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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 Answers 2

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1). Canonical references are Adams' Sobolev space, and Triebel's sequence of books on Function spaces. Check also MacLean's Strongly elliptic systems ..., and Bergh and Lögström's Interpolation spaces.

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.

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    $\begingroup$ Another sample reference: this thread. :) $\endgroup$
    – user31373
    Aug 18, 2012 at 3:38
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Not sure about question 2. For 1, I would check out "Sobolev Spaces" by Robert Alexander Adams, John J. F. Fournier.

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