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By Alessandra Lunardi

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Let n A= Di (aij (x)Dj ) i,j=1 be an elliptic operator with L∞ coefficients in a bounded open set Ω ⊂ Rn with regular boundary. If fji , j = 0, . . , n are in L2 (Ω), a weak solution of the Dirichlet problem   Au = f in Ω,  u = 0 in ∂Ω is any u ∈ H01 (Ω) such that for every ϕ ∈ C0∞ (Ω) it holds n n aij (x)Dj u(x)Di ϕ(x)dx = − Ω i,j=1 f0 (x)ϕ(x)dx + Ω fj (x)Dj ϕ(x)dx. Ω j=1 Using the Lax-Milgram theorem, it is not hard to see that the Dirichlet problem has a unique weak solution u, and u H 1 ≤ C nj=0 fj L2 .

Then v(λ) − x ≤ λR(λ, A)(λR(λ, B)x − x) + λR(λ, A)x − x = λR(λ, A)R(λ, B)Bx + R(λ, A)Ax ≤ λ−1 (M (M + 1) Bx + M Ax ), and v(λ) K2 = v(λ) + A2 v(λ) + ABv(λ) + B 2 v(λ) ≤ M 2 x + 2M (M + 1)λ( Ax + Bx ). Setting λ = t−1/2 we deduce that t−1/2 K(t, x, X, K 2 ) ≤ t−1/2 ( x − v(t−1/2 ) + t v(t−1/2 ) ≤ C( x K1 K2 ) + t1/2 x ), is bounded in (0, 1). We know already that t → t−1/2 K(t, x, X, K 2 ) is bounded in [1, ∞). Therefore K 1 is in the class K1/2 between X and K 2 . 15 For every k ∈ N ∪ {0}, K k+1 ∈ J1/2 (K k , K k+2 )∩ K1/2 (K k , K k+2 ).

Then E ∈ Jβ (X, D(Am )), so that (X, D(Am ))θβ,p ⊂ (X, E)θ,p , for every θ ∈ (0, 1), p ∈ [1, ∞]. 62 Chapter 3 Proof. Let x ∈ D(Am ), λ > 0 and set (λI − A)m x = y. Then x = (R(λ, A))m y so that x= (−1)m−1 dm−1 1 R(λ, A)y = m−1 (m − 1)! 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 .

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