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N consisting of functions ψj ∈ C0∞ (Ωj ) satisfying ∞ ψ1 + · · · + ψN = 1 on Kl . For ϕ ∈ CK (Ω) we set l N u, ϕ Ω N = u, ψj ϕ Ω = j=1 uj , ψj ϕ Ωj . ,N and of the partition of unity {ψj }). ,M is another subfamily covering Kl , and ψ1 , . . , ψM is an associated partition of unity, we have, with uk denoting the distribution given on Ωk : N N uj , ψj ϕ Ωj N M = j=1 uj , ψk ψj ϕ Ωj j=1 k=1 N M = uj , ψk ψj ϕ Ωj ∩Ωk j=1 k=1 M M = uk , ψk ψj ϕ Ωk = j=1 k=1 uk , ψk ϕ Ωk , k=1 since uj = uk on Ωj ∩ Ωk .

For f in C |α| (Ω), (∂ α Λf )(ϕ) = (−1)|α| f ∂ α ϕ dx = Ω (∂ α f )ϕ dx = Λ∂ α f (ϕ), Ω C0∞ (Ω). 23. 29 (Convolution). When ϕ and ψ are in C0∞ (Rn ), then, as noted earlier, ϕ ∗ ψ is in C0∞ (Rn ) and satisfies ∂ α (ϕ ∗ ψ) = ϕ ∗ ∂ α ψ for each α, and the map ψ → ϕ ∗ ψ is continuous. ˇ that For ϕ, ψ and χ in C0∞ (Rn ) we have, denoting ϕ(−x) by ϕ(x), Rn ϕ ∗ ψ(y)χ(y)dy = Rn Rn ψ(x)ϕ(y − x)χ(y)dxdy = Rn ψ(x)χ ∗ ϕ(x)dx ˇ ; therefore we define (ϕ ∗ u)(χ) = u(ϕˇ ∗ χ) , u ∈ D (Rn ) , ϕ, χ ∈ C0∞ (Rn ) ; this makes u → ϕ ∗ u a continuous operator on D (Rn ).

18). Let Ω be an open subset of Rn with C 1 -boundary. The function 1Ω (cf. 27)) has distribution derivatives described as follows: For ϕ ∈ C0∞ (Rn ), ∂j 1Ω , ϕ ≡ − ∂j ϕ dx = Ω νj (x)ϕ(x) dσ. 24) shows precisely how it acts. Another important aspect is that the distributions theory allows us to define derivatives of functions which only to a mild degree lack classical derivatives. Recall that the classical concept of differentiation for functions of several variables only works really well when the partial derivatives are continuous, for then we can exchange the order of differentiation.