Atlas of products for wave-Sobolev spaces on $\mathbf R^{1+3}$

0.3 MB·English

by  Piero D'Ancona

ATLAS OF PRODUCTS FOR WAVE-SOBOLEV SPACES ON R1+3
PIEROD’ANCONA,DAMIANOFOSCHI,ANDSIGMUNDSELBERG
0
1
0 Abstract. Thewave-SobolevspacesHs,bareL2-basedSobolevspacesonthe
2 Minkowskispace-timeR1+n,withFourierweights areadapted tothe symbol

n of the d’Alembertian. They are a standard tool in the study of regularity
a propertiesofnonlinearwaveequations,andinsuchapplicationstheneedarises
J forproductestimatesinthesespaces. Unfortunately,itseemsthatwithevery
new application someestimates come up which have not yet appeared inthe
9
literature,andthenonehastoresorttoasetofwell-establishedproceduresfor
2
proving the missingestimates. To relieve the tedium of having to constantly

fill in such gaps “by hand”, we make here a systematic effort to determine
]
P the complete set of estimates in the bilinear case. We determine a set of
A necessary conditions for a product estimate Hs1,b1 ·Hs2,b2 ֒→ H−s0,−b0 to
hold. These conditions define a polyhedron Ω in the space R6 of exponents
.
h (s0,s1,s2,b0,b1,b2). Wethenshow,inspacedimensionn=3,thatallpoints
t intheinteriorofΩ,andallpointsonthefaces minustheedges,giveproduct
a
estimates. We can also allow some but not all points on the edges, but here
m
wedonotclaimtohave thesharpresult. Thecorrespondingresultforn=2
[ andn=1willbepublishedelsewhere.

1
v
2
7
Contents
3
5
1. Wave-Sobolev spaces 1
.
1 2. The product law 4
0
3. Counterexamples 8
0
4. Notation and preliminaries 10
1
: 5. The case b0 =b1 =0<b2 12
v
6. The case b =0<b ,b 12
i 0 1 2
X 7. The case 0<b ,b ,b 19
0 1 2
r 8. The case b0 <0<b1,b2 26
a
9. Reformulation of the rules on the boundary 30
References 31
1. Wave-Sobolev spaces
Define the Fourier transform of a Schwartz function u (R1+n) by
∈S
u(τ,ξ)= e−i(tτ+x·ξ)u(t,x)dtdx,
ZZ
where(t,x)and(τ,ξ)beelongtoR Rn =R1+n;τ andξ willbecalledthetemporal
×
and spatial frequencies, respectively.
2000 Mathematics Subject Classification. 35L05,46E35.
This paper was written as part of the international research program on Nonlinear Partial
Differential Equations at the Centre for Advanced Study at the Norwegian Academy of Science
andLetters inOsloduringtheacademicyear2008–09.
1
2 P.D’ANCONA,D.FOSCHI,ANDS.SELBERG
Definition 1.1. Given s,b R, the wave-Sobolev space Hs,b =Hs,b(R1+n) is the
completion of (R1+n) with∈respect to to the norm
S
u = ξ s τ ξ bu(τ,ξ) ,
k kHs,b h i h| |−| |i L2
τ,ξ
where =(1+ 2)12. We shall(cid:13)(cid:13)refer to the weigehts ξ(cid:13)(cid:13)s and τ ξ b as elliptic
h·i |·| h i h| |−| |i
and hyperbolic, respectively.
By way of comparison, the elliptic weight is a familiar aspect of the standard
Sobolev space Hs =Hs(Rn), obtained as the completion of (Rn) with respect to
S
the norm f = ξ sf(ξ) , where f(ξ)= e−ix·ξf(x)dx.
k kHs kh i kL2ξ
The hyperbolic weight, on the other hand, reflects the fact that the Hs,b-norm
R
is adapted to the wave obperator, or d’Albembertian, (cid:3) = ∂2+∆ , whose symbol
− t x
is τ2 ξ 2.
−| |
For details about the history of the wave-Sobolev spaces and applications to
nonlinearwaveequations,werefertothesurveyarticle[5]. Intheapplications,the
need frequently arises for product estimates of the form
(1.1) Hs1,b1 Hs2,b2 ֒ H−s0,−b0.
· →
Explicitly, this means that there exists C =C(s ,s ,s ,b ,b ,b ;n) such that
0 1 2 0 1 2
(1.2) uv C u v
k kH−s0,−b0 ≤ k kHs1,b1 k kHs2,b2
for all u,v (R1+n).
∈S
Definition 1.2. If (1.2) holds, we say that the exponent matrix
s s s
0 1 2
b b b
0 1 2
(cid:18) (cid:19)
is a product.
Manyproductestimateshaveappearedintheliterature(see[5]forexamplesand
references), but so far no systematic effort has been made to determine necessary
and sufficient conditions on s0 s1 s2 for it to be a product. The present paper is
b0 b1 b2
the resultofour effortsto fillthis gap. We remarkthatthe utility ofthese product
(cid:0) (cid:1)
estimates is not limited to simple products: Many bilinear null form estimates can
alsobereducedtothisform,sincethe nullsymbolcanbeestimatedintermsofthe
weights appearing in the Hs,b-norm. See, e.g., [1].
It turns out that there are 21 necessary conditions of the form
(1.3) σ s +σ s +σ s +β b +β b +β b 0.
0 0 1 1 2 2 0 0 1 1 2 2
≥
Eachsuchconditiondeterminesahalf-spaceinthespaceR6ofcoefficients s0 s1 s2 .
b0 b1 b2
Takentogether,these21conditions,whicharelistedinthe nextsection,determine
a convex polyhedron Ω in R6. The boundary of Ω consists of faces, (cid:0)which ar(cid:1)e
polyhedrons contained in the hyperplanes corresponding to equality in one of the
conditions of the form (1.3). The intersection of two faces is an edge. Thus, the
boundary of a face consists of edges.
On the positive side, it turns out that almost all the points in Ω are products.
Let us call a subset of Ω admissible if all its points are products. We show:
The interior of Ω is admissible.
•
The faces of Ω, excluding the edges, are admissible.
•
Some but not all edges are admissible.
•
Thisparallelsthe situationforthe productlawforthe standardSobolevspacesHs
(see Theorem 2.2 in the next section).
ATLAS OF PRODUCTS FOR WAVE-SOBOLEV SPACES ON R1+3 3
Concerningtheedges,wedonotclaimtohavetheoptimalresult,however. That
is, there may be some points on the edges which are products but which are not
included in our positive results.
In order to avoid an unduly lengthy paper, we restrict our attention, for the
positive results, to the physical space dimension n = 3, which is of most interest
for applications. The cases n =2 and n =1 will be published in a separate paper
(the 2d case is slightly more involved than the 3d case).
Beforeproceeding to the list of necessaryconditions, we make some preliminary
observations, and introduce notation and terminology.
Itisimportanttonotethatif s0 s1 s2 isaproduct,thensoiseverypermutation
b0 b1 b2
of its columns. This becomes obvious once we restate (1.2) in the following more
(cid:0) (cid:1)
symmetric form: By Plancherel’s theorem and duality, (1.2) is equivalent to the
trilinear integral estimate
(1.4) I . F F F ,
0 1 2
| | k kk kk k
where
F (X )F (X )F (X )δ(X +X +X ) dX dX dX
0 0 1 1 2 2 0 1 2 0 1 2
(1.5) I =
ZZZ hξ0is0hξ1is1hξ2is2h|τ0|−|ξ0|ib0h|τ1|−|ξ1|ib1h|τ2|−|ξ2|ib2
andX =(τ ,ξ ) R1+n forj =0,1,2. Hereδ is the pointmassat0inR1+n, and
j j j
denotes the L2∈norm on R1+n. Without loss of generality we may assume that
k·k
F 0 for j =0,1,2, hence I 0.
j
≥ ≥
Since ξ +ξ +ξ =0 in I, the triangle inequality implies ξ . ξ + ξ for
0 1 2 j k l
h i h i h i
all permutations (j,k,l) of (0,1,2), hence the two largest of ξ , ξ and ξ are
0 1 2
h i h i h i
comparable, so we can split
(1.6) I =I +I +I ,
LHH HLH HHL
where the terms on the right hand side are defined by inserting the characteristic
functions of the following conditions, respectively, in the integral I:
(1.7a) ξ . ξ ξ (LHH)
0 1 2
h i h i∼h i
(1.7b) ξ . ξ ξ (HLH)
1 0 2
h i h i∼h i
(1.7c) ξ . ξ ξ (HHL).
2 0 1
h i h i∼h i
Here the mnemonics in the right hand column refer to the relative sizes of the
spatial frequencies in the order (ξ ,ξ ,ξ ), with “L” and “H” standing for low and
0 1 2
high frequencies, respectively.
Insome situations we alsosplit the I’s depending onthe signs and ofthe
1 2
± ±
temporal frequencies τ and τ . Thus,
1 2
(1.8) I =I(+,+)+I(+,−)+I(−,+)+I(−,−),
where
I(±1,±2)
F (X )F (X )F (X )δ(X +X +X ) dX dX dX
0 0 1 1 2 2 0 1 2 0 1 2
=
ZZZ hξ0is0hξ1is1hξ2is2h|τ0|−|ξ0|ib0h|τ1|−|ξ1|ib1h|τ2|−|ξ2|ib2
±1τ1≥0,±2τ2≥0
F (X )F (X )F (X )δ(X +X +X ) dX dX dX
0 0 1 1 2 2 0 1 2 0 1 2
= ,
ZZZ hξ0is0hξ1is1hξ2is2h|τ0|−|ξ0|ib0h−τ1±1|ξ1|ib1h−τ2±2|ξ2|ib2
±1τ1≥0,±2τ2≥0
and similarly for I etc.
LHH
In conjunction with the splittings (1.6) and (1.8), as well as their combination,
itisconvenienttousethefollowingratherobviousmodificationsoftheterminology
introduced in Definition 1.2: When we say, for instance, that s0 s1 s2 is a
b0 b1 b2 HLH
(cid:0) (cid:1)(cid:12)
(cid:12)
4 P.D’ANCONA,D.FOSCHI,ANDS.SELBERG
product, we mean that (1.4) holds for I , and if we say that s0 s1 s2 (+,+) is
HLH b0 b1 b2 LHH
(+,+)
a product, we mean that (1.4) holds for I , and so on.
LHH (cid:0) (cid:1)(cid:12)
We use x.y as a convenient shorthand for x Cy, where C 1 is a c(cid:12)onstant
≤ ≫
whichmay depend onquantities that areconsideredfixed. Moreover,x y stands
∼
for x.y .x.
2. The product law
2.1. Necessary conditions. Anumberofexplicitexamplesgivenin 3showthat
any product s0 s1 s2 must necessarily satisfy the following 21 condit§ions:
b0 b1 b2
(cid:0) (cid:1) 1
(2.1) b +b +b
0 1 2
≥ 2
(2.2) b +b 0
0 1
≥
(2.3) b +b 0
0 2
≥
(2.4) b +b 0
1 2
≥
n+1
(2.5) s +s +s (b +b +b )
0 1 2 0 1 2
≥ 2 −
n
(2.6) s +s +s (b +b )
0 1 2 0 1
≥ 2 −
n
(2.7) s +s +s (b +b )
0 1 2 0 2
≥ 2 −
n
(2.8) s +s +s (b +b )
0 1 2 1 2
≥ 2 −
n 1
(2.9) s +s +s − b
0 1 2 0
≥ 2 −
n 1
(2.10) s +s +s − b
0 1 2 1
≥ 2 −
n 1
(2.11) s +s +s − b
0 1 2 2
≥ 2 −
n+1
(2.12) s +s +s
0 1 2
≥ 4
n
(2.13) (s +b )+2s +2s LH H
0 0 1 2 ≥ 2 +−
n
(2.14) 2s +(s +b )+2s (cid:0)HLH(cid:1)
0 1 1 2 ≥ 2 + −
n
(2.15) 2s +2s +(s +b ) (cid:0)H H L(cid:1)
0 1 2 2 ≥ 2 +−
(2.16) s +s b (cid:0)LH H(cid:1)
1 2 ≥− 0 ++
(2.17) s0+s2 ≥−b1 (cid:0)+HL+H(cid:1)
(2.18) s0+s1 ≥−b2 (cid:0)+H+HL(cid:1)
(2.19) s1+s2 0 (cid:0)LH H(cid:1)
≥
(2.20) s0+s2 0 (cid:0)HLH(cid:1)
≥
(2.21) s0+s1 0 (cid:0)H HL(cid:1).
≥
The tags in the right hand column have the following mea(cid:0)ning: T(cid:1)he upper row
indicates the spatial frequency interaction (LHH, HLH or HHL) in which the con-
dition is necessary. The lower row,if not empty, indicates whether the signs of the
respectivetemporalfrequenciesareequal(indicatedby ++)oropposite(indicated
by + ). For example, (2.13) [resp. (2.16)] is needed in the LHH interaction with
−
opposite [resp. equal] signs for τ and τ . An empty lower row means, of course,
1 2
that the condition is needed regardless of the signs.
ATLAS OF PRODUCTS FOR WAVE-SOBOLEV SPACES ON R1+3 5
The same qualifications are understood to apply also in Theorem 2.3 below.
Definition 2.1. Let Ω be the convex polyhedron in R6 determined by the above
conditions.
The interior of Ω corresponds to strict inequality in all the conditions. Each
condition determines a face of Ω, corresponding to the case of equality. If at least
two of the conditions are equalities, then we are in an edge.
As we said in the introduction, we shall prove (for n 3) that the interior and
≤
the faces minus the edges are admissible. Moreover, some but not all of the edges
are admissible.
This parallels the situation for the comparatively trivial product law for the
standard Sobolev spaces, which we now recall.
2.2. Comparison with the product law for Hs. This reads as follows:
Theorem 2.2. Let s ,s ,s R. The product estimate
0 1 2
∈
fg C f g
k kH−s0 ≤ k kHs1 k kHs2
holds if and only if
n
(2.22) s +s +s
0 1 2
≥ 2
(2.23) s +s 0
0 1
≥
(2.24) s +s 0
0 2
≥
(2.25) s +s 0
1 2
≥
(2.26) If (2.22) is an equality, then (2.23)–(2.25) must be strict.
Thesimpleproofofthepositivepartwillbeshownlater,sincethesameargument
comes up also in the proof of the wave-Sobolev product law. The negative part of
the above theorem follows by a standard example which we do not repeat here.
Theconditions(2.22)–(2.25)determineaconvexpolyhedronofpoints(s ,s ,s )
0 1 2
in R3. The edges corresponding to equality in (2.22) and one of (2.23)–(2.25) are
not admissible. On the other hand, the edges corresponding to equality in two
of (2.23)–(2.25) are admissible, as long as we stay away from the face given by
equalityin(2.22). Itthereforeseems difficultto write downa simple rule telling us
which edges are admissible.
Butifinsteadoftalkingaboutedgeswetalkaboutequalities,thenwecanmake
a simple rule as follows: Replace (2.22)–(2.26) by
n
(2.27) s +s +s
0 1 2
≥ 2
(2.28) s +s +s max(s ,s ,s ).
0 1 2 0 1 2
≥
where we combined (2.23)–(2.25) into a single condition. Then (2.26) is replaced
by the statement:
(2.29) We do not allow both (2.27) and (2.28) to be equalities.
By comparing (2.27) and (2.28), both with equality assumed, the last rule can
also be reformulated as a list of explicit exceptions as follows:
(2.30) If s = n, then (2.22)=(2.25) must be strict.
0 2
(2.31) If s = n, then (2.22)=(2.24) must be strict.
1 2
(2.32) If s = n, then (2.22)=(2.23) must be strict.
2 2
Here the notation “(2.22)=(2.25)” indicates that the two conditions coincide.
Theseideashelptosystematizethemuchmorecomplicatedexceptionsalongthe
edges of Ω, which we now discuss.
6 P.D’ANCONA,D.FOSCHI,ANDS.SELBERG
2.3. Exceptions on the boundary of Ω. First rewrite (2.1)–(2.4) as
1
(2.33) b +b +b
0 1 2
≥ 2
(2.34) b +b +b max(b ,b ,b )
0 1 2 0 1 2
≥
Then we impose the rule:
(2.35) We do not allow both (2.33) and (2.34) to be equalities.
Next, consider (2.5)–(2.21). By symmetry, it suffices to consider the LHH case,
hence we ignore those conditions among (2.13)–(2.21) which are not tagged LHH.
Moreover, we do not want to compare (2.13) with (2.16) since they have different
sign assumptions, hence we split into LH H and LH H
++ +−
In the case LH H we rewrite the relevant conditions from (2.5)–(2.21) as:
++ (cid:0) (cid:1) (cid:0) (cid:1)
n+1
(2.36) s +(cid:0)s +s(cid:1) (b +b +b )
0 1 2 0 1 2
≥ 2 −
n
(2.37) s +s +s +max( b b , b b , b b )
0 1 2 0 1 0 2 1 2
≥ 2 − − − − − −
n 1 n 3
(2.38) s +s +s − +max b , b , b , −
0 1 2 0 1 2
≥ 2 − − − − 4
(cid:18) (cid:19)
(2.39) s +s +s s +max(0, b ) LH H .
0 1 2 ≥ 0 − 0 ++
Then we impose the rule: (cid:0) (cid:1)
(2.40) We allow at most one of (2.36)–(2.39) to be an equality.
In the case LH H we rewrite the relevant conditions from (2.5)–(2.21) as:
+−
n+1
(2.41) s +(cid:0)s +s(cid:1) (b +b +b )
0 1 2 0 1 2
≥ 2 −
n
(2.42) s +s +s +max( b b , b b , b b )
0 1 2 0 1 0 2 1 2
≥ 2 − − − − − −
n 1 n 3
(2.43) s +s +s − +max b , b , b , −
0 1 2 0 1 2
≥ 2 − − − − 4
(cid:18) (cid:19)
n s b
(2.44) s +s +s + 0− 0 LH H
0 1 2 ≥ 4 2 +−
(2.45) s +s +s s (cid:0)LH H(cid:1),
0 1 2 ≥ 0 +−
and we impose the rule: (cid:0) (cid:1)
(2.46) We allow at most one of (2.41)–(2.45) to be an equality.
An alternative formulation of the above rules is given in Theorem 2.7 below.
The analogous rules for the HLH and HHL cases are obtained by changing the
subscript0intherighthandsideof (2.39),(2.44)and(2.45)toa1or2,respectively.
2.4. The product law for Hs,b. We can now formulate the main result:
Theorem 2.3. Let n = 3. Assume that s ,s ,s ,b ,b ,b R satisfy the condi-
0 1 2 0 1 2
∈
tions (2.1)–(2.21). Moreover, assume that the rules set out in 2.3 are satisfied.
Then s0 s1 s2 is a product. §
b0 b1 b2
Remar(cid:0)k 2.4. F(cid:1)or n=1 and n=2 the same result holds; the proofs will appear in
a separate paper. We expect the same result to hold also for n 4.
≥
Remark 2.5. In the course of the proof, we break Theorem 2.3 down according to
the classification into product types introduced in 2.5 below, and we restate the
§
theoreminamoreexplicitformineachcase. Forpracticaluse,thereadermayfind
these restatements easier to deal with than the general statement in Theorem 2.3.
See 5–8.
§§
ATLAS OF PRODUCTS FOR WAVE-SOBOLEV SPACES ON R1+3 7
Remark 2.6. We are not claiming that the boundary rules are necessary,only that
theyaresufficient. Wedoexpect,however,that(2.35)isnecessary. Thisiscertainly
true in the 1dcase,where itcanbe seenfromthe standardcounterexampleforthe
Hs product law. We also expect (2.40) and (2.46) to be necessary if all the b’s are
nonnegative,butifoneoftheb’sisnegative,thentheycanundercertainconditions
be relaxed somewhat (see Theorem 8.2 below).
By comparing equalities pairwise within the groups (2.33)–(2.34), (2.36)–(2.39)
and (2.41)–(2.45), we can restate the rules (2.35), (2.40) and (2.46) as an explicit
list of exceptions analogous to the list (2.30)–(2.32) for the Hs product law:
Theorem 2.7. Let n = 3. Assume that (2.1)–(2.21) are verified. Then the
rules (2.35), (2.40)and (2.46) for the LHH interaction are equivalent to the follow-
ing list of exceptions:
(2.47) Ifb = 1,then (2.1)=(2.4),(2.5)=(2.8),(2.6)=(2.10)and (2.7)=(2.11)
0 2
must all be strict.
(2.48) If b = 1, then (2.1)=(2.3), (2.5)=(2.7), (2.6)=(2.9) and (2.8)=(2.11)
1 2
must all be strict.
(2.49) If b = 1, then (2.1)=(2.2), (2.5)=(2.6), (2.7)=(2.9) and (2.8)=(2.10)
2 2
must all be strict.
(2.50) If b +b =1, then (2.5)=(2.11) must be strict.
0 1
(2.51) If b +b =1, then (2.5)=(2.10) must be strict.
0 2
(2.52) If b +b =1, then (2.5)=(2.9) must be strict.
1 2
(2.53) If b +b = n−1, then (2.6)=(2.12) must be strict.
0 1 4
(2.54) If b +b = n−1, then (2.7)=(2.12) must be strict.
0 2 4
(2.55) If b +b = n−1, then (2.8)=(2.12) must be strict.
1 2 4
(2.56) If b +b +b = n+1, then (2.5)=(2.12) must be strict.
0 1 2 4
(2.57) If s b = n+2 2(b +b +b ), then (2.5)=(2.13) must be strict.
0− 0 2 − 0 1 2
(2.58) If s b = n 2(b +b ), then (2.6)=(2.13) must be strict.
0− 0 2 − 0 1
(2.59) If s b = n 2(b +b ), then (2.7)=(2.13) must be strict.
0− 0 2 − 0 2
(2.60) If s b = n−2 2b , then (2.9)=(2.13) must be strict.
0− 0 2 − 0
(2.61) If s b = 1, then (2.12)=(2.13) must be strict.
0− 0 2
(2.62) If s b = n 2b , then (2.19)=(2.13) must be strict.
0− 0 2 − 0
(2.63) If one of (2.5)–(2.12) is an equality, then (2.16) and (2.19) must be
strict.
Here the notation “(2.1)=(2.2)” indicates that the two conditions coincide.
TheanalogousexceptionsfortheHLHandHHLcasesareobtainedbypermuting
the subscripts in (2.57)–(2.62).
2.5. Classification of products. By permutation invariance, it suffices to prove
the main result for products of the following special types:
(I) b ,b ,b 0. Then by symmetry it suffices to consider the subtypes
0 1 2
≥
(a) b =b =0<b ,
0 1 2
(b) b =0<b ,b ,
0 1 2
(c) 0<b ,b ,b ,
0 1 2
(II) b <0<b ,b .
0 1 2
2.6. Outline of paper. In 3 the counterexamples which imply the necessary
§
conditions are given. In 4 we make a dyadic decomposition of the integral I and
§
recall the dyadic estimates which are the fundamental building blocks in the proof
8 P.D’ANCONA,D.FOSCHI,ANDS.SELBERG
of the product laws. We also recall the simple proof of the Hs product law, since
that argument is used repeatedly in later sections. The main result, Theorem
2.3, is proved in 5–8, broken into sections according to the classification into
§§
types as in 2.5. In each section we explicitly restate the theorem, and this may
§
be useful also when applying our results, as an alternative to grappling with the
generalformulationabove. The reformulationof the boundary rules, Theorem 2.7,
is proved in 9.
§
3. Counterexamples
To prove the necessity of (2.1)–(2.21) we will estimate the integral I, defined
by (1.5), on examples of the form F = χ , F = χ and F = χ , where
0 −C 1 A 2 B
A,B,C R1+n depend on a parameter λ 1 and are chosen so that A+B C.
⊂ ≫ ⊂
Since X +X +X =0 in I, this ensures that
0 1 2
X A, X B = X = (X +X ) C,
1 2 0 1 2
∈ ∈ ⇒ − ∈−
and then we estimate the weight in I by a power of λ:
ξ s0 ξ s1 ξ s2 τ ξ b0 τ ξ b1 τ ξ b2 λδ(s0,s1,s2,b0,b1,b2),
0 1 2 0 0 1 1 2 2
h i h i h i h| |−| |i h| |−| |i h| |−| |i ∼
where δ is some linear combination of the s’s and b’s. Then I λ−δ A B , while
1 1 1 ∼ | || |
F0 F1 F2 A 2 B 2 C 2. The estimate (1.4) will then imply the condition
λkδ &kkA 12kkB 12kC∼−|12.|I|f w|e|ha|ve an estimate of the form
| | | | | |
(3.1) |A|12|B|21 λd(n),
C 21 ∼
| |
then we deduce the necessary condition δ =δ(s ,s ,s ,b ,b ,b ) d(n).
0 1 2 0 1 2
In the following we split ξ Rn as ξ = (ξ ,ξ′), where ξ′ = (ξ ,≥...,ξ ) Rn−1.
1 2 n
∈ ∈
To avoid any confusion, we emphasize that in this notation the subscript refers to
coordinates, whereas elsewhere we use subscripts to label different vectors.
3.1. Necessity of (2.1). This is obtained by scaling only the temporal variables:
A=B = (τ,ξ): λ τ 2λ, ξ 1 , A = B λ,
{ ≤ ≤ | |≤ } | | | |∼
C = (τ,ξ): 2λ τ 4λ, ξ 2 , C λ,
{ ≤ ≤ | |≤ } | |∼
1
δ =b +b +b , d= .
0 1 2
2
3.2. Necessity of (2.2)–(2.4). Bysymmetry,itsufficesto show (2.3),andforthis
we choose:
A= (τ,ξ): τ 1, ξ 1 , A 1,
{ | |≤ | |≤ } | |∼
B = (τ,ξ): τ λ 1, ξ 1 , B 1,
{ | − |≤ | |≤ } | |∼
C = (τ,ξ): τ λ 2, ξ 2 , C 1,
{ | − |≤ | |≤ } | |∼
δ =b +b , d=0.
0 2
3.3. Necessity of (2.5). This is obtained by scaling all variables:
λ
A=B = (τ,ξ): τ ,λ ξ 2λ, ξ′ λ , A = B λn+1,
1
| |≤ 2 ≤ ≤ | |≤ | | | |∼
(cid:26) (cid:27)
C = (τ,ξ): τ λ,2λ ξ 4λ, ξ′ 2λ , C λn+1,
1
{ | |≤ ≤ ≤ | |≤ } | |∼
n+1
δ =s +s +s +b +b +b , d= .
0 1 2 0 1 2
2
ATLAS OF PRODUCTS FOR WAVE-SOBOLEV SPACES ON R1+3 9
3.4. Necessity of (2.6)–(2.8). Bysymmetry,itsufficesto show (2.7),andforthis
we choose:
5λ λ
A= (τ,ξ): τ ξ 1,λ ξ , ξ′ , A λn,
1
| −| ||≤ ≤ ≤ 4 | |≤ 4 | |∼
(cid:26) (cid:27)
λ 5λ λ
B = (τ,ξ): τ ,2λ ξ , ξ′ , B λn+1,
1
| |≤ 2 ≤ ≤ 2 | |≤ 2 | |∼
(cid:26) (cid:27)
5λ
C = (τ,ξ): τ ,3λ ξ 4λ, ξ′ λ , C λn+1,
1
| |≤ 2 ≤ ≤ | |≤ | |∼
(cid:26) (cid:27)
n
δ =s +s +s +b +b , d= .
0 1 2 0 2
2
3.5. Necessity of (2.9)–(2.11). By symmetry, it suffices to show (2.9), and for
this we choose:
3λ λ
A= (τ,ξ): τ ξ 1,λ ξ , ξ′ , A λn,
1
| −| ||≤ ≤ ≤ 2 | |≤ 2 | |∼
(cid:26) (cid:27)
3λ λ
B = (τ,ξ): τ + ξ 1,λ ξ , ξ′ , B λn,
1
| | ||≤ ≤ ≤ 2 | |≤ 2 | |∼
(cid:26) (cid:27)
3λ
C = (τ,ξ): τ ,2λ ξ 3λ, ξ′ λ , C λn+1,
1
| |≤ 2 ≤ ≤ | |≤ | |∼
(cid:26) (cid:27)
n 1
δ =s +s +s +b , d= − .
0 1 2 0
2
3.6. Necessity of (2.12). This represents the effect of Lorentz transformations
(concentration along null directions):
A=B = (τ,ξ): τ ξ1 1,λ ξ1 2λ, ξ′ √λ , A = B λn+21,
| − |≤ ≤ ≤ | |≤ | | | |∼
C = (τ,ξn): τ ξ1 2,2λ ξ1 4λ, ξ′ 2√λ ,o C λn+21,
| − |≤ ≤ ≤ | |≤ | |∼
n o n+1
δ =s +s +s , d= .
0 1 2
4
3.7. Necessity of (2.13)–(2.15). By symmetry, it suffices to show (2.13), and for
this we choose:
1
A= (τ,ξ): τ ξ 1, ξ λ2 λ, ξ′ λ,ξ λ , A λn,
1 1 2
| − |≤ | − |≤ | |≤ ≥ 2 | |∼
(cid:26) (cid:27)
1
B = (τ,ξ): τ ξ 1, ξ +λ2 λ, ξ′ λ,ξ λ , B λn,
1 1 2
| − |≤ | |≤ | |≤ ≥ 2 | |∼
(cid:26) (cid:27)
C = (τ,ξ): τ ξ 2, ξ 2λ, ξ′ 2λ,ξ λ , C λn,
1 1 2
{ | − |≤ | |≤ | |≤ ≥ } | |∼
n
δ =s +b +2s +2s , d= .
0 0 1 2
2
3.8. Necessity of (2.16)–(2.18). By symmetry, it suffices to show (2.16), and for
this we choose:
A= (τ,ξ): τ λ 1, ξ λ 1, ξ′ 1 , A 1,
1
{ | − |≤ | − |≤ | |≤ } | |∼
B = (τ,ξ): τ λ 1, ξ +λ 1, ξ′ 1 , B 1,
1
{ | − |≤ | |≤ | |≤ } | |∼
C = (τ,ξ): τ 2λ 2, ξ 2, ξ′ 2 , C 1,
1
{ | − |≤ | |≤ | |≤ } | |∼
δ =s +s +b , d=0.
1 2 0
10 P.D’ANCONA,D.FOSCHI,ANDS.SELBERG
3.9. Necessity of (2.19)–(2.21). By symmetry, it suffices to show (2.20), and for
this we choose:
A= (τ,ξ): τ 1, ξ 1, ξ′ 1 , A 1,
1
{ | |≤ | |≤ | |≤ } | |∼
B = (τ,ξ): τ 1, ξ λ 1, ξ′ 1 , B 1,
1
{ | |≤ | − |≤ | |≤ } | |∼
C = (τ,ξ): τ 2, ξ λ 2, ξ′ 2 , C 1,
1
{ | |≤ | − |≤ | |≤ } | |∼
δ =s +s , d=0.
0 2
4. Notation and preliminaries
4.1. Dyadic decompositions. Throughout,M,N andL,aswellastheirindexed
counterparts, denote dyadic numbers of the form 2j, j 0,1,2,... . We rely on
∈ { }
dyadic decompositions with respect to the size of the weights in the Hs,b-norm. In
some cases we also decompose with respect to the sign of the temporal frequency.
Given u Hs,b, we define the L2-function F 0 by
∈ ≥
(4.1) F(X)= ξ s τ ξ bu(X),
h i h| |−| |i
where X =(τ,ξ). We shall use the shorthand FN(X)=χ F(X), FN,L(X)=
hξi∼N
χ FN(X)andFN,L,±(X)=χ FN,L(eX),andcorrespondinglywethen
h|τ|−|ξ|i∼L ±τ≥0
define uN, uN,L and uN,L,± as in (4.1), replacing F there by FN, FN,L and
FN,L,±, respectively. Note that FN 2 F 2, FN,L 2 FN 2 and
N ∼ k k L ∼
FN,L,± 2 . FN 2.
L P (cid:13) (cid:13) P (cid:13) (cid:13) (cid:13) (cid:13)
Defining the trilinear convolution f(cid:13)orm(cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
P (cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
J(F ,F ,F )= F (X )F (X )F (X )δ(X +X +X ) dX dX dX ,
0 1 2 0 0 1 1 2 2 0 1 2 0 1 2
ZZZ
we then obtain
J FN0,L0,FN1,L1,FN2,L2
0 1 2
(4.2) I . ,
(cid:16)Ns0Ns1Ns2Lb0Lb1Lb2 (cid:17)
N,L 0 1 2 0 1 2
X
where I is given by (1.5) and we set N = (N ,N ,N ) and L = (L ,L ,L ). We
0 1 2 0 1 2
usethe shorthandN012 =min(N ,N ,N ),andsimilarlyforthe L’s,andforother
min 0 1 2
indexes than 012.
We also have the analogues of (4.2) for I , I and I , obtained by
LHH HLH HLH
inserting the characteristic functions of the following conditions, respectively, in
the sum on the right hand side of (4.2): N N N (LHH), N N N
0 1 2 1 0 2
≤ ∼ ≤ ∼
(HLH) and N N N (HHL).
2 0 1
Note that if 1≤ A∼<B and a R, then
≤ ∈
Ba if a>0
(4.3) La log B if a=0
∼ hAi
A≤XL≤B Aa if a<0.
We frequently apply the estimate, forany ε>0,
(4.4) log B C Bε for all B 1.
ε
h i≤ ≥
4.2. Hyperbolic Leibniz rule. We recalla well-known“Leibniz rule” for hyper-
bolic weights (a proof can be found, for example, in [5, Lemma 3.4]): Assume that
τ +τ +τ =0 andξ +ξ +ξ =0,asinthe integralI,andlet and denote
0 1 2 0 1 2 1 2
± ±
the signs of τ and τ , respectively. Then
1 2
(4.5) τ ξ . τ ξ + τ ξ +b (ξ ,ξ ,ξ ),
| 0|−| 0| − 1±1| 1| − 2±2| 2| (±1,±2) 0 1 2
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)