An Introduction to Interpolation Theory by Alessandra Lunardi

By Alessandra Lunardi

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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 . 10. Since p0 ≤ q0 and p1 ≤ q1 then p ≤ q, so that Lq,p (Λ) ⊂ Lq,q (Λ) = Lq (Λ). It follows that T is bounded from Lp (Ω) to Lq (Λ), with norm not exceeding C M01−θ M1θ .

IX] or [1]. The second step is a regularity theorem: if fj = 0 for each j and the coefficients aij ∈ C 1 (Ω), then u ∈ H 2 (Ω) and u H 2 ≤ C f0 L2 . See again [8] or [1]. Moreover Campanato in [14, Thm. , each second order derivative Dij u belongs to BM O(Ω), and Dij u BM O,2 ≤ C f0 BM O,2 . 14 with r = 2 to the operators Ti , i = 1, . . , n, defined by Ti (f0 , . . , fn ) = Di u, u being the solution of the Dirichlet problem, we get that if the fj ’s are in Lp (Ω), 2 < p < ∞, then each derivative Di u belongs to Lp (Ω), and Di u Lp ≤ C nj=0 fj Lp .

This also allows to use the Reiteration Theorem to characterize the real interpolation spaces between complex interpolation spaces. We get, for 0 < θ1 < θ2 < 1, 0 < θ < 1, 1 ≤ p ≤ ∞, ([X, Y ]θ1 , [X, Y ]θ2 )θ,p = (X, Y )(1−θ)θ1 +θθ2 ,p . Further reiteration properties are the following. Calderon ([13]) showed that if one of the spaces X, Y is continuously embedded in the other one, or if X, Y are reflexive and X ∩ Y is dense both in X and in Y , then [[X, Y ]θ1 , [X, Y ]θ2 ]θ = [X, Y ](1−θ)θ1 +θθ2 , Lions ([30]) proved that if X and Y are reflexive, then for 0 < θ1 < θ2 < 1, 0 < θ < 1, 1 < p < ∞, [(X, Y )θ1 ,p , (X, Y )θ2 ,p ]θ = (X, Y )(1−θ)θ1 +θθ2 ,p .

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