Timelike Initial Value Problems for Hyperbolic Equations I

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Timelike Initial Value Problems for Hyperbolic Equations I
J M Zimmerman
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O Proof: For any function v(t, x, y) e L^'^ for each y we employ the notation " I |v(t, x, y)|2dt dx = ||v(y)||2 *-• -00 f " lv(t, x, y)l^dx = ltv(t, y)||^ .
Since f, g e B(x, , h) ||u(y)M 1 A ^a||Fll+i5||G|l e ° for all T, a), ;^ y j^ h, where a and ^ are given by a = ^[a(0)]^/' + J[a(0)]-^/\'(0)] P= ^a(0)I-l/^ ^ ^ ■ .
and .^(o), h) = h^E^(Ca3 +D) . Hence from the Plancherel Theorem o we have M(^^, h) 1 -^-^^n (17) l|u(y)||

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