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6(iii), S(RN ) is contained in D(A) and Au = ∆u for u ∈ S(RN ). 16 it is a core for A. Let u ∈ W 2,p (RN ) and let un ∈ S(RN ) be such that un → u in W 2,p (RN ). Then Aun = ∆un → ∆u in Lp (RN ) and, since A is closed, u ∈ D(A) and Au = ∆u. In the case of Cb (RN ) we argue differently because the Schwartz space is not dense ∂ in Cb (RN ) and in Cb2 (RN ). Instead, we use the identities T (t)∆f = ∆T (t)f = ∂t T (t)f N which hold pointwise in (0, +∞) × R . Setting g = f − ∆f we have +∞ R(1, A)g = 0 2 N +∞ e−t T (t)(f − ∆f )dt = e−t (I − ∆)T (t)f dt 0 N We recall that S(R ) is the space of all the functions f : R → R such that |x|α |Dβ f (x)| tends to 0 as |x| tends to +∞ for any multiindices α and β; C0∞ (RN ) is the space of all compactly supported infinitely many times differentiable functions f : RN → R.

1 Let Ω ⊂ RN be a bounded open set with C 2 boundary, and let f ∈ Lp (Ω), λ ∈ (−∞, 0]. 13) p holds, with C depending only upon Ω and λ. The resolvent estimate is much easier. Its proof is quite simple for p ≥ 2, and in fact we shall consider only this case. For 1 < p < 2 the method still works, but some technical problems occur. 2 Let 2 ≤ p < +∞, let λ ∈ C with Re λ ≥ 0 and let u ∈ W 2,p (Ω) ∩ W01,p (Ω), be such that λu − ∆u = f ∈ Lp (Ω). Then u p ≤ 1+ p2 f p . 4 |λ| Proof. To simplify the notation, throughout the proof, we denote simply by · the usual Lp -norm.

X ∂x k k Ω Ω k=1 Notice that ∂ ∂u p−2 ∂u ∂u |u|p−2 u = |u|p−2 + u|u|p−4 u +u . ∂xk ∂xk 2 ∂xk ∂xk Setting |u|(p−4)/2 u ∂u = ak + ibk , k = 1, . . , N, ∂xk with ak , bk ∈ R, we have N Ω k=1 ∂u ∂ |u|p−2 u dx ∂xk ∂xk N (|u|(p−4)/2 )2 uu = Ω k=1 ∂u ∂u ∂u p−2 ∂u ∂u u +u + (|u|(p−4)/2 )2 u ∂xk ∂xk ∂xk 2 ∂xk ∂xk N a2k + b2k + (p − 2)ak (ak + ibk ) dx, = Ω k=1 whence N λ u p N ((p − 1)a2k + b2k )dx + i(p − 2) + Ω k=1 f |u|p−2 u dx. 5. More general operators 37 Taking the real part we get N Re λ u p ((p − 1)a2k + b2k )dx = Re + Ω k=1 f |u|p−2 u dx ≤ f u p−1 , Ω and then (a) Re λ u ≤ f ; N ((p − 1)a2k + b2k )dx ≤ f (b) u p−1 .