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By Lorenzi L., Lunardi A., Metafune G., Pallara D.

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N , both e−λt Di T (t)g ∞ and e−λt Dij T (t)g ∞ are integrable in (0, +∞). 12)(a) we get Dij T (t)g ∞ = Dj T (t/2)Di T (t/2)g c C(α) g 1/2 (t/2) (t/2)1/2−α k g DA (α,∞) . 16) +∞ Therefore, the integral 0 e−λt T (t)g dt is well defined as a Cb2 (RN )-valued integral, so that f ∈ Cb2 (RN ). We could go on estimating the seminorm [Dij T (t)g]C 2α (RN ) , but we get b [Dij T (t)g]C 2α (RN ) ≤ C g DA (α,∞) /t, and it is not obvious that the integral is well defined b as a Cb2+2α (RN )-valued integral.

B) Using the functions uε (x, y) = xy log(ε + x2 + y 2 ), show that there exists no C > 0 such that u C 2 (R2 ) ≤ C( u ∞ + ∆u ∞ ) for any u ∈ C0∞ (R2 ). Deduce that b the domain of the Laplacian in Cb (R2 ) is not Cb2 (R2 ). 4 The Dirichlet Laplacian in a bounded open set Now we consider the realization of the Laplacian with Dirichlet boundary condition in Lp (Ω), 1 < p < +∞, where Ω is an open bounded set in RN with C 2 boundary ∂Ω. Even for p = 2 the theory is much more difficult than in the case Ω = RN .

1: the curve Γε . Mε π ≤ Mε π ≤ +∞ e(ω+ξ cos η)t dξ + a Mε 2π b e(s(A)+ε)t dy −b 1 + b e(s(A)+ε)t . 2) follows for n = 0. Arguing in the same way, for t ≥ 1 we get AetA = ≤ ≤ 1 2πi Mε 2π λetλ R(λ, A)dλ Γε +∞ b e(ω+ξ cos η)t dξ + 2 e(s(A)+ε)t dy −b a Mε Mε (s(A)+2ε)t (| cos η|−1 + b)e(s(A)+ε)t ≤ e . 2) follows also for n = 1. 2) is proved. 2) for t large.

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