ϵ, satisfiesP u∈C0∞(ω) andP u= 0 in [p.3 ↑3 — p.4 ↓15] Proof of Proposition 1.1 should be replaced by: Let ω be the open set in Definition 1.1

Hyperbolic Systems With Analytic Coefficients：正誤表

[p.2 ↑8] with|τ|< ϵ, satisfiesP u= 0 in =⇒with|τ|< ϵ, satisfiesP u∈C₀^∞(ω) andP u= 0 in

[p.3 ↑3 — p.4 ↓15] Proof of Proposition 1.1 should be replaced by: Let ω be the open set in Definition 1.1. Take an open setV such thatK⋐V ⋐ω. Then for any f ∈ C₀^∞(V₋ϵ) there exists a unique u ∈H^∞(ω) satisfying P u =f in ω and vanishing in x0 ≤ −ϵ. Denote by T the map T : C₀^∞(V₋ϵ)3f 7→u∈ H^∞(ω). Note that H^∞(ω) is a Fr´echet space equipped with countable semi- norms k · kH^p(ω), p = 0,1, . . .. Assume that C₀^∞(V₋ϵ) 3 fj → f in C₀^∞(V₋ϵ) and T fj = uj → u in H^∞(ω). SinceP uj = fj it is clear that P u = f and u= 0 in x0 ≤ −ϵ. From the uniqueness of the solution one has T f = uand hence the graph of T is closed. From the Banach’s closed graph theorem it follows thatT is a continuous map. Therefore for anyp∈Nthe inverse image of{u∈H^∞(ω)| kukH^p(ω) <1}, which is a neighborhood of 0 in H^∞(ω), is a neighborhood of 0 inC₀^∞(V_−ϵ), that is there existδ >0 andq∈Nsuch that

f ∈C₀^∞(V_−ϵ), kfkH^q(V)< δ=⇒ kT fkH^p(ω)<1.

For any f ∈ C₀^∞(V_−ϵ) the H^q(V) norm of δf /kfkH^q(V) is less than 1 then from the uniqueness of the solution we conclude that for anyf ∈C₀^∞(V_−ϵ) and u∈H^∞(ω) satisfying P u=f inω and vanishing inx₀≤ −ϵ satisfies

kukH^p(ω)≤δ⁻¹kfkH^q(V).

[p.21 ↑12] polynomial inx=⇒polynomial iny [p.27 ↑10]Proof of Proposition 1.6. =⇒Proof.

[p.29 ↑9]Proof of Proposition 1.6. =⇒Proof.

[p.72 ↑5]hρ(sY −tX) = (−1)^rh_ρ (−sY+tX) = (−1)^rh_ρ (Y)∏

(−s−λj(tX)) =⇒ hρ(sY −tX) = (−1)^rhρ(−sY +tX) = (−1)^rhρ(Y)∏

(−s−λj(tX)) [p.125 ↑5]C|x|^−2lt^∗(x)^{2(Q−q−l−1)}∑

l₁+l₂≤l

∫φ(x)

ε(x) |∂_t^Q+1+l1∂_x^l2f|²dxdt

=⇒C|x|^−2l|r|^2(q−k−l)t^∗(x)^{2(Q−q−l−1)}∑

l1+l2≤l

∫t^∗(x)

ε(x) |∂_t^Q+1+l1∂^l_x²f|²dt [p.125 ↑4]C|t−ε|^2(Q−k)∫t

ε|∂_t^Q+1∂_x^lf|²dxdt

=⇒Delete

[p.125 ↑3] for q+l+ 1≤Q,k+l ≤q and =⇒for|t| ≤t^∗(x),q+l+ 1≤Q, k+l≤qand

[p.130 ↓7〜11] should be replaced by

|∂_t^k∂_x^lr^qFq−1|²

≤C ∑

k1+k2≤k

|r|^{2(q−l−k1)}|x|^−2lt^∗(x)^{2(Q−q−l−k2−1)+1} ∑

l1+l2≤l

∫ t^∗ ε

|∂_t^Q+1+l1∂_x^l2f|²dt

≤C|x|^−2l|r|^2(q−l−k)t^∗(x)^{2(Q−q−l−1)+1} ∑

l1+l2≤l

∫ t^∗ ε

|∂_t^Q+1+l1∂^l_x²f|²dt

hence we conclude the proof.

[p.132 ↑10] Since ∂_t^pu= 0 on t = sν(x) and t = σν+1(x), |x| = δ(T−t) are space-like curves =⇒Since∂_t^pu= 0 ont=sν(x)

[p.164 ↑1]P(x) =⇒P(ξ)

[p.197 ↓10]intervals=⇒neighborhoods