By J. Diestel
ISBN-10: 0387074023
ISBN-13: 9780387074023
ISBN-10: 3540074023
ISBN-13: 9783540074021
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Extra info for Geometry of Banach spaces: Selected topics
Sample text
1. Let 1 L p,λ f∈ (Ω) = Lploc (Ω) 0. The Morrey space Lp,λ (Ω) is defined as ˆ 1 : sup λ r x∈Ω;r>0 |f (y)|p dy < ∞ . 1) B(x,r) This is a Banach space with respect to the norm f Lp,λ (Ω) := sup x∈Ω;r>0 ˆ 1 rλ 1 p |f (y)| dy p . 2) B(x,r) Note that if we want to have an equivalent norm in the form ˆ 1 sup 1 p |f (y)| dy p λ |B(x, r)| n x∈Ω;r>0 , B(x,r) we should assume that Ω satisfies the condition |B(x, r)| Crn , 0 < r < diam (Ω), which is done, for instance, in the book of Kufner, John, and Fuˇc´ık [151].
54) be satisfied. 49), the operator I α(·) is bounded from the space H ω(·) (Ω) to the weighted space H ωα (·) (Ω, α). Proof. 19. 28. 55) reduces to sup [λ(x) + Re α(x)] < θ. 3. Potentials of Constant Order 593 In the following theorem we use the notation ω−α (x, t) = t− Re α(x) ω(x, t) and ω−α (x, h) = sup ω−α (y, h) y:|y−x| 2]. 18. Let ω(x, t) be in ∈ W (0, ) uniformly in x. Then ω(x, t) ∈ Zβ ⇐⇒ M (w) < β. 17. 19. Let ωδ = ω(x,t) tδ(x) and ωβ = ω(x,t) . 36) ω(x, t) ∈ Zβ(·) ⇐⇒ M (wβ ) < 0. 37) take the form ω(x, t) ∈ Zδ(·) ⇐⇒ ω(x, t) ∈ Zβ(·) ⇐⇒ inf [m(w, x) − δ(x)] > 0, x∈Ω sup[M (w, x) − β(x)] < 0. x∈Ω We will make use of the following property of the bounds for functions ω(x, t) ∈ W(T) in terms of their indices: c1 tM(w)+ε ω(x, t) c2 tm(ω)−ε , 0 t (< ∞), where ε > 0 and the constants c1 , c2 may depend on ε, but do not depend on x and t (see Samko [192, Thm.