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A Characterization of Alternating Groups by the Set of by Chen G.

By Chen G.

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Once such estimates are available, a localization argument shows that u is continuous up to t = 0. The uniform gradient estimates can be proved by adapting the Bernstein method and applying it to the functions un . Here, the convexity of Ωn plays a crucial role in making this machinery work. 5). The semigroup {T (t)} is not strongly continuous and, in general, it is not analytic in Cb (Ω). In any case, as in Chapter 2, it is possible to define its weak generator and to give a partial characterization of it.

The Markov process 27 converges to the function R(λ)f as n tends to +∞, locally uniformly in RN . It follows that R(λ)f = u and, consequently, u ∈ D(A). 3. Now, we prove the second part of the proposition. We limit ourselves to showing that “(i) ⇒ (iii)” and “(iii) ⇒ (ii)”, since “(ii) ⇒ (i)” is trivial. “(i) ⇒ (iii)”. 3 it follows immediately that A ⊂ A. Hence, we only need to prove that A ⊂ A. For this purpose, fix u ∈ Dmax (A) and set f = λu − Au and v = R(λ, A)f . Since A ⊂ A, we have λv − Av = f .

To show that Cε is open in [0, +∞), we fix s ∈ Cε,R and prove that, for any δ > 0, [s, δ] ⊂ Cε,R . For this purpose, it suffices to argue as above, observing that T (·)(ϕn − 1l) converges to 0, uniformly in [s, δ] × B(R). Now, since p(t, x; B(m)) ≥ (T (t)ϕm )(x) for any t > 0, any x ∈ RN and any m ∈ N, and Cε,R = [0, +∞), we easily deduce that, for any arbitrarily fixed T > 0 and any R > 0, there exists m ∈ N such that p(t, x; B(m)) ≥ (T (t)1l)(x) − ε = p(t, x; RN ) − ε, t ∈ [0, T ], x ∈ B(R). Therefore, |(T (t)fn )(x)| ≤ |fn (y)|p(t, x; dy) + B(m) ≤ |fn (y)|p(t, x; dy)dy RN \B(m) sup |fn (y)| + p(t, ; RN \ B(m)) y∈B(m) ≤ sup |fn (y)| + ε, y∈B(m) for any t ∈ [0, T ] and any x ∈ B(R).

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