Newton polygons and order functions - General parabolic mixed order systems in Lp and applicati

2.7 Examples

3.1.2 Newton polygons and order functions









−z³, γ∈(0,1), λz√

−z²−z³, γ= 1, λz√

−z², γ∈(1,2), λ²+ωλz, γ= 2, λ², γ∈(2,∞].

The termλz^3/2+ 1is of lower order and therefore it is not existent in the principal part. With this it is obvious that the symbolP is order-representative in the sense of Definition 3.13.

In the set of all order-representative representations of the symbolP we do not have differentγ-orders andγ-principal parts for various representations. This will be stated in the next lemma.

Lemma 3.15. For two order-representative representations P[℘](λ, z) = X

`∈I

χ_`(℘)τ_`(λ, z)ϕ_`(λ)ψ_`(z)

= X

`∈I⁰

χ⁰_`(℘)τ_`⁰(λ, z)ϕ⁰_`(λ)ψ⁰_`(z), (λ, z)∈Lt×Lx, ℘∈K

of the symbolP we haved_γ(P) =d⁰_γ(P)andπ_γP[℘] =π⁰_γP[℘]for all℘∈K, γ∈(0,∞]. Hered⁰_γ(P)and π⁰_γP[℘]denote the γ-order and the γ-principal part with respect to the second representation.

Proof. Let0< γ <∞and(λ, z, ℘)∈Lt×Lx×K be arbitrary. Assumingdγ(P)< d⁰_γ(P)we derive π⁰_γP[℘](λ, z) = lim

η→∞η^d^γ^(P)−d⁰^γ^(P)η^−d^γ^(P)P[℘](η^γλ, ηz) = 0·πγP[℘](λ, z) = 0.

This yieldsπ⁰_γP[℘]≡0, which contradicts (3.2).

The same argument obviously holds if dγ(P) > d⁰_γ(P). Therefore, we get dγ(P) = d⁰_γ(P) and thus π_γP[℘] =π_γ⁰P[℘]for all℘∈K. The case γ=∞can be done in exactly the same way.

Definition 3.16 (Symbol classS[K](L_t×L_x)). For a compact setK⊆C^m we defineS[K](L_t×L_x) as the set of all order-representative symbolsP ∈S[K](L_t×L_x). Note that the concepts of γ-order and γ-principal part are well-defined in this class.

Definition 3.17 (Symbols without compact parameter). Symbols without parameter-dependance can always be interpreted as symbols with parameter by setting K:={0}andχ_`(0) := 1. We define

S(Lt×Lx) := {P ∈S[{0}](Lt×Lx) :χ`(0) := 1for all `∈I}, S(Lt×Lx) := {P ∈S[{0}](Lt×Lx) :χ`(0) := 1for all`∈I}.

3.1.2 Newton polygons and order functions

Here and in the following we only consider symbols with an order-representative representation. To analyze such a symbol it is helpful to define the Newton polygon which arises in the theory of polynomials, cf. [GV92].

We consider Newton polygons defined by a finite set of tuples. Letν := (a_i, b_i)_i=0,...,M ⊆[0,∞)² be arbitrary. Then we define the Newton polygonN(ν)as the convex hull of

ν∪ {(0,0)} ∪

[

i=0

{(ai,0),(0, bi)}.

3.1: Inhomogeneous symbols and the Newton polygon 53

v₂ v3

v_J−1 v_J+1

q₁ q2

q_J−1 qJ

qJ+1

q₀

Figure 3.1: Regular Newton polygonN

Definition 3.18 (Newton polygon). A convex set N ⊆ [0,∞)² is called a Newton polygon if there exists a finite setν ⊆[0,∞)² such thatN =N(ν).

LetN be a Newton polygon. Then we denote byv0:= (r0, s0), . . . , vJ+1:= (rJ+1, sJ+1)the vertices ofN, starting at the origin and indexed in the counter-clockwise direction and set

NV :={vi:i= 0, . . . , J + 1}.

The corresponding weight function is then defined by ΞN(λ, z) := X

(r,s)∈NV

|z|^r|λ|^s, z∈Cⁿ, λ∈C.

For a finite setν ⊆[0,∞)²we also define the weight function Ξ_ν := Ξ_N(ν).

Remark 3.19. Here we want to point to a simple geometric observation. Let N be a Newton polygon and(x, y)∈[0,∞)². Then we have(x, y)∈N if and only if

x+γy≤ max

(r,s)∈N_V

1 γ

r s

for all0< γ <∞.

Especially we define a Newton polygon for each order-representative symbolP ∈S[K](L_t×L_x). For the sake of simplicity we first define

v₁(`) := (r₁(`),s₁(`)) := (N_`+L_`, M_`),

v2(`) := (r2(`),s2(`)) := (L`, N`/ρ+M`), `∈I.

Then we defineN(P) :=N(ν(P))andΞP := Ξ_N(P)where ν(P) :=[

`∈I

{v1(`),v2(`)}.

v₀

Figure 3.2: Newton polygon which is not reg-ular in time

v₀

Figure 3.3: Newton polygon which is not reg-ular in space

Definition 3.20 (Regular). A Newton polygon N is said to be regular if r1 6=r2 andsJ 6=sJ+1 (i.e.

J >0and there are no edges parallel to the axes except the trivial ones). The Newton polygonN is called regular in time (respectively, regular in space) ifr16=r2 (respectively, sJ 6=sJ+1).

A symbolP ∈S[K](Lt×Lx)is called regular/regular in time/regular in space if the associated Newton polygon N(P)is regular/regular in time/regular in space.

Example 3.21.

(i) The symbolP(λ, z) :=λ²+ 3λz+ 5z−z^3/2,(λ, z)∈Sθ×Σδ, is regular.

(ii) The Newton polygon ofP(λ, z) :=−2λ²+zλ²+z²,(λ, z)∈Sθ×Σδ, is regular in time but it is not regular in space.

(iii) The symbolP(λ, z) := 5λz+z−3,(λ, z)∈S_θ×Σ_δ, is neither regular in space nor regular in time.

For a Newton polygonN we can define the exterior normal to the edge [vjvj+1] :={tvj+ (1−t)vj+1:t∈[0,1]}

connecting the verticesv_j andv_j+1 by qj:= (qj,1, qj,2) :=q

1 +γ_j²⁻¹

(1, γj), j= 1, . . . , J−1, qJ:=

((p

1 +γ²_J)⁻¹(1, γJ), sJ6=sJ+1,

(0,1), sJ=sJ+1

where we have defined

γ_j:= rj−rj+1

s_j+1−s_j, j= 1, . . . , J−1, γ_J:=

(_r_j_−r_j+1

sJ+1−sJ , s_J 6=s_J+1,

∞ , s_J =s_J+1. Furthermore, we define

γ0:= 0, q0:= (q0,1, q0,2) := (0,−1), γ_J+1:=∞, q_J+1:= (q_J+1,1, q_J+1,2) := (−1,0),

and the orthogonal vectorsq_j^⊥:= (−qj,2, qj,1), j= 1, . . . , J + 1. We have 0≤γ1<· · ·< γJ ≤ ∞. Note that in general there does not necessarily existsj∈ {1, . . . , J}withγ_j=ρ.

Definition 3.22. Let N be an arbitrary Newton polygon. To avoid some tedious distinctions and to provide a unified treatment we introduce

κ₁(N) :=

(1, ifN is regular in time,

2, ifN is not regular in time, κ₂(N) :=

(J+ 1, ifN is regular in space, J, ifN is not regular in space.

3.1: Inhomogeneous symbols and the Newton polygon 55 Remark 3.23. LetP ∈S[K](Lt×Lx)be an order-representative symbol. Then the following statements are easy to prove:

(i) Geometrical arguments simply show dγ(P) = max

i=1,...,J+1

1 γ

s_i

, 0< γ <∞

and therefore the Newton polygon coincide for all order-representative representations ofP. (ii) If there exists j∈ {κ1(N(P)), . . . , κ2(N(P))} withρ∈(γ_j−1, γj), we haveN`= 0 (i.e.τ`=const)

for all `∈Iρ. Otherwise we get an edge with normal direction (1, ρ). Hence M`=sj andL` =rj

for all`∈Iρ.

(iii) For j= 1, . . . , J and 0< γ_j<∞we have I_γ⁰=

`∈I_γ_j:M_`+χ_(ρ,∞)(γ_j)N_`/ρ=s_j =

`∈I_γ_j:χ_(0,ρ](γ_j)N_`+L_`=r_j , I_γ⁰⁰=

`∈I_γ_j:M_`+χ_[ρ,∞)(γ_j)N_`/ρ=s_j+1 =

`∈I_γ_j: χ_(0,ρ)(γ_j)N_`+L_`=r_j+1 for allγ⁰∈(γ_j−1, γ_j)and all γ⁰⁰∈(γ_j, γ_j+1).

Remark 3.24(cf. [GV92, Lemma 1.2.1]). Let(ri, si)∈[0,∞)² fori= 0, . . . , M and (r, s)∈conv{(ri, si) :i= 0, . . . , M}.

Then we have

x^rt^s≤

i=0

x^rⁱt^sⁱ, x, t≥0.

Definition 3.25(Order functions). (i) A continuous functionO: [0,∞)→Ris called order func-tion if there existM ∈N,γ`>0,`= 1, . . . , M,m`(O), b`(O)∈R,`= 0, . . . , M, with

γ0:= 0< γ1< γ2<· · ·< γM < γM+1:=∞ such thatO(γ) =m`(O)·γ+b`(O)forγ∈[γ`, γ`+1),`= 0, . . . , M.

(ii) An order function Ois called increasing if

m_`−1(O)≤m_`(O), b_`−1(O)≥b_`(O), `= 1, . . . , M.

In this situation we have

O(γ) = max

`=0,...,M{b_`(O) +γ·m_`(O)}, 0< γ <∞.

(iii) An increasing order function O is called positive ifm`(O)≥0 for all`= 0, . . . , M.

(iv) An increasing order function O+ is called strictly positive if m`(O+) ≥0 and b`(O+)≥0 for all

`= 0, . . . , M. In this situation we define the weight function corresponding toO+ by Ξ_O₊(λ, z) := Ξ_ν(O₊₎(λ, z) = Ξ_N(ν(O₊₎₎(λ, z), (λ, z)∈C×Cⁿ where

ν(O+) :={(b`(O+), m`(O+)) :`= 0, . . . , M} ⊆[0,∞)².

(v) Furthermore, an order function O is called decreasing/negative/strictly negative if−O is increas-ing/positive/strictly positive.

(vi) An increasing or decreasing order function O is called monotone.

In this situation we define the weight function corresponding to an increasing order functionO by Ξ_O(λ, z) := Ξ_ν(O₊₎(λ, z)

Ξ_{(β,α)}(λ, z), (λ, z)∈C×Cⁿ with

α:=α_in(O) := |min{0, m0(O)}| ≥0, β :=βin(O) := |min{0, bM(O)}| ≥0 and the strictly positive order function

O+(γ) :=O(γ) +α·γ+β, γ >0.

If O is a decreasing order function then O⁰ := −Ois an increasing order function and we define the weight function corresponding toO by

ΞO(λ, z) := Ξ⁻¹_O0(λ, z) = Ξ_{(β,α)}(λ, z) Ξ_ν(O⁰

+)(λ, z), (λ, z)∈C×Cⁿ where α:=α_de(O⁰) :=α_in(−O)andβ:=β_de(O⁰) :=β_in(−O).

Remark 3.26. Note that the order function O(γ) :=mγ+b,m, b∈R, is an increasing order function as well as a decreasing order function. In general we get

(αin(O), βin(O))6= (αde(O), βde(O))

for such an order function. Nevertheless one easily validates that the weight function is the same either we interpret Oas increasing or decreasing.

Example 3.27. (i) The function

O(γ) := max

2,3 2 +1

2γ,1 + 3 4γ







2, γ∈[0,1),

2+¹₂γ, γ∈[1,2), 1 +³₄γ, γ≥2

is strictly positive with weight function Ξ_O(λ, z) = 1 +|z|²+|λ|^1/2|z|^3/2+|λ|^3/4|z|+|λ|^3/4. (ii) Let P ∈S[K](Lt×Lx) be an order-representative symbol andO+(γ) :=dγ(P),γ >0. Then O+

is a strictly positive order function and Ξ_P = Ξ_O₊.

(iii) We define the decreasing order functionO(γ) := min{0,−γ+ 2}. For the increasing order function O⁰(γ) :=−O(γ) = max{0, γ−2} we haveαin(O⁰) = 0andβin(O⁰) = 2. Hence, we have

O⁰₊(γ) :=O⁰(γ) + 2 = max{2, γ}.

According to Definition 3.25 (vi) we obtain the associated weight function by Ξ_O(λ, z) = Ξ_{(β,α)}(λ, z)

Ξ_ν(O⁰

+)(λ, z) = 1 +|z|²

1 +|λ|+|z|², (λ, z)∈C×Cⁿ. Definition 3.28. (i) LetN be an arbitrary Newton polygon with vertices

NV ={(rj, sj) :j= 0, . . . , J + 1} ⊆[0,∞)², J ∈N⁰

starting at the origin and indexed counter-clockwise. Then we can define the associated strictly positive order function by

ON(γ) := max

j=0,...,J+1{rj+γsj}, γ >0.

3.1: Inhomogeneous symbols and the Newton polygon 57 (ii) For a strictly positive order functionO+ we can define the associated Newton polygon by

N(O+) :=N(ν(O+)) andN_V(O₊)is defined as the set of the vertices of N(O₊).

Remark 3.29. LetO₊ be a strictly positive order function. For(x, y)∈[0,∞)² we have(x, y)∈N(O₊) if and only ifx+γy≤ O₊(γ) for allγ >0.

In the following chapters we often need estimates from above for symbols. This is why we introduce the concept of upper order functions.

Definition 3.30(Upper order function). LetObe an increasing or decreasing order function and let Q∈HP[K](L^◦t×L^◦x)be such that there existC=C(Q,O)>0 andλ0=λ0(Q,O)≥0 with

|Q[℘](λ, z)| ≤C·ΞO(λ, z)

for all(λ, z, ℘)∈L^◦_t×L^◦_x×K with|λ| ≥λ₀. Then Ois called an upper order function of Q.

Definition 3.31(Lower order function). Let Obe an increasing or decreasing order function and let Q∈HP[K](L^◦t×L^◦x)be such that there existC=C(Q,O)>0 andλ0=λ0(Q,O)≥0 with

|Q[℘](λ, z)| ≥C·Ξ_O(λ, z)

for all(λ, z, ℘)∈L^◦t×L^◦x×K with|λ| ≥λ0. Then Ois called a lower order function ofQ.

Example 3.32. (i) The symbol Q(λ, z) :=λ²−3λ+λz+λz²−3z³ for(λ, z)∈S_θ×Σ_δ fulfills

|Q(λ, z)| ≤ 3 |λ|²+|λ|+|λ||z|+|λ||z|²+|z|³

≤ 3 |λ|²+|λ|+|λ||z|+|λ||z|²+|z|³+ 1

≤ 5 |λ|²+|λ||z|²+|z|³+ 1

, (λ, z)∈Sθ×Σδ

by Remark 3.24 and (0,1),(1,1)∈conv{(0,2),(2,1),(3,0),(0,0)}. Defining O(γ) := max{2γ, γ+ 2,3}, γ >0

we get

Ξ_O(λ, z) =|λ|²+|λ||z|²+|z|³+ 1, (λ, z)∈S_θ×Σ_δ. So we obtain thatO is an increasing upper order function of Qwhereλ₀(Q,O) = 0.

(ii) Let ω(λ, z) := q

λ+|z|²₋, (λ, z) ∈ Sθ×Σⁿ_δ, θ > π/2. We have ω ∈ 0Hom(2,1, Sθ×Σⁿ_δ) and therefore Remark 3.4 (i) yields a constant C >0 such that

ω(λ, z) ≥ C(|λ|^1/2+|z|) = C

2(|λ|^1/2+ 2|z|+|λ|^1/2)

≥ C

2(|λ|^1/2+|z|+ 1)

holds for all (λ, z)∈Sθ×Σⁿ_δ with |λ| ≥1. Hence,O(γ) := max{γ/2,1},γ > 0, is a lower order function ofω with λ0(ω,O) = 0.

Lemma 3.33. Let O be an upper or lower order function forQ ∈H_P[K](S_θ×Σⁿ_δ), θ, δ > 0. In this situation there exists a constantµ=µ(λ₀, θ)>0 with |λ+µ| ≥λ₀ for all λ∈S_θ. With this we define the shifted symbol by Q_µ(λ, z) := Q(λ+µ, z) for all (λ, z) ∈ S_θ×Σⁿ_δ. Then O is also an upper or lower order function forQ_µ. In this situation we can even chooseλ₀(Q_µ,O) = 0 in Definition 3.30 and Definition 3.31.

Proof. Since Remark 3.24 and c|λ| ≤ |λ+µ| ≤ |λ|+µ, λ ∈ Sθ, it is easy to show that there exist C⁰, C⁰⁰>0such that

C⁰·Ξ_O(λ, z)≤Ξ_O(λ+µ, z)≤C⁰⁰·Ξ_O(λ, z) (3.3) for all(λ, z)∈Sθ×Σⁿ_δ. Now the assertion directly follows from (3.3) and the choice ofµ.

Lemma 3.34. LetP ∈S[K](Lt×Lx)be an order-representative symbol andO+a strictly positive order function with

d_γ(P)≤ O₊(γ), γ >0. (3.4)

ThenO+ is an upper order function ofP. In particular, this yields

|P[℘](λ, z)| ≤C·ΞP(λ, z), (λ, z)∈Lt×Lx, ℘∈K.

Proof. We have

|P[℘](λ, z)| ≤ X

`∈I

|χ_`(℘)||τ_`(λ, z)||ϕ_`(λ)||ψ_`(z)|

≤ C·X

`∈I

|λ|^M^`^+N^`^/ρ|z|^L^`+|λ|^M^`|z|^L^`^+N^` .

By virtue of (3.4) and Remark 3.19 we can show(L_`, M_`+N_`/ρ),(L_`+N_`, M_`)∈N(O+)for all`∈I.

The claim then follows from Remark 3.24 and Example 3.27 (ii).

Example 3.35. Recall the symbolψ(λ, z) :=√

−z²−√

λ−z²in Example 3.14 (ii). A naive analysis of the order structure of ψ shows that O+(γ) :=dγ(ψ) = max{1, γ/2} is an upper order function of ψ. In the following we show that there exists an upper order functionOwhich characterizes the order structure of ψbetter then O+.

Let ψ⁰(λ, z) :=√

−z²+√

λ−z² for(λ, z)∈Sθ×Σδ. For sufficiently small δwe get ψ⁰∈0Hom(2,1, S_θ×Σ_δ)

and therefore

|ψ⁰(λ, z)| ≥ 1

C ·(|λ|^1/2+|z|)≥ 1

2C ·(|λ|^1/2+|z|+|λ|^1/2)

≥ 1

2C ·(|λ|^1/2+|z|+ 1), (λ, z)∈Sθ×Σδ, |λ| ≥1 due to Remark 3.4. Consideringψ(λ, z)·ψ⁰(λ, z) =−λwe get

|ψ(λ, z)|=|λ/ψ⁰(λ, z)| ≤C· |λ|

|λ|^1/2+|z| ≤2C· 1 +|λ|

|λ|^1/2+|z|+ 1

for all (λ, z)∈Sθ×Σδ with|λ| ≥1 =:λ0. This yields|ψ(λ, z)| ≤C⁰·Ξ_O(λ, z) for the decreasing order function O(γ) := min{γ−1,1/2·γ},γ >0. HenceO is an upper order function ofψ.

In the next lines we want to show the relation between the summation of order functions and the multiplication of the associated weight functions. For this, we provide the following lemma.

Lemma 3.36. Let O+ be a strictly positive order function and letα, β≥0such that O₊⁰ :γ7→ O+(γ)−(β+αγ)

is also strictly positive. In this situation we get a constantC >0such that Ξ_O₊≤Ξ_{(β,α)}·Ξ_O⁰

+≤C·Ξ_O₊.

3.1: Inhomogeneous symbols and the Newton polygon 59 Proof. Due to the definition of ΞO+ and Ξ_O⁰

+ the left inequality is obvious. The right inequality follows from(b`(O+) +jβ, m`(O+) +kα)∈N(O+),j, k= 0,1, and Remark 3.24.

Lemma 3.37. (i) LetO+,1 andO+,2 be strictly positive order functions. ThenO+,1+O+,2 is also a strictly positive order function and there existC, C⁰ >0such that

C⁰·Ξ_O_+,1_+O_+,2(λ, z)≤Ξ_O_+,1(λ, z)·Ξ_O_+,2(λ, z)≤C·Ξ_O_+,1_+O_+,2(λ, z) for all(λ, z)∈C×Cⁿ.

(ii) The two-sided estimate in (i) also holds for increasing order functionsO1 andO2.

Proof. (i) The monotone structure of the strictly positive order functions ensures that the sum of strictly positive order functions is also strictly positive. For(λ, z)∈C×Cⁿ we trivially get

Ξ_O_+,1(λ, z)·Ξ_O_+,2(λ, z) = Ξ_ν(O_+,1₎(λ, z)·Ξ_ν(O_+,2₎(λ, z)

≥ Ξ_ν(O_+,1_+O_+,2₎(λ, z) = Ξ_O_+,1_+O_+,2(λ, z)

because ofν(O+,1+O+,2) ={(b`(O+,1) +b`(O+,2), m`(O+,1) +m`(O+,2)) :`= 0, . . . , M}. Due to Remark 3.24 it is sufficient to show

(b_`(O+,1) +b_p(O+,2), m_`(O+,1) +m_p(O+,2))∈N(O+,1+O+,2) (3.5) for all`, p= 0, . . . , M. Letγ∈[γk, γk+1)be arbitrary. Then we derive

1 γ

b_`(O_+,1) +b_p(O_+,2) m`(O+,1) +mp(O+,2)

= [m_`(O_+,1)γ+b_`(O_+,1)] + [m_p(O_+,2)γ+b_p(O_+,2)]

≤ [mk(O+,1)γ+bk(O+,1)] + [mk(O+,2)γ+bk(O+,2)]

= (O+,1+O+,2)(γ).

Remark 3.19 then yields (3.5).

(ii) The monotone structure of increasing order functions ensures that the sum of increasing order functions is also increasing. Let αk :=αin(Ok)and βk :=βin(Ok), k = 1,2. Defining the strictly positive order functions

O₁⁰(γ) :=O1+β1+α1γ, α12:=α1+α2, O₂⁰(γ) :=O2+β₂+α₂γ, β₁₂:=β₁+β₂ we get

Ξ_O₁·Ξ_O₂ = Ξ_O⁰

1·Ξ_O⁰

Ξ_{(β₁_,α₁_)}·Ξ_{(β₂_,α₂_)}.

Using Lemma 3.37 (i) for the nominator and the denominator we obtain the two-sided estimate C₁·Ξ_O₁·Ξ_O₂ ≤ Ξ_O⁰

1+O⁰₂

Ξ_{(β₁₂,α₁₂)}

≤C₂·Ξ_O₁·Ξ_O₂ (3.6)

forC₁, C₂>0. Due to Lemma 3.36 we get C⁰·Ξ_O₁_+O₂ ≤ Ξ_O⁰

1+O₂⁰

Ξ_{(β₁₂_,α₁₂_)} ≤C⁰⁰·Ξ_O₁_+O₂ (3.7) for certain constantsC⁰, C⁰⁰>0. The assertion now follows from (3.6) and (3.7).

Lemma 3.38. (i) Let O⁰ and O⁰⁰ be increasing order functions with O⁰(γ) ≤ O⁰⁰(γ) for all γ > 0.

Then there existsC >0 with

Ξ_O⁰(λ, z)

Ξ_O⁰⁰(λ, z) ≤C, (λ, z)∈C×Cⁿ.

(ii) LetO be an increasing order function and let{On}n=1,...,m be monotone order functions with

n=1

On(γ)≤ O(γ), γ >0.

Then there existsC >0 such that Qm

n=1Ξ_O_n(λ, z)

ΞO(λ, z) ≤C, (λ, z)∈C×Cⁿ. In generalPm

n=1On is not a monotone order function.

Proof. (i) Letα:= max{α_in(O⁰), α_in(O⁰⁰)}andβ := max{β_in(O⁰), β_in(O⁰⁰)}and define O₁(γ) :=O⁰(γ) +β+α·γ, O₂(γ) :=O⁰⁰(γ) +β+α·γ.

Then Lemma 3.36 yields

Ξ_O⁰(λ, z)

Ξ_O⁰⁰(λ, z)≤C⁰· Ξ_O₁(λ, z)

Ξ_O₂(λ, z), (λ, z)∈C×Cⁿ (3.8) with C, C⁰ >0. The order functionsO1 and O2 are strictly positive with O1(γ)≤ O2(γ), γ > 0.

Remark 3.19 showsN(O1)⊆N(O2)and therefore the assertion follows from (3.8) with Remark 3.24.

(ii) Let

M₊ := {n∈ {1, . . . , m}:On is an increasing order function}, M₋ := {n∈ {1, . . . , m}:On is a decreasing order function}.

Then we have

Qm n=1Ξ_O_n

Ξ_O =

n∈M+ΞOn

j∈M−Ξ_−O_j Ξ_O

. Using Lemma 3.37 we get

n=1Ξ_O_n(λ, z)

Ξ_O(λ, z) ≤C· Ξ_O⁰(λ, z)

Ξ_O⁰⁰(λ, z), (λ, z)∈C×Cⁿ whereO⁰:=P

n∈M₊On andO⁰⁰:=O −P

j∈M−Oj. Note thatO⁰ andO⁰⁰ are both increasing. As a consequence of the assumptions we have O⁰(γ)≤ O⁰⁰(γ) for all γ >0. Therefore we can prove part (ii) by (i).

Definition 3.39(Support and index of an order function). LetO be an order function. Then the support ofO is defined by

suppO:={i∈ {1, . . . , M}: (b_i−1(O), m_i−1(O))6= (bi(O), mi(O))}.

The constant I=I(O) := #(suppO) is called the index ofO. We define i0:= 0 andiI+1 :=M+ 1. If I6= 0, then we choose

iς=iς(O)∈suppO, ς = 1, . . . , I withi0< i1<· · ·< iI < iI+1. Note that we have

(b_i(O), m_i(O)) = (b_j(O), m_j(O)), i_ς≤i, j < i_ς+1

for all ς = 0, . . . , I. The order function O is said to be of trivial index if I(O) = 0, i.e. there exist m, b∈R withO(γ) =mγ+b for allγ >0.

3.1: Inhomogeneous symbols and the Newton polygon 61 Example 3.40. Let γ1:= 4,γ2:= 6,

O1(γ) :=







γ+ 2, γ∈[0, γ1), 3/2γ, γ∈[γ1, γ2), 3/2γ, γ∈[γ2,∞),

O2(γ) :=







1/2γ+ 2, γ∈[0, γ1), γ, γ∈[γ1, γ2), 2γ−6, γ∈[γ2,∞).

Then O₁ is of index 1 and O₂ is of index 2. Furthermore, we have i₁(O₁) = 1, i₁(O₂) = 1, and i2(O2) = 2.

Lemma 3.41. Let O1 andO2 be strictly positive order functions. Then we have (b`(O1) +bκ(O2), m`(O1) +mκ(O2))∈N(O1+O2) for all`, κ∈ {0, . . . , M}.

Proof. Leti∈ {0, . . . , M}andγ∈[γ_i, γ_i+1). Definition 3.25 (ii) implies

[m`(O1) +mκ(O2)]γ+b`(O1) +bκ(O2) =m`(O1)γ+b`(O1) +mκ(O2)γ+bκ(O2)

≤m_i(O₁)γ+b_i(O₁) +m_i(O₂)γ+b_i(O₂) = (O₁+O₂)(γ).

Hence Remark 3.29 completes the proof.

In Lemma 3.41 we have seen that the tuples(b`(O1)+bκ(O2), m`(O1)+mκ(O2))are always contained in the Newton polygon ofO1+O2. The next lemma provides more information on the position of these tuples inN(O1+O2).

Lemma 3.42. Let O1 andO2 be strictly positive order functions and let

`∈ {0, . . . , M}, ς ∈ {0, . . . , I(O₁)}, i_ς(O₁)≤` < i_ς+1(O₁).

Define

µ1 := max{µ∈ {0, . . . , I(O2) + 1}:iµ(O2)≤iς(O1)}, µ2 := min{µ∈ {0, . . . , I(O2) + 1}:iµ(O2)≥iς+1(O1)},

andN:=N(O1+O2). Furthermore, we defineΓ as the set of allx∈R² lying on a non-horizontal and non-vertical line between two vertices ofN.

(i) For allκ∈ {0, . . . , M} with iµ₁(O2)≤κ < iµ₂(O2)we have

(b`(O1) +bκ(O2), m`(O1) +mκ(O2)) = (bj(O1+O2), mj(O1+O2))∈NV

where

j:=







iς(O1), κ < iς(O1),

κ, i_ς(O1)≤κ < i_ς+1(O1), iς+1(O1)−1, κ≥iς+1(O1).

(ii) If µ16= 0 andiµ₁(O2) =iς(O1), then we have

(b_`(O₁) +b_κ(O₂), m_`(O₁) +m_κ(O₂))∈Γ\N_V for allκ∈ {0, . . . , M} with iµ₁−1(O2)≤κ < iµ₁(O2).

More precisely,(b_`(O₁) +b_κ(O₂), m_`(O₁) +m_κ(O₂))lies on the edge between the vertices (b_j−1(O1+O2), m_j−1(O1+O2))and(bj(O1+O2), mj(O1+O2))where j:=iς(O1).

(iii) Ifµ26=I(O2) + 1andiµ2(O2) =iς+1(O1), then we have

(b`(O1) +bκ(O2), m`(O1) +mκ(O2))∈Γ\NV

for all κ∈ {0, . . . , M} withi_µ₂(O₂)≤κ < i_µ₂₊₁(O₂).

More precisely, (b`(O1) +bκ(O2), m`(O1) +mκ(O2))lies on the edge between the vertices (bj−1(O1+O2), mj−1(O1+O2))and(bj(O1+O2), mj(O1+O2))wherej:=iς+1(O1).

(iv) For all κ∈ {0, . . . , M} not covered by the conditions of (i)-(iii) we have (b`(O1) +bκ(O2), m`(O1) +mκ(O2))∈N(O1+O2)\Γ.

Proof. (i) For κ and j as above we get the two equalities (b`(O1), m`(O1)) = (bj(O1), mj(O1))and (bκ(O2), mκ(O2)) = (bj(O2), mj(O2)). Then we have

(O1+O2)(γ) = [mj(O1) +mj(O2)]γ+bj(O1) +bj(O2)

= [m_`(O₁) +m_κ(O₂)]γ+b_`(O₁) +b_κ(O₂), γ∈(γ_j, γ_j+1).

This yields (b`(O1) +bκ(O2), m`(O1) +mκ(O2))∈NV.

(ii) Let iµ1−1(O2) ≤κ < iµ1(O2), µ1 6= 0, and iµ1(O2) = iς(O1) =: j >0. Then we simply obtain (b_j(O1), m_j(O1)) = (b_`(O1), m_`(O1)),(b_j−1(O2), m_j−1(O2)) = (b_κ(O2), m_κ(O2)), and

(O1+O2)(γj) = [mj(O1) +mj(O2)]γj+bj(O1) +bj(O2)

= [m_`(O₁) +m_κ(O₂)]γ_j+b_`(O₁) +b_κ(O₂) due tomj−1(O2)γj+bj−1(O2) =mj(O2)γj+bj(O2).

(iii) Let i_µ₂(O2)≤κ < i_µ₂₊₁(O2), µ₂ 6=I(O2) + 1, and i_µ₂(O2) = i_ς+1(O1) =: j < M+ 1. Then we have (b_j−1(O1), m_j−1(O1)) = (b_`(O1), m_`(O1)),(b_j(O2), m_j(O2)) = (b_κ(O2), m_κ(O2)), and

(O1+O2)(γj) = [mj(O1) +mj(O2)]γj+bj(O1) +bj(O2)

= [m_`(O₁) +m_κ(O₂)]γ_j+b_`(O₁) +b_κ(O₂) due tomj−1(O1)γj+bj−1(O1) =mj(O1)γj+bj(O1).

(iv) This can be seen by the same arguments as in the proof of the parts (i)-(iii).

The last two lemmas help us to understand the arithmetic of Newton polygons. Especially the last characterization turns out to be helpful when we consider compatibility conditions in Chapter 4.

Im Dokument General parabolic mixed order systems in Lp and applications (Seite 62-72)