An Introduction to Interpolation Theory by Alessandra Lunardi

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By Alessandra Lunardi

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Real interpolation 31 s (Rn ) is the Besov space defined as follows: if s is not an integer, let [s] and Here Bp,q s (Rn ) consists of {s} be the integral and the fractional parts of s, respectively. Then Bp,q the functions f ∈ W [s],p (Rn ) such that s = [f ]Bp,q q/p dh Rn |α|=[s] 1/q |Dα f (x + h) − Dα f (x)|p dx |h|n+{s}q Rn s (Rn ) = W s,p (Rn ). is finite. In particular, for p = q we have Bp,p k (Rn ) consists of the functions f ∈ W k−1,p (Rn ) such that If s = k ∈ N, then Bp,q [f ]Bp,q = k |α|=[s]−1 Rn dh |h|n+q q/p 1/q |Dα f (x + 2h) − 2Dα f (x + h) + Dα f (x)|p dx Rn is finite.

N, defined by Tij f0 = Di u, we get that if f0 ∈ Lp (Ω), 2 < p < ∞, then each derivative Dij u belongs to Lp (Ω), and Dij u Lp ≤ C f0 Lp . 3 Exercises 1) Show that for 0 < θ < 1, θ = 1/2, (Cb (Rn ), Cb2 (Rn ))θ,∞ = Cb2θ (Rn ). 8, then use the Reiteration Theorem with E = Cb1 (Rn ). 1 find some interpolation couple (X, Y ) and intermediate spaces in the classes J0 , J1 , K1 between X and Y that do not coincide with X or Y . (b) Give a direct proof of statement (ii) of the Reiteration Theorem in the case (θ0 , θ1 ) = (0, 1).

It is not hard to see that F(X, Y ) and F0 (X, Y ) are Banach spaces. Indeed, if fn is a Cauchy sequence, the maximum principle gives, for all z ∈ S, fn (z) − fm (z) X+Y ≤ max{supt∈R fn (it) − fm (it) X+Y , supt∈R ≤ max{supt∈R fn (it) − fm (it) X , supt∈R fn (1 + it) − fm (1 + it) fn (1 + it) − fm (1 + it) X+Y } Y }. Therefore for every z ∈ S there exists f (z) = limn→∞ fn (z) in X + Y , and it is easy to see that f ∈ F(X, Y ). Since t → fn (it) converges in Cb (R; X) and t → fn (1 + it) converges 38 Chapter 2 in Cb (R; Y ), then fn converges to f in F(X, Y ).

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