Table Of ContentAsymptotic stability of ground states in some
2 Hamiltonian PDEs with symmetry
1
0
2 Dario Bambusi
b
e 24.02.12
F
4
2 Abstract
Weconsideragroundstate(soliton)ofaHamiltonianPDE.Weprove
]
S thatifthesoliton isorbitally stable,thenit isalso asymptotically stable.
D The main assumptions are transversal nondegeneracy of the manifold of
. the ground states, linear dispersion (in the form of Strichartz estimates)
h
andnonlinearFermiGoldenRule. Weallowthelinearizationoftheequa-
t
a tionatthesolitontohaveanarbitrarynumberofeigenvalues. Thetheory
m is tailor made for the application to the translational invariant NLS in
space dimension 3. The proof is based on the extension of some tools of
[
the theory of Hamiltonian systems (reduction theory, Darboux theorem,
3 normal form) to the case of systems invariant under a symmetry group
v with unboundedgenerators.
5
3
8 1 Introduction
5
.
7 In this paper we study the asymptotic stability of the ground state in some
0
dispersive Hamiltonian PDEs with symmetry. We will prove that, in a quite
1
generalsituation, an orbitally stable ground state is also asymptotically stable.
1
: In order to describe the main result of the paper we concentrate on the specific
v
modelgivenbythe translationallyinvariantsubcriticalNLSinspacedimension
i
X 3, namely
r
a ψ =i∆ψ+iβ′(ψ 2)ψ , β(k)(u) C u 1+p−k , β′(0)=0 . (1.1)
t k
| | | |≤ h i
p< 2, x R3. It is well known that, under suitable assumptions on β, such an
3 ∈
equationhasafamilyofgroundstateswhichcantravelatanyvelocityandwhich
are orbitally stable (see e.g. [FGJS04] for a review). Consider the linearization
of the NLS at the soliton, and let L be the linear operator describing such
0
a linearized system. Due to the symmetries of the system, zero is always an
eigenvalue of L with algebraic multiplicity at least 8. In the case where this
0
is the exact multiplicity of zero and L has no other eigenvalues, asymptotic
0
stability was proven in [BP92, Cuc01] (see also [Per11]). Here we tackle the
casewhereL hasanarbitrarynumberofeigenvalues,disjointfromtheessential
0
spectrum,andprovethat,assumingasuitableversionoftheFermiGoldenRule
1
(FGR),the groundstateis (orbitally)asymptoticallystable. We recallthatthe
importance ofthe FGRin nonlinearPDEs wasunderstoodby Sigal[Sig93]and
shown to have a crucial role in the study of asymptotic stability in [SW99].
Similar conditions have been used and generalizedby many authors. The FGR
thatweusehereisageneralizationofthatof[GW08](seealso[BC11,Cuc11a]).
The presentpaperis a directdevelopmentof[BC11] and[Cuc11a], which in
turn are strongly related to [GS07, GNT04, CM08, GW08]. We recall that in
[BC11]Hamiltoniananddispersivetechniqueswereusedtoprovethattheempty
stateofthenonlinearKleinGordonequationisasymptoticallystableeveninthe
presenceofdiscretespectrumofthelinearizedsystem. Then[Cuc11a]extended
the techniques of [BC11] to the study of the asymptotic stability of the ground
state in the NLS with a potential.
The mainnoveltyofthe presentpaper is thatwe dealhere withthe transla-
tional invariant case. The new difficulty one has to tackle is related to the fact
that the groupof the translations ψ(.) ψ(. te ) is generatedby ∂ which
7→ − i − xi
is an unbounded operator: it turns out that this obliges to use non smooth
maps in order to do some steps of the proof. To overcome this problem we
introduceandstudy asuitableclassofmaps,thatwecall“almostsmooth”(see
in particular sect.3.2). We use them to develop Hamiltonian reduction theory,
Darboux theorem and also canonical perturbation theory.
The fact that the generator of the translations is not smooth causes some
difficulties also in the use of Strichartz estimates, but such difficulties were
already overcome by Perelman [Per11] (see also [Bec11]), so we simply apply
her method to our case.
We now describe the proof. First, we use Marsden Weinstein reduction
procedure in order to deal with the symmetries. In order to overcome the
problems related to the fact that the generators of the symmetry group are
unbounded, we fix a concrete local model for the reduced manifold and work
in it. The local model is a submanifold contained in the level surface of the
integrals of motion. The restriction of the Hamiltonian and of the symplectic
form to such a submanifold give rise to the Hamiltonian system one has to
study. The advantage of such an approach is that the ground state appears
as a minimum of the Hamiltonian, so one is reduced to study the asymptotic
stability of an elliptic equilibrium, a problem close to that stidued in [BC11].
Howevertheapplicationofthemethodsof[BC11,Cuc11a]tothepresentcaseis
far from trivial, since the restriction of the symplectic form to the submanifold
turns out to be in noncanonical form, and to have non smooth coefficients
(some “derivatives” appear). So, we proceed by first proving a suitable version
of the Darboux theorem which reduces the symplectic form to the canonical
one. This requires the use of non smooth transformations. We point out that
a key ingredient of our developments is that the ground state is a Schwartz
function, and this allows to proceed by systematically moving derivatives from
the unknown function to the ground state.
ThenwestudythestructureoftheHamiltonianintheDarbouxcoordinates
2
andprovethatithasaprecise(andquitesimple)form. Subsequently,following
[BC11, Cuc11a], we develop a suitable version of normal form theory in order
to extract the essential part of the coupling between the discrete modes and
the continuous ones. Here we greatly simplify the theory of [BC11, Cuc11a].
In particular we think that we succeeded in developing such a theory under
minimal assumptions. We also point out that in the present case the canonical
transformations putting the system in normal form are not smooth, but again
almost smooth.
Finally, following the scheme of [GNT04, CM08, BC11, Cuc11a], we use
Strichartz estimates in order to prove that there is dispersion, and that the
energy in the discrete degrees of freedomgoes to zero as t . As we already
→∞
remarked there are some difficulties in the linear theory, difficulties that we
overcomeusingthe methods of[Per11]. Inthis part,wemadeaneffortto point
out the properties that the nonlinearity has to fulfill in order to ensure the
result. Thus we hope to have proved a result which can be simply adapted to
different models.
We now discuss more in detail the relation with the paper [Cuc11a]. In
[Cuc11a]CuccagnastudiedthecaseofNLSwithapotentialandprovedaresult
similar to the present one. Here we generalize Cuccagna’s result in several
aspects. The first one is that we allow the system to have symmetry groups
with more than one dimension, but the main improvement we get consists of
the fact that we allow the symmetries to be generated by unbounded operators
(as discussed above). Furthermore we work in an abstract framework.
Finally, we work here on the reduced system (according to Marsden-Wein-
stein theory), but we think that all the arguments developed in such a context
could be reproduced also working in the original phase space. We also expect
that the same (maybe more) difficulties will appear also when working in the
original phase space.
Three days before the first version of this paper was posted in Arxiv, the
paper [Cuc11b] was also posted there. The paper [Cuc11b] deals exactly with
the same problem. The result of [Cuc11b] is very close to the present one, but
weaker: the result of such a paper is valid only for initial data of Schwartz
class, while the control of the difference between the soliton and the solution
is obtained in energy norm, and no decay rate is provided. Such a kind of
conclusions is usual for initial data in the energy space, while the typical result
valid for solutions corresponding to initial data decaying in space also controls
the rate of decay of the solution to the ground state. On the contrary, in the
present paper we give a result valid for any initial datum of finite energy (and
of course we do not deduce a decay rate).
A further difference between the two papers is that, here a large part of the
proof is developed in an abstract framework, thus we expect our result to be
simply applicable also to different systems. We are not aware of other papers
in the domain of asymptotic stability in dispersive Hamiltonian PDEs in which
the proof is developed in an abstract framework.
3
Our proof is also much simpler than that of [Cuc11b], indeed in order to
generate the flow of the transformation introducing Darboux coordinates (and
the transformations putting the system in normal form) we use a techinique
coming from the theory of semilinear PDEs, while [Cuc11b] uses techniques
coming from quasilinear PDEs.
A further difference, is that we work using Marsden Weinstein reduction,
while [Cuc11b] works in the original phase space.
Thepaperisorganizedasfollows: insect. 2westateourmainresultforthe
NLS; in sect. 3 we set up the abstract framework in which we work and state
andprove the Darboux theoremmentioned above;in sect. 4 we study the form
of the Hamiltonian in the Darboux coordinates. In sect. 5 we use canonical
perturbation theory in order to decouple as far as possible the discrete degrees
of freedom from the continuous ones; in sect. 6 we prove that the variables
corresponding to the continuous spectrum decay dispersively and the variables
corresponding to the discrete spectrum decay at zero; here the main abstract
theorem6.1is statedandproved;insect. 7we applythe abstracttheoryto the
NLS. In the first Appendix we prove that the dynamics of the reduced system,
while in the second one, we reproduce Perelman’s Lemma on the dispersion of
the linear system.
Acknowledgments. First, I would like to warmly thank Galina Perelman for
sending me her notes on asymptotic stability of solitons in energy space. Dur-
ing the preparation of this paper I benefit of the constructive criticism and of
the suggestions by many persons. In particular I would like to thank N. Burq,
P. Gerard, S. Gustafson, T. Kappeler, E. Terraneo. In the second version of
[Cuc11b] some criticisms are raised on a previous version of the present paper.
The anaylisis stimulated by such criticisms led me to a considerable simplifica-
tion of the proof.
2 Asymptotic stability in NLS
We state here our result on the NLS eq. (1.1). We assume
(H1) There exists an open interval R such that, for the equation
I ⊂ E ∈I
∆b β′(b2)b + b =0 , (2.1)
− E − E E E E
admitsaC∞ familyofpositive,radiallysymmetricfunctionsb belonging
E
to the Schwartz space.
d
(H2) One has b 2 >0, .
d k EkL2 E ∈I
E
Then one can construct traveling solitons, which are solutions of (1.1) of the
form
ψ(x,t)=e−i(cid:18)E−|v4|2(cid:19)te−iv2·xbE(x vt) . (2.2)
−
4
(H3) Consider the operators
A := ∆+ β′(b2) , A := ∆+ β′(b2) 2β′′(b2)b2 , (2.3)
+ − E − E − − E − E − E E
then the Kernel of the operator A is generated by b and the Kernel of
+ E
the operator A is generated ∂ b , j =1,2,3.
− j E
Remark 2.1. Under the aboveassumptions the solutions (2.2)areorbitally sta-
ble (see e.g. [FGJS04]).
In order to state the assumptions on the linearization at the soliton insert
the following Ansatz in the equations
ψ(x,t)=e−i(cid:18)E−|v4|2(cid:19)te−iv2·x(bE(x vt)+χ(x vt)) , (2.4)
− −
and linearize the so obtained equation in χ. Then one gets an equation of the
form χ˙ = L χ with a suitable L . It can be easily proved that the essential
0 0
spectrum of L is i[ ,+ ) and that 0 is always an eigenvalue. The rest
0 ±± E ∞
ofthe spectrumconsistsofpurely imaginaryeigenvalues iω ,thatwe orderas
j
S ±
follows 0<ω ω ... ω . We assume that
1 2 K
≤ ≤ ≤
(H4) ω < . Furthermore, let r be the smallest integer number such that
K t
r ω >E, then we assume ω k = , k ZK : k 2r .
t 1 t
E · 6 E ∀ ∈ | |≤
(H5) i are not resonances of L .
0
±E
(H6) The Fermi Golden Rule (6.52) holds.
The main theorem we are now going to state refers to initial data ψ which
0
are sufficiently close to a ground state. In its statement we denote by ǫ the
quantity below
ǫ:=q04∈R,q0∈Ri3n,vf0∈R3,E0∈I(cid:13)ψ0−e−iq04e−iv02·xbE0(x−q0)(cid:13)H1 (2.5)
(cid:13) (cid:13)
Theorem 2.2. Assume ǫ is small e(cid:13)nough, then there exist C1 fu(cid:13)nctions
(t),v(t),q4(t),q(t),y4(t),y(t) ,
E
and ψ H1 such that the solution ψ(t) with initial datum ψ admits the
+ 0
∈
decomposition
ψ(x,t)=e−iq4(t)e−iv(t2)·xbE(t)(x q(t))+e−iy4(t)χ(x y(t),t) (2.6)
− −
and
lim χ(t) eit∆ψ =0 . (2.7)
t→+∞ − + H1
Furthermore the functions (t)(cid:13)(cid:13),v(t),q˙4(t),y˙4((cid:13)(cid:13)t),q˙(t),y˙(t) admit a limit as t
E →
+ .
∞
The rest of the paper is devoted to the proof of an abstract version of this
theorem.
5
3 General framework and the Darboux theorem
Consider a scale of Hilbert spaces k, k Z. The scalar product in 0 will
H ∈ H
be denoted by .;. ; such a scalar product is also the pairing between k and
h i H
−k. We will denote ∞ := k, and −∞ := k. Let E : k k, k
k k
H H ∩ H H ∪ H H →H ∀
be a linear continuous operator skewsymmetric with respect to .;. . Assume
h i
it is continuously invertible. Let J : k k be its inverse (Poisson tensor).
H → H
We endow the scale by the symplectic form ω(U ,U ) := EU ;U , then the
1 2 1 2
h i
Hamiltonian vector field X of a function H is defined by X = J H,where
H H
∇
H is the gradient with respect to the scalar product of 0.
∇ H
Remark 3.1. In the application to dispersive equations one has to deal with
weighted Sobolev space Hk1,k2, which are labeled by a couple of indexes. All
what follows holds also in such a situation provided one defines the notation
(k ,k )>(l ,l ) by k >l and k l .
1 2 1 2 1 1 2 2
≥
For j = 1,...,n, let Aj : k k−dj, k Z and some dj 0, be n
H → H ∀ ∈ ≥
bounded selfadjoint (with respect to .;. ) linear operators, and consider the
h i
Hamiltonian function (u) := A u;u /2. Then X = JA generates a flow
Pj h j i Pj j
in 0 denoted by etJAj.
H
Remark 3.2. In the case of multiple indexes the index d represents the loss of
j
smoothnessandalwaysactsonlyonthe firstindex,namelyonehasAj k1,k2
H ⊂
k1−dj,k2.
H
Remark 3.3. The operatorsJA willplaythe roleofthe generatorsofthe sym-
j
metriesoftheHamiltoniansystemwewillstudy. Correspondinglythefunctions
will be integrals of motion.
j
P
We denote d := max d . For i,j = 1,...,n we assume that, on ∞
A j=1,...,n j
H
one has
(S1) [A ,E]=0,
j
(S2) A JA =A JA which implies , =0= A u;JA u .
i j j i j i j i
{P P } h i
(S3) For any t R the map etJAj leaves invariant ∞.
∈ H
Let A be a linear operator with the same properties of the A ’s. Assume
0 j
d d . The Hamiltonian we will study has the form
0 A
≥
1
H(u)= (u)+H (u) , (u)= u;A u , (3.1)
0 P 0 0
P P 2h i
where H is a nonlinear term on which we assume
P
(P1) There exists k0 and an open neighborhood of zero k0 k0 such that
HP C∞( k0,R). U ⊂ H
∈ U
We also assume that (on ∞)
H
(S4) H and Poisson commutes with each one of the functions :
P 0 j
P P
; = H ; =0 , j =1,...,n (3.2)
0 j P j
{P P } { P }
6
Weareinterestedinboundstatesη,namelyinphasepointssuchthatu(t):=
etλjJAjη is a solution of the Hamilton equations of H. Here and below we use
Einstein notation according to which sum over repeated indexes is understood.
Theindexes willalwaysrunbetween1 andn. Thenη hastofulfill the equation
A η+ H (η) λjA η =0 . (3.3)
0 P j
∇ −
We assume
(B1) There exists an open set I Rn and a C∞ map
⊂
I p (η ,λ(p)) ∞ Rn ,
p
∋ 7→ ∈H ×
s.t. (η ,λ(p)) fulfills equation (3.3). Furthermore the map p λ is 1 to
p
→
1.
(B2) Foranyfixedp∈I,thesetC := q∈RneqjJAjηpisasmoothndimensional
submanifold of ∞.
H S
(B3) The manifold η is isotropic,namely the symplectic formω vanishes
p∈I p
on its tangent space.
S
By (B1) it is possible to normalize the values of p in such a way that
j
(η )=p ,Formnowonwewillalwaysassumesuchaconditiontobesatisfied.
j p j
P
Remark 3.4. By the proof of Arnold Liouville’s theorem, the manifold of
hypothesis (B2) is diffeomorphic to Tk Rn−k, where T=R/2πZ. C
×
Consider the symplectic manifold
:= eqjJAjη ,
p
T
q∈Rn,p∈I
[
namely the manifold of bound states; its tangent space is given by
∂η
p
T :=span JA η , (3.4)
ηpT j p ∂p
(cid:26) j(cid:27)
and its symplectic orthogonal Tω is given by
ηpT
Tω
ηpT
∂η ∂η
= U −∞ :ω(JA η ;U)= A η ;U =ω p;U = E p;U =0
j p j p
∈H h i ∂p h ∂p i
(cid:26) (cid:18) j (cid:19) j (cid:27)
Lemma 3.5. One has −∞ =Tω T . Explicitly the decomposition of a
H ηpT ⊕ ηpT
vector U −∞ is given by
∈H
∂η
U =P p +QjJA η +Φ , (3.5)
j j p p
∂p
j
7
with
∂η
Qj = E p;U , P = A η ;U (3.6)
j j p
−h ∂p i h i
j
and Φ Tω given by
p ∈ ηpT
∂η ∂η
p p
Φ =Π U :=U A η ;U + E ;U JA η . (3.7)
p p j p j p
−h i∂p h ∂p i
j j
Proof. The first of (3.6) is obtained taking the scalar product of (3.5) with
E∂ηp, and exploiting
− ∂pj
∂η ∂η
p p
E , =0 (3.8)
h ∂p ∂p i
j k
which is equivalent to (B3). Taking the scalar product of (3.5) with A η we
j p
get the second of (3.6). Then (3.7) immediately follows.
Remark 3.6. A key point in all the developments of the paper is that the pro-
jector Π defined by (3.7) is a smoothing perturbation of the identity, namely
p
1l Π C∞(I,B( −k, l)), k,l,where B( −k, l) is the spaceof bounded
p
− ∈ H H ∀ H H
operatorsfrom −k to l. Inparticularonehasthat ∂Πp C∞(I,B( −k, l))
H H ∂pi ∈ H H
An explicit computation shows that the adjoint of Π is given by
p
∂η ∂η
Π∗U :=U p;U A η + JA η ,U E p . (3.9)
p −h∂p i j p h j p i ∂p
j j
Some useful formulae are collected below
∂Π ∂Π2 ∂Π ∂Π
EΠ =Π∗E , JΠ∗ =Π J , p = p =Π p + pΠ . (3.10)
p p p p ∂p ∂p p ∂p ∂p p
j j j j
∂Π ∂Π ∗ ∂Π∗ ∂Π ∂Π∗
p p p p p
Π Π =0 , = , E = E . (3.11)
p p
∂p ∂p ∂p ∂p ∂p
j (cid:18) j (cid:19) j j j
In the following we will work locally close to a particular value
p I. Thus we fix it and define
0
∈
k :=Π k (3.12)
V p0H
which we endow by the topology of k. Similarly we define ( k)∗ := Π∗ k.
H V p0H
When we do not put an exponent we mean 0.
V
Remark 3.7. For any positive k,l, one has
(Π Π Π )u C p p′ u , (3.13)
k p p′ − p′ kHk ≤ k,l| − |k kH−l
and, by the first of (3.11), for φ ∞, one has
∈V
∂Π
p
Π φ C p p φ . (3.14)
p ∂p ≤ kl| − 0|k kH−l
(cid:13) j (cid:13)Hk
(cid:13) (cid:13)
(cid:13) (cid:13)
(cid:13) (cid:13)
8
Remark 3.8. Consider the operator Π : −∞ Π −∞; it has the structure
p p
V → H
1l+(Π Π ), and one has
p− p0
(Π Π )φ C p p φ . (3.15)
k p− p0 kHk ≤ | − 0|k kH−l
−1 −1
Thus,by Neumannformulathe inverseΠ ofΠ has the formΠ = 1l+S
p p p
with S sulfilling (3.15).
f f
3.1 Reduced manifold
We introduce now the reduced symplectic manifold obtained by exploiting the
symmetry. In the standard case where the generators of the symmetry group
are smooth (i.e. d =0) the construction is standard and goes as follows.
j
Fix p I as above and define a surface = u : (u)=p , then pass
0 j 0j
to the quot∈ientwith respectto the groupactSiono{fRn oPn defined}by (q,u)
S 7→
eqjJAju,obtainingthereducedphasespaceM. AlocalmodelofMclosetoηp0
isobtainedby takinga codimensionnsubmanifoldof transversalto theorbit
S
of the group. Here we proceed the other way round: we choose a submanifold
of codimension n, transversal to the orbit of the group at η , and we
M ⊂ S p0
study the Hamiltonian system obtained by restricting the Hamiltonian to .
M
Consider the map
I (p,φ) i (p,φ):=η +Π φ ; (3.16)
0 p p
×V ∋ 7→
we will use the implicit function theorem(see lemma 3.11)in order to compute
p =p (φ) in such a way that the image of the map
j j
φ i(φ):=η +Π φ , (3.17)
p(φ) p(φ)
V ∋ 7→ ⊂S
is the wanted local model of , and i is a local coordinate system in it. In
M
studying this map we will use a class of maps which will play a fundamental
role in the whole paper. In the correspondingdefinitions we will consider maps
from Rn k to some space. By this we always mean a map defined in an
×V
open neighborhood of the origin. Since the width of the neighborhood does not
play any role in the future we avoid to specify it.
Definition 3.9. A map S : dA/2 ∞ will be said to be of class i if there
V →H Sj
exists a smooth map S˜ : Rn −∞ ∞ such that S(φ) = S˜( (φ),φ), and
the map S˜ fulfills ×H → H P
S˜(N,φ) C N i φ j , m,k 0 . (3.18)
Hm ≤ mk| | k kH−k ∀ ≥
(cid:13) (cid:13)
In the case o(cid:13)f maps t(cid:13)aking values in Rn we give an analogous definition.
(cid:13) (cid:13)
Definition 3.10. AmapR: dA/2 Rn willbe saidtobe ofclass i ifthere
V → Rj
exists a smooth map R˜ : Rn −∞ Rn such that R(φ) = R˜( (φ),φ), and
the map R˜ fulfills ×H → P
R˜(N,φ) C N i φ j , k 0 . (3.19)
≤ k| | k kH−k ∀ ≥
(cid:13) (cid:13)
(cid:13) (cid:13)
(cid:13) (cid:13)
9
The functions belonging to the above classes will be called smoothing.
In the following we will identify a smoothing function S (or R) with the
corresponding function S˜ (or R˜). Most of the times functions of class k ( l
Sl Rk
resp.) will be denoted by Sk (Rk resp.). Furthermore, since the only relevant
l l
propertyofsuchfunctionsaregivenbytheinequalities(3.18)and(3.19)wewill
use the same notation for different smoothing functions. For example we will
meet equalities of the form
S1+S1 =S1 (3.20)
1 2 1
where obviously the function S1 at r.h.s. is different from that at l.h.s.
1
Finally, we always consider functions and vector fields as functions of N,φ,
with the idea that, at the end of the procedure we will put N = (φ).
j j
P
Lemma 3.11. There exists a smoothing map p 0 with the following prop-
∈ R0
erties
(1) For any j =1,...,n, and for φ dA/2, one has
∈V
(η +Π φ)=p ;
j p(P(φ),φ) p(P(φ),φ) 0j
P
(2) there esist R1 1 s.t. p=p N +R1(N,φ);
2 ∈R2 0− 2
∂Π
(3) Define the matrix M = M(N,φ), by (M−1) =δ + Π φ;A pφ
jk jk p j
h ∂p i
k
(evaluated at p=p(N,φ)), then the gradient of p( (φ),φ) is given by
P
p = M Π∗ A φ . (3.21)
∇ j − jk p0 k
k
X
Proof. First remark that one has
(η +Π φ)=p + (Π φ)=p + (φ)+V (p,φ) , (3.22)
j p p j j p j j j
P P P
where V (p,φ) := (Π φ) (φ) extends to a smooth map on −k k and
j j p j
P −P V ∀
fulfills
V (p,φ) C p p φ 2 , k (3.23)
| j |≤ k| − 0|k kH−k ∀
We apply the implicit function theorem to the system of equations
0=F (p,N,φ):=p +N +V (p,φ) p . (3.24)
j j j j 0j
−
Using (3.22) one gets
∂F ∂Π
j =δk+ Π φ;A pφ =(M−1) .
∂p j h p j ∂p i jk
k k
Since ∂Fj (M−1) is invertible, the implicit function theorem ensures the
∂pk ≡ jk
existenceofasmoothfunctionp=p(N,φ)fromRn −k toRn solving(3.24).
×H
Thentheestimateensuringp (p N) 1followsfromthefactthatp(p ,φ)
− 0− ∈R2 0 −
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