By Mischa Cotlar

ISBN-10: 0720420660

ISBN-13: 9780720420661

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Example text

We then define Kf := lim Kfj in j→∞ C 0 (G). 2. The integral operator K : D → C 0 (G) with the singular kernel K ∈ Sα (G) and α ∈ [0, n) can be uniquely continued to the bounded linear operator K : Lq (G) → C 0 (G) satisfying Kf C 0 (G) ≤ C(q) f Lq (G) , f ∈ Lq (G) (17) n , +∞), according to (16). Here we have chosen the constant for each q ∈ ( n−α C = C(q) ∈ (0, +∞) appropriately. Remark: In the case n ≥ 3, Green’s function of the Laplace operator H = H(x, y) belongs to the class Sn−2 (G) which means α = n−2 and q ∈ n2 , +∞ .

D. 5. Let the sets Ω1 , Ω2 ⊂ B be bounded, open and disjoint. · Furthermore, we define Ω := Ω1 ∪ Ω2 . Then we have δB (ϕf , Ω) = δB (ϕf , Ω1 ) + δB (ϕf , Ω2 ). Proof: Take with g : Ω → Bg ⊂ B an admissible approximation of f satisfying Ωi ∩ Bg = ∅ for i = 1, 2. 5 from Section 2 yields δB (ϕf , Ω) = δBg (ϕg , Ω ∩ Bg ) = δBg (ϕg , Ω1 ∩ Bg ) + δBg (ϕg , Ω2 ∩ Bg ) = δB (ϕf , Ω1 ) + δB (ϕf , Ω2 ). d. 6. Let U = U (z) ⊂ B denote an open neighborhood of the point z and f : U (z) → B a completely continuous mapping.

Therefore, K : D → C 0 (G) represents a bounded linear operator, where D is endowed with the Lq (G)-norm (see Chapter 2, Section 6). 1 from Section 8 in Chapter 2, we can now continue K to the operator K : Lq (G) → C 0 (G) (15) on the Banach space Lq (G). The set C0∞ (G) ⊂ D is dense in the space Lq (G), and for each f ∈ Lq (G) we have a sequence {fj }j=1,2,... ⊂ C0∞ (G) satisfying f − fj Lq (G) → 0 (j → ∞). We then define Kf := lim Kfj in j→∞ C 0 (G). 2. The integral operator K : D → C 0 (G) with the singular kernel K ∈ Sα (G) and α ∈ [0, n) can be uniquely continued to the bounded linear operator K : Lq (G) → C 0 (G) satisfying Kf C 0 (G) ≤ C(q) f Lq (G) , f ∈ Lq (G) (17) n , +∞), according to (16).

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An Introduction to Functional Analysis (North-Holland Texts in Advanced Mathematics) by Mischa Cotlar


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