# Elliptic regularity in Sobolev spaces of negative order

I am having some trouble with Sobolev spaces of negative order. More precisely I am considering the space $W^{-1,p}(\mathbb{R}^2),$ considered as 'the' dual space of $W^{1,q}(\mathbb{R}^2).$

Question 1: Is there a nice reference for Sobolev spaces of negative order, for $1<p<\infty.$

Question 2: Suppose $f\in W^{-1,p}(\mathbb{R}^2,\mathbb{C})$ is a weak solution to the inhomogeneous Cauchy-Riemann equation, i.e. $\left\langle f,\overline{\partial} g+Sg \right\rangle$ for all smooth and compactly supported $g,$ where $\overline{\partial}$ is the Cauchy-Riemann operator and $S$ is smooth. Is it then true that $f$ is itself smooth?

I know the case $S=0$ is sometimes called Weyl's Lemma.

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2). This is true by elliptic regularity. You can even take $f$ to be a distribution solving the equation in the sense of distribution. If $S$ is analytic $f$ will also be analytic. Sample references would be Folland's Introduction to PDE, and Taylor's PDE I.