Distributions, Partial Differential Equations, and Harmonic Analysis by Dorina Mitrea

Author:Dorina Mitrea [Mitrea, Dorina]
Language: eng
Format: epub, pdf
Tags: Differential Equations, Vector Analysis, Probability & Statistics, Mathematical Analysis, Stochastic Processes, Functional Analysis, Mathematics, General
ISBN: 9781461482086
Google: 2ZC4BAAAQBAJ
Publisher: Springer Science & Business Media
Published: 2013-09-20T00:00:00+00:00

Proof.

The fact that u ∈ C ∞ (Ω) is a consequence of the hypoellipticity of the operator P(∂) (cf. Definition 6.1 and Remark 6.2). As regards (6.3.2), pick an arbitrary x 0 ∈ Ω, fix , and let be such that near . Then and since P(∂)E = δ in we may write, for each point x ∈ B(x 0, r ∕ 2),

(6.3.4)

where we have used (6.3.1), part (e) of Theorem 2.87, and that P(∂)u = 0 in Ω. Note that for each β > 0 the function ∂ β is compactly supported in . Keeping this in mind, we may then integrate by parts in order to further write the last expression in (6.3.4) as

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