3.3 Proof of Theorems 3.2.2 and 3.2.4
3.3.3 Proof of Theorem 3.2.4: Sufficiency part
For the first identity, see [8], and for the second, we refer the reader to the work of Merucci [77]. Finally, we note that with the above choices ofκandθ, we have
(L2σt 0Lrx∗, LκtLrx∗)θ
2,2=L2σt ∗,2Lrx∗.
Here, note that the second exponent in the mixed norm is fixed, so we may treat the mixed norm as a norm in a vector-valued space. Hence, by Proposition 3.3.3,T : (W1, W2)7→ {W1SjW2}j is bounded according to (3.3.11); this implies (3.3.7) and hence Theorem 3.2.2 ford≥3.
Finally, we remark that whend= 2 the proof needs a very minor modification since (3.1.3) fails when (1q,1p) = (0,1) and thus the proof of the claimed estimates at Q0 andQ00 in Lemma 3.3.4 does not work as it stands. However, we may apply (3.1.3) with (1q,1p) = (ε,1−ε) for all ε∈ (0,1) and inserting such estimates means that we can obtain the analogous estimates at pointsQ0,ε= (12−ε,0) and Q00,ε= (0,12−ε) for sufficiently smallε >0. The same argument as used above then yields Theorem 3.2.2 ford= 2.
The above theorem provides sharp estimates on all of the admissible region except the line segment [A, D], where the estimate (3.3.12) is equivalent to the corresponding (strong-type) estimate without the localisation operator P via a simple scaling argument; in this case, the strong-type estimate at the criticalα=premains open (see Theorems 3.2.4 and 3.2.2 below).
The crucial part of this Theorem is the following estimate atF = (d−12d ,12).
Proposition 3.3.6. Let d≥2. For any orthonormal system (fj)j inL2,
X
j
λj|eit∆P fj|2 2,2d
d−1
.kλk`2, or equivalently,
T0
C2 .kW1k4,4d
d+1kW2k4,4d d+1, where the operator T0 is determined by
T0F(x, t) =W1(x, t) Z
R
ei(t−s)∆P2[F(·, s)W2(·, s)](x)ds.
Proof. The kernel of the operatorT0 can be written by W1(x, t)ei(t−s)∆h
dφ02i
(x−y)W2(y, s).
So, our goal is to show Z
W1(x, t)ei(t−s)∆h dφ02i
(x−y)W2(y, s)
2
dxdtdyds.kW1k24, 4d d+1
kW2k24, 4d d+1
.
To show this, it suffices to show Z
U1(x, t)
ei(t−s)∆h dφ02i
(x−y)
2
U2(y, s) dxdtdy.kU1k2,2d
d+1kU2k2,2d
d+1, (3.3.13) for any positiveU1, U2. By using the bump functionψon [−1,1], let us decompose the left-hand side of (3.3.13) into
X
j∈Z
Tj(U1, U2) :=X
j∈Z
Z
ψ(2−j(t−s))U1(x, t)
ei(t−s)∆h cφ20i
(x−y)
2
U2(y, s)dxdtdyds
=X
j∈Z
(Tj,1(U1, U2) +Tj,2(U1, U2)), where
Tj,1(U1, U2) :=
Z
ψ(2−j(t−s))χ|x−y|≤C2jU1(x, t)
ei(t−s)∆h dφ02i
(x−y)
2
U2(y, s) dxdtdy and C is a large constant. The term Tj,2(U1, U2) can be treated easily because of the factor χ|x−y|≥C2j ∼χ|x−y|≥C|t−s|. 2
2Thanks to the restriction|x−y| ≥C|t−s|, one has
|ei(t−s)∆h φd02i
(x−y)|.(1 +|x−y|)−N for any largeN which implies
Tj,2(U1, U2).min (2j,2−jN)kU1k2, 2d d+1
kW2k2, 2d d+1
and one can sum-up.
1/r
1 1/s
r∗ 1
r∗
1 r0= 1
s0 1
r0
1 r1 = 1
s1 1
r1
1
0 1
1 2 1
2
A0×B0→C0 A0×B1→C1
A1×B0→C1
Figure 3.2: Bilinear real interpolation, r1∗ = d+12d . On the shaded region, the estimate (3.3.14) is valid.
So, we focus on the estimate of Tj,1(U1, U2). From the Young convolution inequality with the fact that|ei(t−s)∆h
dφ02i
(x−y)|.|t−s|−d2, we have
Tj,1(U1, U2).2j(d+2−d(1r+1s)−1u−1v)kU1ku,rkU2kv,s
for anyu, v, r, s≥1 and 1u+1v,1r+1s ≥1. Especially foru=v= 2,
Tj,1(U1, U2).2j(d+1−d(1r+1s))kU1k2,rkU2k2,s. (3.3.14) Now, we putβ(r, s) =d+ 1−d(1r+1s) and noticeβ(r∗, r∗) = 0 if and only if r1∗ =d+12d . Then chooser0, r1 close enough tor∗ so thatr0< r∗< r1 and sets0=r0, s1=r1. Further put
A0=L2tLrx0, A1=L2tLrx1, B0=L2tLsx0, B1=L2tLsx1,
C0=`β(r∞ 0,s0), C1=`β(r∞ 1,s0)=`β(r∞ 0,s1). The picture is as follows.
Formally, write B ={Tj,1}j∈Z which is a bilinear operator. Also, notice that β(r0, s0) <
0 < β(r1, s0) = β(r0, s1). So, from the bilinear real interpolation, for any θ ∈ (0,1) and θ0, θ1∈(0,1) such thatθ=θ0+θ1,
B={Tj,1}j∈Z: (L2tLrx0, L2tLrx1)θ0,2×(L2tLsx0, L2tLsx1)θ1,2→(`β(r∞0,s0), `β(r∞0,s1))θ,1
=L2tLrxθ/2×L2tLrθ/2→`1,
where we choseθ so thatβ(r0, s0)(1−θ) +β(r0, s1)θ= 0 andθ0=θ1= θ2. It can be checked thatrθ/2=r∗=d+12d , which says that{Tj,1}j∈Z:L2tLrx∗×L2tLr∗→`1, namely
X
j∈Z
Tj,1(U1, U2).kU1k2,r∗kU2k2,r∗.
1/p
1/q D
C
E
A
F
d−1 2d
B 1
12
0 d−1d+1 1
d d+1
This completes the proof.
By employing Stein’s analytic interpolation argument, we may extend the above argument further. We give the picture below.
Proposition 3.3.7. Let d≥3. For any(1p,1q)∈(F, A),
X
j
λj|eit∆P fj|2 Lp
tLqx
.φ0 kλk`p.
Proof of Proposition 3.3.7. Suppose 1p = (d−1)qd and 1q ∈ [d−12d ,d−1d+1). First we use the duality in Proposition 3.3.1 and relabel (σ=p0 andr= 2q0) to re-write the goal as
kW1T0W2kCσ .kW1ku,rkW2ku,r (3.3.15) forσ∈[2, d+ 1), u= 2σand 2dr =d−1σ + 1. Here,T0 is the operator given by
T0F(x, t) = Z
R
ei(t−t0)∆P2F(·, t0)(x) dt0. We decompose this operator dyadicallyT0=P
j∈ZTj by writing TjF(x, t) =
Z
R
ψj(t−t0)ei(t−t0)∆P2F(·, t0)(x) dt0 whereψj =ψ(2·j). Here, we chooseψ∈Cc∞(R) to be supported in [12,2],P
j∈Zψ(2tj) = 1 for t6= 0, and ψ(0) = 0; the existence of such a function is easily justified.b
Also, we write
β(r, s) = d+ 1 2 −d
1 r +1
s
.
We establish estimates at σ= 2 andσ=∞, beginning in the former case with the claim
W1
X
j∈Z
2−β(r,r)jTjW2
C2 .kW1k4,rkW2k4,r (3.3.16) for each r ∈ (2,4). For this, we break-up each Tj one stage further by writingTjF(x, t) = Tj,0F(x, t) +Tj,1F(x, t), where
Tj,1F(x, t) = Z
R
ψj(t−t0)ei(t−t0)∆P2[χB(x,C2j)F](·, t0)(x) dt0
and B(x, C2j) is the ball centred at x with radius C2j, where the constant C is chosen mo-mentarily to be sufficiently large.
The contribution from Tj,0 should be considered as an error term which is more easily handled, so we begin with this part. Observe if Ψ∈C∞(Rd) has the compact Fourier support in the ballB(0,2) and|x| ≥100|t|, then by repeated integration by parts we can obtain
|eit∆Ψ(x)| ≤ CN
(1 +|x|)N (3.3.17)
for anyN ≥0. Since the kernel of W1Tj,0W2 at (x, t, y, t0) is given by W1(x, t)χB(0,C2j)c(x−y)ψj(t−t0)ei(t−t0)∆Ψ(x−y)W2(y, t0) withΨ =b φ2, it follows from (3.3.17) (for an appropriate choice ofC) that
X
j∈Z
2−β(r,r)jkW1Tj,0F W2kC2
≤CN
X
j∈Z
2−β(r,r)j (1 + 2j)N
Z
R2
Z
R2d
|ψj(t−t0)|2|W1(x, t)|2|W2(y, t0)|2
(1 +|x−y|)N dxdydtdt0 1/2
.
Young’s convolution inequality implies X
j∈Z
2−β(r,r)jkW1Tj,0F W2kC2
≤CN
X
j∈Z
2−β(r,r)j (1 + 2j)N
Z
R2
|ψj(t−t0)|2kW1(·, t)k2rkW2(·, t0)k2rdtdt0 1/2
as long asr∈[2,4] andNis sufficiently large, and a further application of the Young inequality gives
X
j∈Z
2−β(r,r)jkW1Tj,0F W2kC2 ≤CN
X
j∈Z
21−β(r,r)j
(1 + 2j)NkW1k4,rkW2k4,r
and hence the desired estimate X
j∈Z
2−β(r,r)jkW1Tj,0W2kC2 .kW1k4,rkW2k4,r.
RegardingTj,1, we first proceed by using the Cauchy–Schwarz inequality to obtain
X
j∈Z
2−β(r,r)jW1Tj,1W2
2
C2
.X
j∈Z
2−2β(r,r)j Z
R2
Z
|x−y|.2j
|ψj(t−t0)||W1(x, t)|2|W2(y, t0)|2
|ei(t−t0)∆Ψ(x−y)|2dxdydtdt0
and to estimate this bykW1k24,rkW2k24,r for eachr∈(2,4), it suffices to prove that X
j∈Z
|Tj,r(V1, V2)|.kV1k2,r/2kV2k2,r/2 (3.3.18) for eachr∈(2,4), whereTj,r is the bilinear operator given by
Tj,r(V1, V2) := 2−2β(r,r)j
Z
R2
Z
|x−y|.2j
|ψj(t−t0)||ei(t−t0)∆Ψ(x−y)|2V1(x, t)V2(y, t0) dxdydtdt0. We shall establish (3.3.18) by bilinear interpolation, so we proceed by fixing r∗ ∈ (2,4) and establishing a range of asymmetric estimates on eachTj,r∗. First, we use the dispersive estimate for the Schr¨odinger propagator to obtain
|Tj,r∗(V1, V2)|.X
j∈Z
2−2β(r∗,r∗)j2−dj Z
R2
Z
|x−y|.2j
|ψj(t−t0)||V1(x, t)||V2(y, t0)|dxdydtdt0 and then, continuing in a similar manner to the case of Tj,0, we use Young’s convolution inequality to deduce that
|Tj,r∗(V1, V2)|
.2−2β(r∗,r∗)j2(2d(1−1r−1s)−d)j Z
R2
|ψj(t−t0)|kV1(·, t)kr/2kV2(·, t0)ks/2dtdt0
for anyr, s≥2 such that 1r+1s ≥ 12. Another use of Young’s convolution inequality allows us to obtain
|Tj,r∗(V1, V2)|.2γ(r,s)jkV1k2,r/2kV2k2,s/2
where
γ(r, s) := 2β(r, s)−2β(r∗, r∗) = 2d 2
r∗ − 1
r +1 s
.
It follows from these estimates that the vector-valued bilinear operatorT ={Tj,r∗}jis bounded between the spaces
A0×B0→C0, (3.3.19)
A0×B1→C1, (3.3.20)
A1×B0→C1, (3.3.21)
where
A0=B0=L2tLrx0/2, A1=B1=L2tLrx1/2 and
C0=`∞γ(r
0,r0), C1=`∞γ(r
1,r0) (3.3.22)
withr0, r1∈(2,4). Specifically, we chooser0, r1∈(2,4) such that r1
0 =r1
∗+δand r1
1 = r1
∗−2δ, whereδ >0 is sufficiently small. This choice ofr0andr1ensures thatγ(r0, r0)<0< γ(r1, r0) = γ(r0, r1).
In the above,k(aj)jk`∞β(X)= supj∈Z2jβkajkX and whenp <∞we have k(aj)jk`p
β(X)=
X
j∈Z
2jβkajkpX 1/p
.
In this notation, (3.3.18) will follow once we prove thatT is bounded as follows:
(L2tLr0/2, L2tLr1/2)1
3,2×(L2tLr0/2, L2tLr1/2)1
3,2→`10. (3.3.23) Indeed (L2tLr0/2, L2tLr1/2)1
3,2=L2tLr∗/2,2 and fromr∗<4 it follows thatL2tLrx∗/2is embedded inL2tLrx∗/2,2; hence we obtain (3.3.18) for allr=r∗∈(2,4).
One can check that by Proposition 3.3.3 with θ = 23 we immediately obtain (3.3.23) and this completes our proof of (3.3.16).
From the above argument, it is clear that
W1X
j∈Z
2zjTjW2 C2
.kW1k4,rkW2k4,r (3.3.24) wheneverz∈Cis such that Re(z) =−β(r, r) andr∈(2,4). Note that Re (z)∈(−12,d−12 ) for this range ofr.
Turning to the caseσ=∞, we claim that
W1
X
j∈Z
2zjTjW2
C∞ .kW1k∞,∞kW2k∞,∞ (3.3.25) wheneverz∈Cis such that Re(z) =−1, and for this, it suffices to prove
X
j∈Z
2zjTjF L2
x,t
.kFkL2
x.t. (3.3.26)
We have
TdjF(ξ, τ) =C2jψ(2b j(τ+|ξ|2))φ0(ξ)2Fb(ξ, τ) and since|ψ(τ)|b .min{|τ|,|τ|−1}it follows that
X
j∈Z
|ψ(2b j(τ+|ξ|2))|.1
uniformly in (ξ, τ). Hence (3.3.26) immediately follows when Re(z) =−1, yielding (3.3.25).
Finally, we use complex interpolation between the estimates (3.3.24) and (3.3.25). Atz= 0 this gives the goal (3.3.15), where the range ofσ∈[2, d+ 1) arises because the exponent rin (3.3.24) is strictly less than 4.
Thanks to Propositions 3.3.6 and 3.3.7, it suffices to show (3.3.12) with p=q=α=∞to obtain Theorem 3.3.5 sinceα∗(p, q) =q+12q on (A, B]. This estimate has already been obtained by Frank-Sabin [43]. However, we give a simple proof by taking advantage of the orthonormality of (fj)j explicitly. The goal is to show
X
j
λj|eit∆P fj|2 L∞
t L∞x
.kλk`∞ (3.3.27)
holds for all orthonormal systems (fj)jinL2(Rd). Recall, thatPis given byP f(ξ) =c φ0(ξ)fb(ξ), whereφ0∈Cc(Rd). Firstly, it suffices to consider the case whereλj= 1 for allj. Now fix any (x, t)∈Rd×Rand define ψx,t(ξ) =e−i(x·ξ−t|ξ|2)φ0(ξ). Then we see from the orthonormality of (fj)j in L2 and Bessel’s inequality that
X
j
|eit∆P fj(x)|2=X
j
|hψdx,t, fjiL2|2≤ kψdx,tk2L2=kφ0k2L2
from which (3.3.27) clearly follows.
Step 2: Restricted-weak type estimate
We first prove the weak type version of (3.1.4) by employing Proposition 3.3.2.
Proposition 3.3.8. Letd≥1,(1q,1p)∈intOAB,2s=d−(2p+dq)andα∗(p, q)be determined by α∗(p,q)d =p1+dq. Then for any (possibly infinite) coefficient λ= (λj)j and any orthonormal system(fj)j inH˙s,
X
j
λj|eit∆fj|2 Lp,∞
t Lqx
≤Ckλk`α,1, (3.3.28)
holds true whenever α≤α∗(p, q).
To this end, by rescaling Theorem 3.3.5 we have the following.
Lemma 3.3.9. Let (1q,1p) ∈ intOAB, α = α∗(p, q) and s ∈ R be arbitrary. Then for any (possibly infinite) coefficientsλ= (λj)j⊂Cand any orthonormal system(fj)j inL2,
X
j
λj|eit∆PkD−sfj|2 Lp
tLqx
.2k(d−2s−2p−dq)kλk`α, (k∈Z). (3.3.29)
Proof of Proposition 3.3.8. Fix anyq∗∈(1,∞) and definep∗,s∗ by 1
p∗ =σ∗(1− 1
q∗), 2s∗=d−(2 p∗ + d
q∗)
whereσ∗∈(0,d2) is arbitrary such that (q1∗,p1∗)∈intOAB. So, our goal is to show
X
j
λj|eit∆D−s∗fj|2 Lp∗,∞
t Lqx∗
.kλk`α, α=α∗(p∗, q∗) (3.3.30)
for any coefficientsλ= (λj)j and any orthonormal system (fj)j inL2. Next, we chooseσ0,σ1
to sandwich the critical caseσ∗. Namely, we take any σ0,σ1 such that 0< σ1 < σ∗< σ0 <d2 andσ0 is any close toσ∗. Then fori= 0,1, we definepi and αi by
1 pi
=σi(1− 1 q∗), d
αi
= 1 pi
+ d q∗.
Again remark that (q1∗,p1
i)∈intOAB as long asσ0 is close toσ∗ enough.
Since the parameters (p0, q∗, α0, s∗) and (p1, q∗, α1, s∗) satisfy the condition in Lemma 3.3.9 with eachσ0 andσ1, we arrive at fori= 0,1,
X
j
λj|eit∆PkD−s∗fj|2 Lpi
t Lqx∗
.2k(d−2s
∗−2
pi−qd∗)
kλk`αi.
Here, by inserting the definition ofpi ands∗ to the exponent of 2k, one notices that d−2s∗− 2
pi
− d
q∗ = 2(σ∗−σi)(1− 1 q∗).
Recall the relation σ1 < σ∗ < σ0. So, if one setε0 =−2(σ∗−σ0)(1− q1∗) andε1 = 2(σ∗− σ1)(1− q1∗) then ε0, ε1 > 0 and it holds for any k ∈ Z, any λ= (λj) and any orthonormal
system (fj)j inL2 that
X
j
λj|eit∆PkD−s∗fj|2 Lp0
t Lqx∗
.2−kε0kλk`α0, (3.3.31)
X
j
λj|eit∆PkD−s∗fj|2 Lp1
t Lqx∗
.2kε1kλk`α1. (3.3.32) By the estimates (3.3.31) and (3.3.32), we may employ Proposition 3.3.2 withgj=eit∆D−s∗fj to see
X
j
λj|eit∆D−s∗fj|2 Lp,∞
t Lqx∗
.kλk`α∗(p∗,q∗),1,
for anyλ= (λj)j and any orthonormal system (fj)j inL2, where 1/p=θ/p0+ (1−θ)/p1. By θ= εε1
0+ε1, we noticep=p∗ for any choice ofσ0, σ1. Step 3: Upgrade to strong type estimate
To prove Theorem 3.2.4, we have to replace Lp,∞t -norm in the left-hand side of (3.3.28) by Lpt-norm. To do this, we give further observation, namely, the Lorentz improvement of the orthonormal Strichartz inequality, Theorem 3.1.1.
Proposition 3.3.10. Suppose that the parameters p, q, r, αsatisfy d−1
d+ 1 < 1 q < d
d+ 2, 1 p= d
2(1−1
q), r=α= 2q q+ 1. Then for anyλ= (λj)j⊂Cand any orthonormal system(fj)j inL2,
X
j
λj|eit∆fj|2 Lp,r
t Lqx
.kλk`α. (3.3.33)
One may notice thatr= q+12q < pif 1q > d+1d−1. Hence, Proposition 3.3.10 improves Theorem 3.1.1 although we do not knowr=q+12q is sharp or not. It may be interesting to determine the sharprfor the inequality of (3.3.33) but we do not go to in this direction. It is also remarkable that such a Lorentz improvement is not so surprising. Indeed, all we have to do to show Proposition 3.3.10 is just to use the Lorentz improvement of the Hardy-Littlewood-Sobolev inequality due to O’Neil [81]: for any non-negative measurable functionsf1andf2
Z
Rn
Z
Rn
f1(x1)f2(x2)
|x1−x2|λ dx1dx2≤Ckf1kLp1,r1kf2kLp2,r2 (3.3.34) wheren≥1,λ∈(0, n), pj ∈(1,∞) satisfy p1
1 +p1
2 +λn = 2, and r1
1 +r1
2 ≥1, instead of the classical Hardy-Littlewood-Sobolev inequality. A similar Lorentz improvement can be found in [3, 6].
Proof of Proposition 3.3.10. Since the proof is almost the same to the one given by Frank and Sabin [42], we give the sketch of its proof. Corresponding to the duality principle [42, Lemma 3], we have its Lorentz space version. Namely, the Lorentz orthonormal Strichartz inequality (3.3.33) holds if and only if
kW eit∆(eit∆)∗WkCα0 .kWk2
L2pt 0,2r0L2qx0 (3.3.35)
for anyW ∈L2pt 0,2r0L2qx0. To deduce (3.3.35) we consider the family of operatorsTzdefined by F[Tzφ](τ, ξ) =Gz(τ, ξ) ˆφ(τ, ξ), (τ, ξ)∈R×Rd
whereGz(τ, ξ) = (τ− |ξ|2)z+/Γ(z+ 1) and d+12 <−Re(z)< d+22 . Note thatT−1=eit∆(eit∆)∗. For this family of operators, by using (3.3.34) instead of the classical Hardy-Littlewood-Sobolev inequality, we obtain
W1TzW2
2
C2 ≤C(Im(z))kW1k2
Lp,4t˜ L2xkW2k2
Lp,4t˜ L2x,
where 2p˜ =−Re(z)−d2 and d+12 <−Re(z)< d+22 . Here C(Im(z)) grows sufficiently slowly to use the analytic interpolation. On the other hand, we have from the Plancherel theorem that
W1TzW2
C∞ ≤C(Im(z))kW1kL∞t L∞x kW2kL∞t L∞x ,
if Re(z) = 0. By the analytic interpolation [42], we obtain the desired inequality (3.3.35) or equivalently (3.3.33) withα=r=q+12q .
Combining Propositions 3.3.8 and 3.3.10, let us complete the proof of Theorem 3.2.4. We first upgrade Proposition 3.3.8 to the restricted type estimate as follows:
Proposition 3.3.11. Supposed≥1 and let(1q,1p)belong tointOGA, where G= (d+2d ,d+2d ).
If2s=d−(2p+dq)andα=α∗(p, q), then
X
j
λj|eit∆fj|2 Lp
tLqx
.kλk`α,1
holds for all orthonormal sequences (fj)j inH˙s(Rd) and all sequencesλ= (λj)j in`α,1(C).
Proof. We fix (1q,1p) in intOGAand let (q1
0,p1
0) be the intersection point of the line through the origin and (1q,1p) with the line segment (A, G). Next we defineε0=12(d+1q
0 −(d−1))∈(0,d+21 ) andθ= 1−1p(p1
0+ε0)−1∈(0,1).Finally, we define (p1
1,q1
1) by 1p = 1−θp
0 +pθ
1 and 1q =1−θq
0 +qθ
1. One can check that (p1
1,q1
1) belongs to intOAB, and therefore an application of Proposition 3.3.8 gives
X
j
λj|eit∆D−s1fj|2 Lp1,∞
t Lqx1
.kλk`α1,1, whereα1=α∗(p1, q1) and 2s1=d−(p2
1+qd
1). Also, we have
X
j
λj|eit∆D−s0fj|2 Lp0,r0
t Lqx0
.kλk`α0,1, where α0 = r0 = α∗(p0, q0) and p2
0 + qd
0 = d, by Proposition 3.3.10. We claim that using complex interpolation between these two estimates withθ above gives the estimate
X
j
λj|eit∆D−sfj|2 Lp
tLqx
.kλk`α∗(p,q),β
for someβ ≥1 and where 2s=d−(2p+dq) (hence slightly stronger than the desired estimate).
Indeed, it is clear that 2s= 2s1θ=d−(2p+dq) and αgiven by α1 = 1−θα
0 +αθ
1 coincides with α∗(p, q). Finally, a computation shows that ifr is given by 1r = 1−θr
0 , thenr=p.
Finally, we upgrade Proposition 3.3.11 to the strong type estimate. Observe that it is sufficient to show the desired estimate for all (1q,1p) belonging to intOGA. Indeed, once this is established, we may employ complex interpolation once again with the estimates in Theorem 3.1.1 on the line segment [B, A) to obtain desired estimates for all (1q,1p) belonging to intOAB.
As we shall soon see, the advantage of first considering the regionOGAis that
α∗(p, q)≤p≤q (3.3.36)
whenever (1q,1p) belongs to OGA. Indeed, one easily sees that α∗(p, q) ≤ pis equivalent to
1
p ≤ (d−1)qd which means below the line [O, A] (andp≤qobviously holds inOGAsinceGlies on the diagonal 1q =1p).
Since we wish to use real interpolation, we fixs ∈(0,d2) and take any two points (q1
i,p1
i) from intOGAsuch that p2
i +qd
i =d−2sfor i= 0,1. From Proposition 3.3.11 we know that Os(pi, qi; (α∗(pi, qi),1)):
X
j
λj|eit∆D−sfj|2 Lpi
t Lqix
.kλk`α∗(pi,qi),1
holds fori= 0,1. This means that if we fix an orthonormal system (fj)j in the common space H˙s, then real interpolation, (3.3.4) and (3.3.5) yield
X
j
λj|eit∆fj|2 Lp
tLq,px
.kλk`α∗(p,q),p (3.3.37)
with 1p = 1−θp
0 + pθ
1, 1q = 1−θq
0 + qθ
1 and any θ ∈ (0,1); that is, (3.3.37) holds for all (1q,1p) belonging to intOGA. Thanks to (3.3.36), we may deduce from the nesting of Lorentz spaces
that
X
j
λj|eit∆fj|2 Lp
tLqx
.kλk`α∗(p,q)
and therefore desired estimates holds for all (1q,p1) belonging to intOGAwith 2s= 2p+dq, as claimed.