

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Gehring Lemma in dimension 2

In Iwaniec’s paper presenting the Gehring Lemma, the embedding used is $W^{1,p}hookrightarrow L^2$ with $p=frac{second}{d+2}$. What about dimension 2 : can we really ebb all the way down to $p=1$?

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