Download Dynamical systems : lectures given at the C.I.M.E. summer by Shui-Nee Chow; Centro internazionale matematico estivo.; et PDF

By Shui-Nee Chow; Centro internazionale matematico estivo.; et al

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Extra info for Dynamical systems : lectures given at the C.I.M.E. summer school held in Cetraro, Italy, June 19-26, 2000

Example text

1. The space i7jf (IRn) was introduced and studied in detail in Dubin­ skii [4,5]. We note that for n = 1 this space was used by Miranker [1] for studying the heat equation backwards in time. n) as follows: 34 DIFFERENTIAL OPERATORS OF INFINITE ORDER Let n(t\ a(0 = \J £&■« •"**"' ^ 5 * ' + 0O, £,$BR be such that

Weak Compactness. /,} is a subsequence of the sequence {um}. We have the following result. 1. Let the operator A be satisfied Conditions I and II. 1) has at least one solution u in X. The proof of this theorem can be found in Lions [1] and Dubinskii [7]. Chapter 2 PSEUDO-DIFFERENTIAL OPERATORS W I T H REAL A N A L Y T I C S Y M B O L S It is well known that the classical pseudo-differential (PD) operators can be repre­ sented, with the aid of the Fourier transform, in the form L(D)u(x) = J L j j L{Z)u{ty*d£.

However, ( W m L { $ a , G} j consists only of point functions. If $ 0 are the complementary Nfunctions to $ a , then (wmL{^>a,G}J = W~mE{$a,G}. More precisely, we set W""E{K, G] := | h | h(x) = £ (-l)^Daha(x) I |o|

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