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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 ). 13 If 1 < p ≤ 2, F is a bounded operator from Lp (Rn ) to Lp (Rn ), p = p/(p − 1), and 1 F L(Lp ,Lp ) ≤ .

Since Lq (Λ) = Lq,q (Λ) ⊂ Lq,∞ (Λ), then any operator of strong type (p, q) is also of weak type (p, q). 12 Let T : L1 (Ω) + L∞ (Ω) → L1 (Λ) + L∞ (Λ) be of weak type (p0 , q0 ) and (p1 , q1 ), with constants M0 , M1 respectively, and 1 ≤ p0 , p1 ≤ ∞, 1 < q0 , q1 ≤ ∞, q 0 = q 1 , p0 ≤ q 0 , p 1 ≤ q 1 . For every θ ∈ (0, 1) define p and q by 1 1−θ θ 1 1−θ θ = + , = + . p p0 p1 q q0 q1 Then T is of strong type (p, q), and there is C independent of θ such that Tf Lq (Λ) ≤ CM01−θ M1θ f Lp (Ω) , f ∈ Lp (Ω).

Dλ (m − 1)! ∞ e−λs sm−1 T (s)y ds, 0 so that for every λ > 0 x = E ≤ C (m − 1)! ∞ e−λs sm(1−β)−1 ds y = 0 CΓ(m(1 − β)) mβ−m λ (m − 1)! m CΓ(m(1 − β)) mβ−m λ y (m − 1)! m m−r λ (−1)r Ar x ≤ C r r=0 m λmβ−r Ar u . r=0 Let us recall that D(Ar ) belongs to Jm/r (X, D(Am )) so that there is C such that r/m 1−r/m x D(Ar ) ≤ C x D(Am ) x X . Using such inequalities and then ab ≤ C(ap + bp ) with p = n/r, p = r/(n − r) we get x E ≤ Cλmβ (λ−m u D(Am ) + u ), λ > 0, 1−β β D(Am ) so that taking the minimum for λ > 0 x E ≤C u u and the statement holds.

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An Introduction to Interpolation Theory by Alessandra Lunardi

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