Amenability by Alan L. T. Paterson

By Alan L. T. Paterson

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Then a = limk→∞ (vnk , Avnk )V,V for some sequence {nk }. 7) we have 0 ≤ (vnk , Avnk )V,V + (vnl , Avnl )V,V − (vnk , Avnl )V,V − (vnl , Avnk )V,V ≤ ε ˜ Thus, if we let l → ∞ in the last relation, we obtain that for all k, l ≥ N. 0 ≤ (vnk , Avnk )V,V + a − (v, Avnk )V,V − (vnk , f )V,V ≤ ε ˜ Therefore after limit transition k → ∞ we obtain that for all k ≥ N. 0 ≤ 2a − 2(v, f )V,V ≤ ε for any ε > 0. 6) holds. 5) we find that (v − w, f − Aw)V,V ≥ 0 for any w ∈ D(A) and hence, by maximality of A, v ∈ D(A) and f = Av.

The following notations are used. u s≡ u H s (Ω ) , u ≡ u L2 (Ω ) , and (u, v) ≡ (u, v)L2 (Ω ) . We begin by collecting several estimates for products of functions from Sobolev spaces. There are many known “product” estimates that determine Sobolev-type regularity of the product of two (or more) functions. 4 Properties of the von Karman bracket 39 to [229, 250, 266] and others. However, in the context of von Karman brackets more specific product estimates are needed (which not always can be derived from more general rules).

Therefore using H¨older’s inequality we have f ·g −1+s ≤ C f ·g L2/(2−s) ≤ C f L2p/(2−s) · g L2q/(2−s) , where p−1 + q−1 = 1. 6). 6) follow from the same inequalities established for Ω = R2 and for f and g lying in C0∞ (R2 ). 4) for Ω = R2 . 7) −1 where p−1 j + q j = 1 (p j = ∞ is allowed) and h s,p = h Lp + R2 dy | y |2+2s R2 | h(x + y) − h(x) | p dx 2/p 1/2 is the norm in the Besov space Bsp,2 (R2 ) (cf. 7)). 10) we have g s,2q1 ≤ C g 1+s−1/q1 , f s,2q2 ≤ C f 1+s−1/q2 , q j ≥ 1. 8) Let p2 = σ −1 and q1 = (s + σ )−1 .

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