Elliptic regularity in Sobolev spaces of negative order

analysis partial-differential-equations reference-request sobolev-spaces

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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