In Iwaniec's paper presenting the Gehring Lemma, the embedding used is $W^{1,p}\hookrightarrow L^2$ with $p=\frac{2d}{d+2}$.
Question. What about dimension 2: can we actually go down to $p=1$?
In Iwaniec's paper presenting the Gehring Lemma, the embedding used is $W^{1,p}\hookrightarrow L^2$ with $p=\frac{2d}{d+2}$.
Question. What about dimension 2: can we actually go down to $p=1$?
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: