By Alessandra Lunardi
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Additional info for An Introduction to Interpolation Theory
3]. 11) that mθ (Rn ). Therefore (Lp (Rn ), W m,p (Rn ))θ,p = Bp,p [Lp (Rn ), W m,p (Rn )]θ = (Lp (Rn ), W m,p (Rn ))θ,p unless p = 2. 11 we get the Riesz–Thorin theorem, as stated at the beginning of the chapter. 11 is modeled on the proof of the Riesz–Thorin theorem, so that this has not to be considered an alternative proof. 11) is the theorem of Hausdorff and Young on the Fourier transform in Lp (Rn ). We set, for every f ∈ L1 (Rn ), (Ff )(k) = 1 (2π)n/2 e−i x,k f (x)dx, Rn k ∈ Rn . As easily seen, Ff L2 = f L2 for every f ∈ C0∞ (Rn ), so that F is canonically extended to an isometry (still denoted by F) to L2 (Rn ).
Taking into account that X belongs to J0 (X, D(A2 )) ∩ K0 (X, D(A2 )) and D(A) belongs to J1/2 (X, D(A2 )) ∩ K1/2 (X, D(A2 )), and applying the Reiteration Theorem with E0 = X, E1 = D(A) we get DA (α, p) = (X, D(A))α,p = (X, D(A2 ))α/2,p , 0 < α < 1, and setting α = 2θ the statement follows for 0 < θ < 1/2. Taking into account that D(A2 ) belongs to J1 (X, D(A2 )) ∩ K1 (X, D(A2 )) and D(A) belongs to J1/2 (X, D(A2 )) ∩ K1/2 (X, D(A2 )), and applying the Reiteration Theorem with E0 = D(A), E1 = D(A2 ) we get DA (α + 1, p) = (D(A), D(A2 ))α,p = (X, D(A2 ))(α+1)/2,p , 0 < α < 1, and setting α + 1 = 2θ the statement follows for 1/2 < θ < 1.
32 Chapter 1 Proof. For i = 1, 2, T is bounded from Lpi (Ω) to Lqi ,∞ (Λ), with norm not exceeding CMi . 6, T is bounded from (Lp0 (Ω), Lp1 (Ω))θ,p to (Lq0 ,∞ (Λ), Lq1 ,∞ (Λ))θ,p , and T (Lp0 (Ω),Lp1 (Ω))θ,p ,(Lq0 ,∞ (Λ),Lq1 ,∞ (Λ))θ,p ) ≤ CM01−θ M1θ . 10 that (Lp0 (Ω), Lp1 (Ω))θ,p = Lp,p (Ω) = Lp (Ω), and Lqi ,∞ (Λ) = (L1 (Λ), L∞ (Λ))1−1/qi ,∞ , i = 1, 2 (it is here that we need qi > 1: L1,∞ (Λ) is not a real interpolation space between L1 (Λ) and L∞ (Λ)), so that by the Reiteration Theorem (Lq0 ,∞ (Λ), Lq1 ,∞ (Λ))θ,p = (L1 (Λ), L∞ (Λ))(1−θ)(1−1/q0 )+θ(1−1/q1 ),p = (L1 (Λ), L∞ (Λ))1−1/q,p .
An Introduction to Interpolation Theory by Alessandra Lunardi