By Atle Selberg
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Extra resources for Atle Selberg Collected Papers
Consider ut = A(t)w, - U . L E R, t 2 0. 4. ,O). 8) with q = 1. Now let us discuss the given variable-coefficient problem. The transformed function P(L, t ) = u-l(t)U ( X . t ) satisfies the constant-coefficient equation discussed above. Thus the ,wequation can show arbitrarily fast exponential growth, and - transforming back - the same holds for the given u-equation. t)lJ ! 8) with q 2 1 is not useful if variable-coefficient problems are to be treated via localization. 5. *Extension of the Solution Operator Sdt) Up to this point we have only allowed initial data in Mo.
B2 denote normed spaces, let M denote a dense subspace of B1, and let B2 be complete. f E M . The operator S is called the extension of So. By our construction, the generalized solution u(z, t) is just an L2-function with respect to IC for each fixed t. It is often possible, however, to obtain smoothness properties of u ( z ,t) with respect to z and t by further investigations. Let us consider two simple examples. Example I . 2. I . 43 Discontinuous initial function. 15 1, 0 for 1x1 > 1. (d)dd.
We define the positive definite Hermitian matrix H = S*Sand rewrite the above matrix inequality as H A + A*H = S * S A + A*S*S = S * ( S A S - ' + S * - ' A * S * ) S 2 6 H . 2. Now consider the symbol P(iw) = - w 2 A + i w B + C. We obtain HP(iw) + P*(iw)HI: -w26H + const ((wl+ l ) H 5 2aH, . with a independent of w To finish the proof of the theorem, we prove a lemma on matrix exponentials eP'. It can be applied to each symbol P = P(iw) separately. 4. Let P E C'L-'L, and let 1 -I Ii If H P + P*H 5 2aH.