In [Iwaniec's paper](https://link.springer.com/content/pdf/10.1007/978-1-4612-0605-7_12.pdf) presenting the Gehring Lemma, the embedding used is $W^{1,p}\hookrightarrow L^2$ with $p=\frac{2d}{d+2}$.



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> **Question.** What about dimension 2: can we actually go down to $p=1$?
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Gehring Lemma in dimension 2
The embedding is just the standard Sobolev embedding theorem. And yes, $p = 1$ is ok.


However, I am guessing you feel confused because many presentations of the embedding theorem use the Hardy-Littlewood-Sobolev lemma in the course of the proof, and the argument therefore fails for $p = 1$. Instead, you should follow the version of proof given by Nirenberg, where $p = 1$ is the simplest and first case to be proved.


The Nirenberg paper is:


* Louis Nirenberg, On elliptic partial differential equations, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze Ser. 3, 13 (1959), no. 2, 115–162 (en). MR 109940


