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In the proof of the next theorem we use the constants Mk,α := tk−α Ak etA sup 0

Note that the embeddings D(A) ⊂ Xα ⊂ X imply that t → etA is analytic in (0, +∞) with values in L(Xα ), hence etA L(Xα ) is bounded by a constant independent of t if t runs in any interval [a, b] ⊂ (0, +∞), but it could blow up as t → 0. 3 1. Let A : D(A) → X be a sectorial operator. Prove that D(A) is of class J1/2 between X and D(A2 ). 19) to get Ax ≤ M1 x /t + M0 t A2 x for each t > 0 and then take the minimum for t ∈ (0, +∞). If ω > 0, replace A by A − ωI. . ]. 2. 2. Prove that D(A) is of class J1/2 between X and D(A2 ).

The verification that (T (t))t≥0 is a semigroup is left as an exercise. 7) and defines an analytic semigroup whose generator is a sectorial realization of the Laplacian in X. For clarity reason, we split the proof in several steps. 3. The Laplacian in RN 31 (a) Let us first notice that T (t)f = Gt f , where Gt (x) = and |x|2 1 − 4t e , (4πt)N/2 Gt (x)dx = 1, t > 0, RN denotes the convolution. By Young’s inequality, T (t)f p ≤ f t > 0, 1 ≤ p ≤ +∞. 9) Since Gt and all its derivatives belong to C ∞ (RN ) ∩ Lp (RN ), 1 ≤ p ≤ +∞, it readily follows that the function u(t, x) := (T (t)f )(x) belongs to C ∞ ((0, +∞) × RN ), because we can differentiate under the integral sign.

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