Table Of ContentAn improved local well-posedness result for
8 the one-dimensional Zakharov system
0
0
2
Hartmut Pecher
n
Fachbereich Mathematik und Naturwissenschaften
a
J Bergische Universit¨at Wuppertal
3 Gaußstr. 20
2
D-42097 Wuppertal
] Germany
P
e-mail [email protected]
A
.
h
t
a Abstract
m
The 1D Cauchy problem for the Zakharovsystem is shown to be locally
[
well-posed for low regularity Schr¨odinger data u0 ∈ Hk,p and wave data
2
v (n0,n1) ∈ Hl,p ×Hl−1,p under certain assumptions ondthe parameters k,l
1 and 1 < p ≤d2, wherde ku0kHk,p :=khξiku0kLp′ , generalizing the results for
1 p = 2 by Ginibre, Tsutsumi, and Velo. Especially we are able to improve
8 c
the results from the scalingdpoint of view, and also allow suitable k < 0 ,
11 l < −1/2 , i.e. data u0 6∈ L2 and (n0,n1) 6∈ H−1/2 ×H−3/2, which was
6 excluded in the case p=2.
0
/
h
t 1 Introduction and main results
a
m
: Consider the (1+1)-dimensional Cauchy problem for the Zakharov system
v
i
X iu +u = nu (1)
t xx
ar ntt−nxx = (|u|2)xx (2)
u(0) = u , n(0) = n , n (0) = n (3)
0 0 t 1
where u is a complex-valued und n a real-valued function defined for (x,t) ∈
R×R+.
The Zakharov system was introduced in [Z] to describe Langmuir turbulence
in a plasma.
The Zakharov system (1),(2),(3) can be transformed into a first order system
in t as follows: With
1
n := n±iA−1/2n ,i.e.n = (n +n ), 2iA−1/2n = n −n , A:= −∂2
± t 2 + − t + − x
2000 Mathematics Subject Classification: 35Q55, 35L70
Key words and phrases: Zakharov system, well-posedness, Fourier restriction norm method
1
we get
1
iu +u = (n +n )u (4)
t xx + −
2
in ∓A1/2n = ±A1/2(|u|2) (5)
±t ±
u(0) = u , n (0) = n ±iA−1/2n =:n . (6)
0 ± 0 1 ±0
This problem was considered for data in L2-based Sobolev spaces in detail in the
last decade, especially low regularity local well-posedness results were given by
Ginibre, Tsutsumi, and Velo [GTV] for data u ∈ Hk , n ∈ Hl , n ∈ Hl−1
0 0 1
under the following assumptions:
1 1
− < k−l ≤ 1 , 2k ≥ l+ ≥ 0.
2 2
In this paper there were also given corresponding results in arbitrary space
dimension. It was also shown that these results are sharp within the used
method, namely the Fourier restriction norm method initiated by Bourgain and
Klainerman-Machedon and further developed by Kenig, Ponce, Vega and others.
It could also be shown by Colliander, Holmer, and Tzirakis [CHT], that global
well-posedness in the case k = 0 , l = −1/2 holds true. Holmer [H] was able to
show that the one-dimensional local well-posedness theory is sharp in the sense
that the problem is locally ill-posed in some cases, where the assumptions on k,l
in [GTV] are violated, more precisely: if 0 < k <1 and 2k >l+1/2 , or, if k ≤ 0
and l > −1/2, or, if k = 0 and l < −3/2. Moreover, the mapping data upon
solution is not C2, if k ∈ R , l < −1/2. Ill-posedness for k < 0 and l ≤ −3/2 was
shown by Biagioni and Linares [BL].
The minimal values k = 0 , l = −1/2 are far from critical, if one compares
them with those being critical for a scaling argument, namely k = −1 and l =
−3/2 . The heuristic scaling argument here is the following (for details we refer
1
to [GTV]): Ignoring the term A2n± in equation (5) the system (4),(5),(6) is
invariant under the dilation
u(x,t) −→ uµ(x,t) = µ32u(µx,µ2t) (7)
n (x,t) −→ n (x,t) = µ2n (µx,µ2t). (8)
± ±µ ±
Because
ku (x,0)k = µk+1ku k (9)
µ H˙k 0 H˙k
and
kn±µ(x,0)kH˙l = µl+23kn±0kH˙l (10)
thesystemiscriticalfork = −1andl = −3 . Ifnamelythelifespanof(u,n ,n )
2 + −
wereT thelifespanof(u ,n ,n )wouldbeTµ−2. So,ifk < −1orl < −3,one
µ +µ −µ 2
would have both the norm of the data and the lifespan of the solution (u,n ,n )
+ −
going to zero as µ → ∞ , which strongly indicates ill-posedness.
It is interesting to compare the situation with the corresponding problem for
the cubic Schro¨dinger equation
iu +u +|u|2u= 0 , u(0) = u (11)
t xx 0
2
which is known to be (globally) well-posed for data u ∈ Hs , s ≥ 0 [Y] (cf. also
0
[CW]) , and locally ill-posed for s < 0 [KPV3], whereas scaling considerations
suggest as the critical value s = −1/2. This problem is of special interest also for
the Zakharov system, because the cubic Schro¨dinger equation is the formal limit
for c→ ∞ of the Zakharov system modified by replacing ∂2−∂2 by c−2∂2−∂2.
t x t x
Now, for nonlinear Schro¨dinger equations it was suggested to leave the Hs-scale
of the data by Cazenave, Vega, and Vilela [CVV] and Vargas and Vega [VV]. For
the cubic Schro¨dinger equation local (and even global) well-posedness has been
shown for data with infinite L2-norm. A. Gru¨nrock [G2] was able to show in this
case local well-posedness for data u ∈ Hs,r , where
0
d
ku0kHs,r := khξisu0kLr′ , 1/r+1/r′ = 1,
ξ
c
if s ≥ 0 and 1 < r <d∞ . Moreover, he could show global well-posedness for
2 ≥ r ≥ 5/3 , u ∈ H0,r , and also local ill-posedness for the cubic Schro¨dinger
0
equation in Hs,r for any 1 < r < ∞ and −1/r′ < s < 0 . The well-posedness
d
results were proven by a modified Fourier restriction norm method (for p 6= 2),
d
which was developed by A. Gru¨nrock in [G1], where these ideas were applied to
the modified KdV equation.
The aim of the present paper is to prove local well-posedness results for the
Zakharov system with data u ∈ Hk,p , n ∈ Hl,p , n ∈ Hl−1,p under suitable
0 0 1
assumptions on k,l,p , which allow to weaken the assumptions on the data from
d d d
the scaling point of view, thus improving the L2-based results in this sense, and
also allow to get results for certain data u 6∈ L2 and (n ,n ) 6∈ H−1/2 ×H−3/2.
0 0 1
Details are given in section 2 and 3. Especially we can show that local well-
posedness holds for data (u ,n ,n ) ∈ Hk,p ×Hl,p ×Hl−1,p for suitable k < 0,
0 0 1
l < −1/2 , and 1 < p < 2 in contrast to the above-mentioned ill-posedness
d d d
results of Holmer [H] for theZakharov system, and also in contrast to Gru¨nrock’s
ill-posedness results for the cubic Schro¨dinger equation, so that the limit of the
c-dependent Zakharov system as c → ∞ must be singular. We are also able to
choose k = 0 and l > −1/2 , a choice which was not possible in the L2-case (cf.
[H] again).
WeproveourresultsbyamodificationoftheFourierrestrictionnormmethod,
originally due to J. Bourgain [B1],[B2], and derive the crucial estimates for the
nonlinearities using a variant of the Schwarz method introduced by Kenig, Ponce
and Vega [KPV1],[KPV2] adapted to the Lp-theory. In principle these estimates
are proven along the lines of [GTV].
We recall the modified Fourier restriction norm method in the following. For
details we refer to the paper of A. Gru¨nrock (cf. [G1], Chapter 2). Our solution
spaces are the Banach spaces
Xl,b := {f ∈S′(R2): kfk < ∞},
r Xrl,b
where l,b ∈ R , 1 < r < ∞ , 1/r+1/r′ = 1 and
1/r′
kfk := dξdτhξilr′hτ +φ(ξ)ibr′|fˆ(ξ,τ)|r′ ,
Xrl,b (cid:18)Z (cid:19)
3
where φ : R → R is a given smooth function of polynomial growth. The dual
space of Xl,b is X−l,−b , and the Schwartz space is densein Xl,r . Theembedding
r r′ p
Xl,b ⊂ C0(R,Hl,r) is true for b> 1/r . We have
r
kfkd = dξdτhξilr′hτibr′ F(e−itφ(−i∂x)f)(ξ,τ) r′ 1/r′
Xrl,b (cid:18)Z (cid:12) (cid:12) (cid:19)
(cid:12) (cid:12)
(cid:12) (cid:12)
and
kψeitφ(−i∂x)u k ≤ c ku k
0 Xrl,b ψ 0 Hl,r
for any ψ ∈ C∞(R ) .
0 t d
If v is a solution of the inhomogeneous problem
iv −φ(−i∂ )v = F , v(0) = 0
t x
and ψ ∈ C∞(R ) with suppψ ⊂ (−2,2) , ψ ≡ 1 on [−1,1] , ψ(t) = ψ(−t) ,
0 t
ψ(t) ≥ 0 , ψ (t) := ψ(t) , 0 < δ ≤ 1 , we have for 1 < r < ∞ , b′+1 ≥ b ≥ 0 ≥
δ δ
b′ > −1/r′ :
kψδvkXrl,b ≤ cδ1+b′−bkFkXrl,b′ .
For the reduced wave part φ(ξ) = ±|ξ| we use the notation Xl,b instead of Xl,b ,
±,r r
whereas for the Schro¨dinger part φ(ξ) = ξ2 we simply use Xl,b . We also use the
r
localized spaces
Xrl,b(0,T) := {f = f˜|[0,T]×R :f˜∈ Xrl,b},
where
kfkXrl,b(0,T) := inf{kf˜kXrl,b :f = f˜|[0,T]×R}.
Especially we use [G1], Theorem 2.3, which we repeat for convenience.
Theorem 1.1 Consider the Cauchy problem
u −iφ(−i∂ )u = N(u) , u(0) = u ∈ Hs,r, (12)
t x 0
d
where N is a nonlinear function of u and its spatial derivatives. Assume for given
s ∈ R , 1 < r < ∞ , α ≥ 1 there exist b > 1/r , b−1 < b′ ≤ 0 such that the
estimates
kN(u)kXrs,b′ ≤ ckukαXrs,b
and
kN(u)−N(v)kXrs,b′ ≤ c(kukαX−rs,1b +kvkαX−rs,1b)ku−vkXrs,b
are valid. Then there exist T = T(ku k ) > 0 and a unique solution u ∈
0 Hs,r
Xs,b[0,T] of (12). This solution belongs to C0([0,T],Hs,r) , and the mapping
r
u 7→ u , Hs,r → Xs,b(0,T ) is locally Lipsdchitz continuous for any T < T .
0 r 0 d 0
The madin result of this paper is the following
Theorem 1.2 Let 1 < p ≤ 2 , 1 + 1 = 1 , 1 ≥ b,b > 1 .
p p′ 1 p
• In the case k ≥ 0 assume
1 1 1
l ≥ − , k−l < 2(1−b ), l ≤2k− , l+1−k < +2(1−b), l+1−k ≤ 2b .
p 1 p′ p 1
4
• In the case k < 0 assume
1 1 1 1 1
k ≥ − , l ≥ − , l+k > −2b ,l+k > −2b, l+k > − −2(1−b ),
1 1
p p p p p
1 1
k−l < 2(1−b ), 2k > −b , 2k ≥ l+ , 2k >−(1−b).
1 p 1 p′
Let u ∈ Hk,p , n ∈ Hl,p . Then the Cauchy problem (4),(5),(6) is locally well-
0 ±0
posed, i.e.dthere exists da unique local solution u ∈ Xpk,b1(0,T) , n± ∈ X±l,b,p(0,T).
This solution satisfies u∈ C0([0,T],Hk,p) , n ∈ C0([0,T],Hl,p) , and the map-
±
ping data upon solution is locally Lipschitz continuous.
d d
The estimates for the nonlinearities are given in section 3 and the short proof
of this theorem as a consequence of these estimates in section 4.
Remark: The assumption n ∈ Hl,p requires n ,A−1/2n ∈ Hl,p. This last
±0 0 1
assumption on n1 can also be replaceddby the condition n1 ∈Hl−1,dp. One way to
see this is to modify the transformation of the original Zakharov system into the
d
first order system in t as follows: replace the wave equation by n −n +n =
tt xx
(|u|2) +n and define n := n±iA˜−1/2n , where A˜ := −∂2 +1. This leads to
xx ± t x
the modified reduced wave equation:
in ∓A˜1/2n = ±AA˜−1/2(|u|2)∓(1/2)A˜−1/2(n +n ).
±t ± + −
Now it is easy to see that this modified nonlinear term can be estimated exactly
in the same way as the original term A1/2(|u|2) , and also the additional linear
term is harmless. This remark was already used by [GTV].
Thus we have
Theorem 1.3 Let k,l,b,b ,p fulfill the assumptions of Theorem 1.2. Let u ∈
1 0
Hk,p , n ∈ Hl,p , n ∈ Hl−1,p . Then the Cauchy problem (1),(2),(3) is locally
0 1
well-posed, i.e. there exists a unique solution
d d d
u ∈Xk,b1(0,T), n ∈ Xl,b (0,T)+Xl,b (0,T), n ∈ Xl−1,b(0,T)+Xl−1,b(0,T).
p +,p −,p t +,p −,p
This solution satisfies
u∈ C0([0,T],Hk,p), n ∈C0([0,T],Hl,p), n ∈ C0([0,T],Hl−1,p),
t
d d d
and the mapping data upon solution is locally Lipschitz continuous.
We use the notation hλi := (1 + λ2)1/2 , and a± to denote a number slightly
larger (resp., smaller) than a .
2 Comparison with earlier results
It is interesting to compare our results with those of [GTV] for the case p = 2.
The lowest admissible choice in this case was k = 0 , l = −1/2 , p = 2. This is
contained in our results, too.
5
• A choice, which improves this result from the scaling point of view for the
Schro¨dinger part is k = 0 , p = 1+ǫ , −2 < l ≤ −1 (with b = b = 1+)
p′ p′ 1 p
and ǫ > 0 small. It is easily checked that this choice is admissible due to
Theorem 1.2.
Hk,p scales like Hσ , whereσ = k−1+1 , here: σ = 1− 1 → −1 (ǫ → 0),
p 2 2 1+ǫ 2
Hdl,p scales like Hλ , where λ = l− 1 + 1 , here: λ → −1 (ǫ → 0).
p 2 2
Tdhat Hk,p scales like Hσ here just means that (cf. (7) and (9)):
d
k|ξ|kuµ(ξ,0)kLpξ′ =:kuµ(x,0)kH˙k,p = µk−p1+32ku0kH˙k,p
c
d d
and
ku (x,0)k = µσ+1ku k
µ H˙σ 0 H˙σ
and the exponents of µ here coincide.
• Another admissible choice improving the result from the scaling point of
view for the wave part is k = 0 , l = −1 (with b = b = 1+) and 2 ≥ p > 3.
p 1 p 2
The conditions of Theorem 1.2 are fulfilled:
1. k−l < 2(1−b ) ⇔ 1 < 2(1− 1) ⇔ p > 3 ,
1 p p 2
2. l ≤ 2k− 1 ⇔ p ≤ 2 ,
p′
3. l+1−k < 1 +2(1−b) ⇔ −1 +1< 1 +2(1− 1) , which is fulfilled,
p p p p
and
4. l+1−k ≤ 2b ⇔ −1 +1 ≤ 2 ⇔ 1 ≤ 3 .
1 p p p
Hk,p scales like Hσ with σ = −1 + 1 → −1 (p → 3) , Hl,p scales like Hλ
p 2 6 2
wdith λ = −2 + 1 → −5 (p → 3) . d
p 2 6 2
It is also interesting to remark that it is possible to choose k < 0 and l < −1
2
(with a suitable 1 < p < 2) , and nevertheless achieve local well-posedness for
the Zakharov system (see details below). In this situation Holmer [H] proved in
the L2-case that themappingdata uponsolution in not C2, so that a contraction
mapping method as in our case cannot beapplied. Moreover, the cubicnonlinear
Schro¨dinger equation (11) is known to beill-posed for suitable data u ∈ Hk,p for
0
any −1 < k < 0 and p >1 (cf. [G2]). This equation, as already remarked in the
p′ d
introduction, is the formal limit as c → ∞ of a sequence of velocity-dependent
Zakharovsystems(replacing∂2−∂2 byc−2∂2−∂2). Sothislimitmustbesingular
x t x t
in some sense.
In order to determine the minimal k, which fulfills all the assumptions in
Theorem 1.2 we argue as follows:
1. The conditions 2k > 1 − b and k < l + 2(1 − b ) require 1 − 1b <
p 1 1 2p 2 1
l+2−2b ⇔ b < 2(l+2)− 1 .
1 1 3 3p
2. Theconditions2k ≥ l+ 1 andk < l+2(1−b )require l+ 1 < l+2−2b ⇔
p′ 1 2 2p′ 1
b < l + 3 + 1 .
1 4 4 4p
6
Thus b has to be chosen such that 1 < b < min(2(l+2)− 1 , l + 3 + 1 ) , so
1 p 1 3 3p 4 4 4p
that the condition 2k > 1 −b can only be fulfilled, if
p 1
1 2 1 4 2
2k > − (l+2)+ = − (l+2) (13)
p 3 3p 3p 3
and
1 l 3 1 3 1
2k > − − − = − (l+3). (14)
p 4 4 4p 4p 4
Moreover we need
1 1
2k ≥ l+ = l+1− . (15)
p′ p
The lower bound for 2k in (13) and (15) is minimized, if
4 2 1 1 5
− (l+2) = l+1− ⇔ = l+1. (16)
3p 3 p p 7
One easily checks that under this assumption all 3 lower bounds for 2k coincide.
Thus we end up with (from (13)):
4 5 2 2 l
2k > ( l+1)− (l+2) = l ⇔ k > .
3 7 3 7 7
The minimal and optimal choice for l here is l = −1 (because l ≥ −1), which
p p
means by (16): p = 12 , and thus k > − 1 (from k > l) , and by (16):
7 12 7
5 1 5 7
l = −1 = − ⇔ l = − .
7 p 12 12
Moreover, we should choose b < 2(l+2)− 1 = 3.
1 3 3p 4
Itis now completely elementary toseethat thatthechoice k = − 1 +ǫ , l = − 7 ,
12 12
b = b = 3 −ǫ , p = 12 (ǫ > 0 small) meets all the assumptions of Theorem 1.2.
1 4 7
In this situation we have: Hk,p scales like Hσ with σ = −1 +ǫ , and Hl,p
6
scales like Hλ with λ = −2 .
3 d d
ThisisanimprovementfromthescalingpointofviewforboththeSchro¨dinger
andthewavepart,comparedtotheL2-resultof[GTV],whereσ = 0andλ = −1.
2
3 Nonlinear estimates
In order to estimate the nonlinearities we use the following simple application of
H¨older’s inequality.
Lemma 3.1 For 1/p+1/p′ = 1 , 1 < p < ∞ , the following estimate holds:
| v(ζ)v (ζ )v (ζ )K(ζ ,ζ )dζ dζ |
1 1 2 2 1 2 1 2
Z Z
b c c 1/p
≤ sup(cid:18)Z |K(ζ1,ζ2)|pdζ2(cid:19) kv1kLpkvkLp′kv2kLp′ ,
ζ1
c b c
where ζ := ζ −ζ .
1 2
7
Proof:
| v(ζ)v (ζ )v (ζ )K(ζ ,ζ )dζ dζ |
1 1 2 2 1 2 1 2
Z Z
≤ kbv kc( |c v(ζ −ζ )v (ζ )K(ζ ,ζ )dζ |p′dζ )1/p′
1 Lp 1 2 2 2 1 2 2 1
Z Z
′
≤ kvc1kLp{ [( b|v(ζ1−ζ2c)v2(ζ2)|p′dζ2)( |K(ζ1,ζ2)|pdζ2)pp ]dζ1}1/p′
Z Z Z
≤ kcv k (sup |Kb (ζ ,ζ )|pcdζ )1/p( |v(ζ −ζ )v (ζ )|p′dζ dζ )1/p′
1 Lp 1 2 2 1 2 2 2 2 1
Z Z Z
ζ1
c 1/p b c
≤ sup(cid:18)Z |K(ζ1,ζ2)|pdζ2(cid:19) kv1kLpkvkLp′kv2kLp′ .
ζ1
c b c
Remark: Similarly one can prove
| v(ζ)v (ζ )v (ζ )K(ζ ,ζ )dζ dζ |
1 1 2 2 1 2 1 2
Z Z
b c c 1/p
≤ sup(cid:18)Z |K(ζ +ζ2,ζ2)|pdζ2(cid:19) kvkLpkv1kLp′kv2kLp′ .
ζ
b c c
Our first aim is to estimate the nonlinearity f = n u in Xk,−c1 for given
± p
n ∈ Xl,b and u ∈ Xk,b1 . We estimate f(ξ′,τ ) = (n ∗u)(ξ′,τ ) in terms of
± ±,p p 1 1 ± 1 1
n (ξ,τ) and u(ξ′,τ ) , where ξ = ξ′ −ξ′ , τ = τ −τ . We also introduce the
± 2 2 1 2 b 1 2c b
variables σ = τ +ξ′2 , σ = τ +ξ′2 , σ = τ ±|ξ| , so that
c 1 b 1 1 2 2 2
z := ξ′2−ξ′2∓|ξ|= σ −σ −σ. (17)
1 2 1 2
Define v2 = hξ2′ikhσ2ib1u and v = hξilhσibn± , so that kukXpk,b1 = kv2kLp′ and
kn±kX±l,cb,p = kvkLp′ . Inborder tbo estimate fcin Xpk,−c1 we take its scalacr product
with a functiobn in Xp−′k,c1 with Fourier transform hξ1′ikhσ1i−c1v1 with v1 ∈ Lp .
In the sequel we want to show an estimate of the form
c c
|S| ≤ ckvkLp′kv1kLpkv2kLp′ ,
b c c
where
|vv v |hξ′ik
S := 1 2 1 dξ′dξ′dτ dτ .
Z hσibhσ1bicc1hcσ2ib1hξ2′ikhξil 1 2 1 2
This directly gives the desired estimate
kn±ukXpk,−c1 ≤ ckn±kX±l,b,pkukXpk,b1 . (18)
Proposition 3.1 The estimate (18) holds under the following assumptions:
k ≥ 0, l ≥ −1/p, k−l ≤ 2c , k−l ≤ 2/p,
1
where c ≥ 0, b> 1/p, b > 1/p, 1< p ≤ 2 .
1 1
8
Remark: We simplify (17) as follows. If (17) holds with the minus sign and if
ξ′ ≥ ξ′ (resp., ξ′ ≤ ξ′) , we have
1 2 1 2
z = ξ′2−ξ′2−|ξ′ −ξ′|= (ξ′ ∓1/2)2 −(ξ′ ∓1/2)2 = ξ2−ξ2
1 2 1 2 1 2 1 2
where ξ = ξ′ ∓1/2 . Thus the region ξ′ ≥ ξ′ (resp., ξ′ ≤ ξ′) of S is majorized
i i 1 2 1 2
by
|v(ξ,τ)v (ξ ±1/2,τ )v (ξ ±1/2,τ )|hξ ik
1 1 1 2 2 2 1
S = c dξ dξ dτ dτ ,
Z b chσibhσ1ic1hσ2cib1hξ2ikhξil 1 2 1 2
where now
z = ξ2−ξ2 = σ −σ −σ , ξ = ξ −ξ , τ = τ −τ (19)
1 2 1 2 1 2 1 2
σ =τ +(ξ ±1/2)2 , σ = τ ±|ξ| = τ ±|ξ −ξ |.
i i i 1 2
Also, theplussignin(17)canbetreated similarlybyagaindefiningξ = ξ′±1/2.
i i
If one wants to estimate S by ckvkLp′kv1kLpkv2kLp′ , the variables ξi and ξi±1/2
are completely equivalent, thus we do not distinguish between them.
b c c
Proof of Proposition 3.1: According to Lemma 3.1 we have to show
dξ dσ
Cp := suphσ i−c1phξ ikp 2 2 < ∞.
ξ1,σ1 1 1 Z hσibphσ2ib1phξilphξ2ikp
Case 1: |ξ |≤ 2|ξ | (⇒ |ξ| ≤ 3|ξ |) .
1 2 2
If |ξ |≤ 1 we have hξi ∼ 1 and thus
2
∞
Cp ≤ c sup dξ hσ i−b1pdσ < ∞,
2 2 2
Z Z
ξ1,σ1 |ξ2|≤1 0
because b > 1/p . If |ξ | ≥ 1 we get from (19) for ξ ,σ ,σ fixed: dσ = 2ξ ,
1 2 1 1 2 dξ2 2
and thus
Cp ≤ c sup hξi−lphξ i−1hσi−bphσ i−b1pdσdσ .
2 2 2
Z
ξ1,σ1
For l ≥ 0 we immediately have
Cp ≤ c hσi−bpdσ hσ i−b1pdσ < ∞,
2 2
Z Z
whereas for l ≤ 0 we use our assumption l ≥ −1/p and get the same bound by
using hξi−lphξ i−1 ≤ chξ i−lp−1 ≤ c .
2 2
Case 2: |ξ |≥ 2|ξ | (⇒ |ξ| ∼ |ξ |) .
1 2 1
From (19) we conclude ξ2 ≤ c(|σ |+|σ |+|σ|) and distinguish three cases.
1 1 2
Case 2a: |σ | dominant, i.e. |σ | ≥ |σ |,|σ| , (⇒ ξ2 ≤ c|σ |) .
1 1 2 1 1
This implies, using our assumption k−l ≤ 2c :
1
Cp ≤ c suphξ i(k−l−2c1)p dξ dσ hσi−bphσ i−b1p
1 2 2 2
Z
ξ1,σ1
≤ c sup dξ dσ hσi−bphσ i−b1p.
2 2 2
Z
ξ1,σ1
9
If |ξ | ≤ 1 this is immediately bounded by c dξ dσ hσ i−b1p < ∞ ,
2 |ξ2|≤1 2 2 2
whereas for |ξ | ≥ 1 we use dσ = 2ξ ∼ 2hξ i agaRin and geRt the bound
2 dξ2 2 2
c dσhσi−bp dσ hσ i−b1p < ∞,
2 2
Z Z
using b,b > 1/p .
1
Case 2b: |σ | dominant ( ⇒ ξ2 ≤ c|σ | ).
2 1 2
We have
Cp ≤ c suphξ i(k−l)p dξ dσ hσi−bphσ i−b1phξ i−kp.
1 2 2 2 2
Z
ξ1,σ1
The case k ≤ l is simple and leads to
Cp ≤ c sup dξ dσ hσi−bphσ i−b1p,
2 2 2
Z
ξ1,σ1
which can be handled as in case 2a.
The case k > l is treated as follows:
Cp ≤ c sup dξ2dσ2hσi−bphσ2i−b1p+(k−2l)phξ2i−kp.
Z
ξ1,σ1
Substituting y = ξ2 , thus dξ = dy , leads to
2 2 2|y|1/2
Cp ≤ c sup |y|−12hyi−k2p hσi−bphσ2i−b1p+(k−2l)pdσ2dy.
Z Z
ξ1,σ1
From (19) we have hσi = hσ −(σ −ξ2+y)i , and thus by [GTV], Lemma 4.2,
2 1 1
using b,b > 1/p :
1
dσ2hσi−bphσ2i−b1p+(k−2l)p ≤ chσ1−ξ12+yi−1+(k−2l)p−,
Z
(k−l)p
because −b p+ < −1+1 = 0 using our assumptions b > 1/p and k−l ≤
1 2 1
2/p. Thus
Cp ≤ c sup Z |y|−21hyi−k2phσ1−ξ12+yi−1+(k−2l)p−dy.
ξ1,σ1
The supremum occurs for σ = ξ2 by [GTV], Lemma 4.1, so that
1 1
Cp ≤ c |y|−12hyi−k2phyi−1+(k−2l)p−dy = |y|−12hyi−1−l2p−dy < ∞,
Z Z
because l ≥ −1/p .
Case 2c: |σ| dominant.
This case can be treated like case 2b, which completes the proof.
It is also possible to prove (18) in certain cases where k is negative. This is done
in the following
10
