AN ALGEBRA FOR CRITICAL REGULARITY SOLUTIONS TO THE FREE WAVE EQUATION

TERENCE TAO

Abstract. We present a Banach space V ∩ Ḣ^{n/2,1/2} ∩ L^{∞} of functions in R^{n+1} which is closed under multiplication and which contains all bounded free Ḣ^{n/2} solutions to the free wave equation, when n > 2.

1. Introduction

(Note: This is NOT an expository article, but rather a technical calculation which seems to have some mild interest to the critical regularity wave map problem. I haven’t been able to take this idea much further than what’s in these notes, so feel free to do something with it.)

Let n > 2. Let F denote the Banach space of all free Ḣ^{n/2} solutions to the wave equation, and let F ∩L^{∞} denote the subspace of those solutions which remain bounded. In a recent article of Klainerman and Machedon, it is shown (among other things) that polynomial expressions of F ∩ L^{∞} always stay in the space Ḣ^{n/2,1/2} of functions ϕ on R^{n+1} such that

D_{+}^{n/2}D_{−}^{1/2}ϕ ∈ L^{2},

where D_{+} and D_{−} are the Fourier multipliers corresponding to |τ|+|ξ| and ||τ| − |ξ|| respectively. We adopt the convention that functions in F are also in Ḣ^{n/2,1/2}; such functions are in the weak closure of Ḣ^{n/2,1/2}, and in practice one can perform a smooth time localization to some large compact time interval [−T, T], using the ideas first used by Bourgain, to make the imbedding more precise.

This result would be easy to show if Ḣ^{n/2,1/2} ∩ L^{∞} were an algebra, but this is unlikely to be the case; certainly for n = 1 this space is not closed under multiplication. However in this note we show that there exists a subspace V ∩ Ḣ^{n/2,1/2} ∩ L^{∞} of this space which contains F and which does turn out to be an algebra, thus recovering Klainerman and Machedon’s result. The algebra involves Besov-like decompositions into frequency-localized pieces, and uses pointwise majorization in physical space as well as in frequency space.

Unfortunately there are technical difficulties when n = 2, which probably require a modification of the definition of the space V, but they do not appear insurmountable. It would of course also be very nice if the space is closed under the bilinear operation (ϕ, ψ) → □^{−1}(ϕ□ψ). Although this operation does seem to map to Ḣ^{n/2,1/2} ∩ V if the dimension is sufficiently high, it does not map into L^{∞}, so this space is not quite powerful enough to resolve a critical well-posedness problem.

We begin the construction of the algebra. For any integer j, let Q_{j}^{+} denote the Littlewood-Paley projection to the region of frequency space where D_{+} ∼ 2^{j}, in such a way that ∑_{j} Q_{j}^{+} is the identity. Let F_{j} = Q_{j}^{+}F denote the Banach space of functions of functions of the form Q_{j}^{+}ϕ, where ϕ is in F; this is the space of all Ḣ^{n/2} free solutions with frequencies ∼ 2^{j}.

1991 Mathematics Subject Classification. 35L70.

Consider the set F_{j} · L^{∞}, the set of all functions which are pointwise dominated by a function in F_{j}. This set is not closed under addition and is therefore not a Banach space; however we can complete it by taking the elements of that set to be atoms. More precisely, we define V_{j} to be the closure of the normed space whose unit ball is

{∑_{α} λ_{α}Q_{j}^{+}(ϕ_{α})χ_{α} : ∑_{α} |λ_{α}| ≤ 1, ∥ϕ_{α}∥_{F} ≤ 1, ∥χ_{α}∥_{∞} ≤ 1}

and α can range over any finite index set. It is easy to see that V_{j} is a Banach space.

The space F can be recovered from its components F_{j} by the formula

∥ϕ∥_{F} ∼ (∑_{j} ∥Q_{j}^{+}ϕ∥_{F_{j}}^{2})^{1/2}.

In analogy with this formula, we define the Banach space V by

∥ϕ∥_{V} ∼ (∑_{j} ∥Q_{j}^{+}ϕ∥_{V_{j}}^{2})^{1/2}.

Since V_{j} clearly contains F_{j}, we see that V contains F. Thus to complete the claim of this note, we need to show

Proposition 1.1. The space V ∩ Ḣ^{n/2,1/2} ∩ L^{∞} is an algebra.

We now begin the proof. Let ϕ, ψ be functions such that

∥ϕ∥_{V} , ∥ϕ∥Ḣ^{n/2,1/2}, ∥ϕ∥_{∞}, ∥ψ∥_{V} , ∥ψ∥Ḣ^{n/2,1/2}, ∥ψ∥_{∞} ≲ 1.

We have to show that

∥ϕψ∥_{∞} ≲ 1.  (1)
∥ϕψ∥Ḣ^{n/2,1/2} ≲ 1  (2)
∥ϕψ∥_{V} ≲ 1  (3)

The estimate (1) is trivial from the L^{∞} control on ϕ, ψ, so it remains to show (2) and (3).

2. Proof of (2).

We use the formula

∥ϕ∥Ḣ^{n/2,1/2} ∼ (∑_{j,k}(2^{nj/2}2^{k/2}∥Q_{j}^{+}Q_{k}^{−}ϕ∥_{2})^{2})^{1/2} ,  (4)

where Q_{k}^{−} is the Fourier projection to the region of frequency space in which D_{−} ∼ 2^{k}. We thus consider the problem of controlling

∥Q_{j}^{+}Q_{k}^{−}(ϕψ)∥_{2}  (5)

for some integers j, k, which we now fix. We may of course assume j ≥ k − C since the expression vanishes otherwise.

We break up ϕ, ψ dyadically as

ϕ = P_{k−C}^{+}ϕ + ∑_{j_{1}>k−C} Q_{j_{1}}^{+}ϕ

ψ = P_{k−C}^{+}ψ + ∑_{j_{2}>k−C} Q_{j_{2}}^{+}ψ

where P_{k}^{+} = ∑_{j≤k} Q_{j}^{+}, and consider the various contributions of the summands to (5) and hence to (4).

We first consider the contribution of P_{k−C}^{+}ϕ. The only terms in the expansion of ψ which give a nonvanishing contribution to (5) are the ones in which 2^{j_{2}} ∼ 2^{j}. In fact one can see from an examination of frequency supports that

Q_{j}^{+}Q_{k}^{−}((P_{k−C}^{+}ϕ)ψ) = ∑_{j′=j+O(1)} ∑_{k′=k+O(1)} Q_{j}^{+}Q_{k}^{−}((P_{k−C}^{+}ϕ)Q_{j′}^{+}Q_{k′}^{−}ψ).

We can thus essentially reduce to estimating

Q_{j}^{+}Q_{k}^{−}((P_{k−C}^{+}ϕ)Q_{j}^{+}Q_{k}^{−}ψ).

Since Q_{j}^{+}Q_{k}^{−} is bounded on L^{2}, the contribution of P_{k−C}^{+}ϕ to (5) is thus majorized by

∥(P_{k−C}^{+}ϕ)Q_{j}^{+}Q_{k}^{−}ψ∥_{2}.

Since ϕ is in L^{∞} by hypothesis and P_{k−C}^{+} is an approximation to the identity, P_{k−C}^{+}ϕ is in L^{∞} and we can therefore estimate the above by

∥Q_{j}^{+}Q_{k}^{−}ψ∥_{2}.

By (4), the contribution of the above to (2) is thus majorized by

(∑_{j,k}(2^{nj/2}2^{k/2}∥Q_{j_{2}}^{+}Q_{k_{2}}^{−}ψ∥_{2})^{2})^{1/2} .

But by (4) this is comparable to ∥ψ∥Ḣ^{n/2,1/2}, which is acceptable by hypothesis.

A similar argument disposes of the contribution of P_{k−C}^{+}ψ. Note that the simultaneous contribution of P_{k−C}^{+}ϕ and P_{k−C}^{+}ψ to (5) vanishes.

By the triangle inequality, the remaining contribution to (5) is majorized by

∑_{j_{1}>k−C} ∑_{j_{2}>k−C} ∥Q_{j}^{+}Q_{k}^{−}(Q_{j_{1}}^{+}ϕQ_{j_{2}}^{+}ψ)∥_{2}.

By symmetry we can restrict ourselves to the case j_{1} ≥ j_{2}. By inspecting the frequency support of Q_{j_{1}}^{+}ϕQ_{j_{2}}^{+}ψ, we see that the summand vanishes unless the two largest members in {j, j_{1}, j_{2}} are within O(1) of each other. We may therefore estimate the above sum by

∑_{j_{1}>j−C} ∑_{j_{2}=j_{1}+O(1)} + ∑_{j_{1}=j+O(1)} ∑_{k−C<j_{2}<j+C} ∥Q_{j}^{+}Q_{k}^{−}(Q_{j_{1}}^{+}ϕQ_{j_{2}}^{+}ψ)∥_{2}.  (6)

We may drop the operator Q_{j}^{+}Q_{k}^{−} since it is bounded in L^{2}.

We now use

Proposition 2.1. Let j ≥ k − C. If ϕ ∈ V_{j}, ψ ∈ V_{k}, and n > 2, then

∥ϕψ∥_{2} ≲ 2^{−nj/2}2^{−k/2}∥ϕ∥_{V_{j}}∥ψ∥_{V_{k}}.

Proof We may assume that ϕ, ψ are atoms of V_{j}, V_{k} respectively. Since multiplication by L^{∞} functions has no harmful effect on ∥ϕψ∥_{2}, it thus suffices to show that

∥Q_{j}^{+}(ϕ)Q_{k}^{+}(ψ)∥_{2} ≲ 2^{−nj/2}2^{−k/2}∥ϕ∥_{F_{j}}∥ψ∥_{F_{k}}

for all ϕ ∈ F_{j}, ψ ∈ F_{k}.

By Plancherel’s theorem, this is equivalent to showing that

∥fdσ_{j} ∗ gdσ_{k}∥_{2} ≲ 2^{−nj/2}2^{−k/2}2^{nj/2}∥f∥_{2}2^{nk/2}∥g∥_{2}

for all f ∈ L^{2}(S_{j}), g ∈ L^{2}(S_{k}), where S_{j} is the surface {(τ, ξ) : |τ| = |ξ| ∼ 2^{j}} with surface measure dσ_{j}, and similarly for S_{k}. By interpolation it suffices to prove the estimates

∥fdσ_{j} ∗ gdσ_{k}∥_{1} ≲ ∥f∥_{1}∥g∥_{1}

and

∥fdσ_{j} ∗ gdσ_{k}∥_{∞} ≲ 2^{−nj}2^{−k}2^{nj}∥f∥_{∞}2^{nk}∥g∥_{∞}.

The former estimate is just Young’s inequality, whereas the latter follows from the easily verified calculation

∥dσ_{j} ∗ dσ_{k}∥_{∞} ≲ 2^{(n−1)k}.

Note that this last calculation requires n > 2.

From this proposition, we can bound (6) by

∑_{j_{1}>j−C} ∑_{j_{2}=j_{1}+O(1)} + ∑_{j_{1}=j+O(1)} ∑_{k−C<j_{2}<j+C} 2^{−nj_{1}/2}2^{−j_{2}/2}∥ϕ∥_{V_{j_{1}}} ∥ψ∥_{V_{j_{2}}}

This in turn is easily seen to be majorized by

2^{−nj/2}2^{−k/2}(∑_{j_{1}} 2^{−|j_{1}−j|/2}∥ϕ∥_{V_{j_{1}}} )(∑_{j_{2}} 2^{−|j_{2}−k|/2}∥ψ∥_{V_{j_{2}}} ).

By (4), the contribution to (2) is thus majorized by

(∑_{j,k}(∑_{j_{1}} 2^{−|j_{1}−j|/2}∥ϕ∥_{V_{j_{1}}} )^{2}(∑_{j_{2}} 2^{−|j_{2}−k|/2}∥ψ∥_{V_{j_{2}}} )^{2})^{1/2} .

This factors as

(∑_{j}(∑_{j_{1}} 2^{−|j_{1}−j|/2}∥ϕ∥_{V_{j_{1}}} )^{2})^{1/2} (∑_{k}(∑_{j_{2}} 2^{−|j_{2}−k|/2}∥ψ∥_{V_{j_{2}}} )^{2})^{1/2} .

By Young’s inequality for sums, and the fact that the kernel 2^{−|m|/2} is summable in m, this is majorized by

(∑_{j_{1}} ∥ϕ∥_{V_{j_{1}}}^{2} )^{1/2} (∑_{j_{2}} ∥ψ∥_{V_{j_{2}}}^{2} )^{1/2} = ∥ϕ∥_{V} ∥ψ∥_{V} ,

which is acceptable by hypothesis.

3. Proof of (3)

To conclude the proof of the Proposition, we need to show that

(∑_{j} ∥Q_{j}^{+}(ϕψ)∥_{V_{j}}^{2})^{1/2} ≲ 1.  (7)

We split Q_{j}^{+}(ϕψ) as the sum of

Q_{j}^{+}(ϕ)P_{j}^{+}(ψ) + P_{j}^{+}(ϕ)Q_{j}^{+}(ψ)  (8)

and

Q_{j}^{+}(ϕψ) − Q_{j}^{+}(ϕ)P_{j}^{+}(ψ) − P_{j}^{+}(ϕ)Q_{j}^{+}(ψ).  (9)

First consider (8). Since ψ, ϕ are in L^{∞} by hypothesis, and P_{j} is an approximation to the identity, we have that P_{j}^{+}(ψ), P_{j}^{+}(ϕ) are in L^{∞}. Since the spaces V_{j} are clearly closed under multiplication by L^{∞} functions, we thus have

∥Q_{j}^{+}(ϕ)P_{j}^{+}(ψ) + P_{j}^{+}(ϕ)Q_{j}^{+}(ψ)∥_{V_{j}} ≲ ∥Q_{j}^{+}(ϕ)∥_{V_{j}} + ∥Q_{j}^{+}(ψ)∥_{V_{j}}.

Thus the contribution of (8) to (7) is majorized by

(∑_{j} ∥Q_{j}^{+}(ϕ)∥_{V_{j}}^{2})^{1/2} + (∑_{j} ∥Q_{j}^{+}(ψ)∥_{V_{j}}^{2})^{1/2} = ∥ϕ∥_{V} + ∥ψ∥_{V} ,

which is acceptable by hypothesis.

It remains to control the error term (9). We will in fact prove that

\begin{matrix}∥Q_{j}^{+}(ϕψ) − Q_{j}^{+}(ϕ)P_{j}^{+}(ψ)− & P_{j}^{+}(ϕ)Q_{j}^{+}(ψ)∥_{2} ≲ 2^{−nj/2}2^{−j/2} \\ (∑_{j_{1}} 2^{−|j_{1}−j|/2}∥Q_{j_{1}}^{+}ϕ∥_{V_{j_{1}}} )(∑_{j_{2}} 2^{−|j_{2}−j|/2}∥Q_{j_{2}}^{+}ψ∥_{V_{j_{2}}} ). \end{matrix} (10)

We will prove this estimate momentarily. To apply (10), we need the following

Lemma 3.1. If ϕ has frequency support in the region |ξ| + |τ| ∼ 2^{j}, then

∥ϕ∥_{V_{j}} ≲ 2^{nj/2}2^{j/2}∥ϕ∥_{2}.

Proof By partitioning ϕ and multiplying by a harmless phase function to translate the frequency support, we may assume that ϕ has Fourier transform on the set {(τ, ξ) : τ = |ξ| + O(2^{j}), ξ ∈ A_{j}}, where A_{j} is an annulus on which the symbol of Q_{j}^{+} is bounded from below. We then apply the formula

ϕ̂(τ, ξ) = ∫_{|λ|≲2^{j}} ϕ̂(|ξ| + λ, ξ)δ(τ − λ − |ξ|) dλ.

The inverse Fourier transform of ϕ̂(|ξ|+λ, ξ)δ(τ −λ−|ξ|) is equal to a phase function multiplied by a solution to the free wave equation, with Ḣ^{n/2} norm comparable to

2^{nj/2}(∫_{A_{j}} |ϕ̂(|ξ| + λ, ξ)|^{2} dξ)^{1/2}.

Thus by the integral form of the triangle inequality, we have

∥ϕ∥_{V_{j}} ≲ 2^{nj/2} ∫_{|λ|≲2^{j}}(∫_{A_{j}} |ϕ̂(|ξ| + λ, ξ)|^{2} dξ)^{1/2} dλ.

By Cauchy-Schwarz, the right-hand side is majorized by

2^{nj/2}2^{j/2}(∫_{|λ|≲2^{j}} ∫_{A_{j}} |ϕ̂(|ξ| + λ, ξ)|^{2} dξ dλ)^{1/2},

and the claim follows from a change of variables and Plancherel’s theorem.

From this lemma, we see that the contribution of (9) to (7) is majorized by

(∑_{j}(∑_{j_{1}} 2^{−|j_{1}−j|/2}∥Q_{j_{1}}^{+}ϕ∥_{V_{j_{1}}} )^{2}(∑_{j_{2}} 2^{−|j_{2}−j|/2}∥Q_{j_{2}}^{+}ψ∥_{V_{j_{2}}} )^{2})^{1/2}.

By Cauchy-Schwarz, this is majorized by

(∑_{j}(∑_{j_{1}} 2^{−|j_{1}−j|/2}∥Q_{j_{1}}^{+}ϕ∥_{V_{j_{1}}} )^{4})^{1/4}(∑_{j}(∑_{j_{2}} 2^{−|j_{2}−j|/2}∥Q_{j_{2}}^{+}ψ∥_{V_{j_{2}}} )^{4})^{1/4}.

By Young’s inequality for sums and the fact that 2^{−|m|/2} is in l^{4/3}, this is majorized by

(∑_{j_{1}} ∥Q_{j_{1}}^{+}ϕ∥_{V_{j_{1}}} )^{2})^{1/2}(∑_{j_{2}} ∥Q_{j_{2}}^{+}ψ∥_{V_{j_{2}}} )^{2})^{1/2} = ∥ϕ∥_{V} ∥ψ∥_{V} ,

which is acceptable by hypothesis.

It remains to prove (10). We decompose this expression dyadically as

Q_{j}^{+}(ϕψ)−Q_{j}^{+}(ϕ)P_{j}^{+}(ψ) − P_{j}^{+}(ϕ)Q_{j}^{+}(ψ) =

∑_{j_{1}} ∑_{j_{2}} Q_{j}^{+}(Q_{j_{1}}^{+}ϕQ_{j_{2}}^{+}ψ) − Q_{j}^{+}(Q_{j_{1}}^{+}ϕ)P_{j}^{+}(Q_{j_{2}}^{+}ψ) − P_{j}^{+}(Q_{j_{1}}^{+}ϕ)Q_{j}^{+}(Q_{j_{2}}^{+}ψ).

It suffices to prove (10) for a single pair of indices j_{1}, j_{2}, with an exponential decay in |j_{1} − j|, |j_{2} − j|. More precisely, it suffices to show that

\begin{matrix}∥Q_{j}^{+}(Q_{j_{1}}^{+}ϕQ_{j_{2}}^{+}ψ) & − Q_{j}^{+}(Q_{j_{1}}^{+}ϕ)P_{j}^{+}(Q_{j_{2}}^{+}ψ) − P_{j}^{+}(Q_{j_{1}}^{+}ϕ)Q_{j}^{+}(Q_{j_{2}}^{+}ψ)∥_{2} ≲ \\ 2^{−|j_{1}−j|/2}2^{−|j_{2}−j|/2}2^{−nj/2}2^{−j/2}∥Q_{j_{1}}^{+}ϕ∥_{V_{j_{1}}} ∥Q_{j_{2}}^{+}ψ∥_{V_{j_{2}}} , \end{matrix} (11)

since (10) follows by summing in j_{1}, j_{2}, then using the triangle inequality and Cauchy-Schwarz.

We may apply the rescaling ϕ̃(t, x) = ϕ(2^{−j}t, 2^{−j}x), ψ̃(t, x) = ψ(2^{−j}t, 2^{−j}x) and assume that j = 0. By symmetry we may take j_{1} ≥ j_{2}.

We first consider the case when j_{1} ≥ j_{2} > −C for some large C. We then estimate (11) crudely as

∥Q_{0}^{+}(Q_{j_{1}}^{+}ϕQ_{j_{2}}^{+}ψ)∥_{2} + ∥Q_{0}^{+}(Q_{j_{1}}^{+}ϕ)P_{0}^{+}(Q_{j_{2}}^{+}ψ)∥_{2} + ∥P_{0}^{+}(Q_{j_{1}}^{+}ϕ)Q_{0}^{+}(Q_{j_{2}}^{+}ψ)∥_{2}.

Discarding the Q_{0}^{+} projection from the first term, we apply Proposition 2.1 to estimate this by

2^{−nj_{1}/2}2^{−j_{2}/2}∥Q_{j_{1}}^{+}ϕ∥_{V_{j_{1}}} ∥Q_{j_{2}}^{+}ψ∥_{V_{j_{2}}} ,

which implies (11) for this case.

We can therefore assume that j_{2} < −C; this implies j_{1} = O(1) since (11) vanishes otherwise. The left-hand side of (11) then simplifies to

∥Q_{0}^{+}(Q_{j_{1}}^{+}ϕQ_{j_{2}}^{+}ψ) − Q_{0}^{+}(Q_{j_{1}}^{+}ϕ)Q_{j_{2}}^{+}ψ∥_{2}.  (12)

The idea is to exploit the cancellation between the two terms on the multiplier level. The expression inside the norm can be written as

[Q_{0}^{+}(Q_{j_{1}}^{+}ϕQ_{j_{2}}^{+}ψ) − Q_{0}^{+}(Q_{j_{1}}^{+}ϕ)Q_{j_{2}}^{+}ψ](x̃) = ∫_{|ξ̃|∼2^{j_{1}}} ∫_{|η̃|∼2^{j_{2}}} e^{2πix̃·(ξ̃+η̃)}m(ξ̃, η̃,)ϕ̂(ξ̃)ψ̂(η̃) dξ̃dη̃ (13) where to avoid notational complexity we have used spacetime variables ξ̃, η̃, x̃ in the Euclidean space R^{n+1}, as opposed to the more usual separate variables for space and time; m is the multiplier

m(ξ̃, η̃) = (ψ(ξ̃+ η̃) − ψ(ξ̃))ψ(2^{−j_{1}} ξ̃)ψ(2^{−j_{2}} η̃),

and ψ is the symbol of the Littlewood-Paley projection Q_{0}^{+}.

Since η̃ is constrained to be about 2^{j_{2}}, the expression ψ(ξ̃+ η̃) − ψ(ξ̃) is O(2^{j_{2}}). In fact, we have the symbol bounds

∥∂α ; ξ̃∂β ; η̃m∥_{∞} ≤ C_{α,β}2^{j_{2}}2^{−j_{1}|α|}2^{−j_{2}|β|}

for all α, β. We can therefore decompose m as a Fourier series in the box ξ̃ ∈ [−C2^{j_{1}}, C2^{j_{1}}]^{n+1}, η̃ ∈ [−C2^{j_{2}}, C2^{j_{2}}]^{n+1} as

m(ξ̃, η̃) = 2^{j_{2}} ∑_{q∈Z^{n+1}} ∑_{r∈Z^{n+1}} c_{q,r}e^{2πiC^{−1}2^{−j_{1}} q·ξ̃}e^{2πiC^{−1}2^{−j_{2}} r·η̃}ψ(2^{−j_{1}} ξ̃)ψ(2^{−j_{2}} η̃)

where the Fourier coefficients c_{q,r} are rapidly decreasing. Inserting this into (13) and simplifying using the Fourier inversion formula, we can rewrite (13) as

Q_{0}^{+}(Q_{j_{1}}^{+}ϕQ_{j_{2}}^{+}ψ)−Q_{0}^{+}(Q_{j_{1}}^{+}ϕ)Q_{j_{2}}^{+}ψ = 2^{j_{2}} ∑_{q} ∑_{r} c_{q,r}(τ_{C^{−1}2^{−j_{1}} q}Q_{j_{1}}^{+}ϕ)(τ_{C^{−1}2^{−j_{2}} r}Q_{j_{2}}^{+}ψ),

where τ_{y} denotes a translation in physical spacetime by y.

We now apply the triangle inequality and Proposition 2.1 to estimate (12) by

2^{j_{2}} ∑_{q} ∑_{r} |c_{q,r}|2^{−nj_{1}/2}2^{−j_{2}/2}∥τ_{C^{−1}2^{−j_{1}} q}Q_{j_{1}}^{+}ϕ∥_{V_{j_{1}}} ∥τ_{C^{−1}2^{−j_{2}} r}Q_{j_{2}}^{+}ψ∥_{V_{j_{2}}} .

The norms V_{j} are easily seen to be translation invariant, so we can rewrite this as

2^{−nj_{1}/2}2^{j_{2}/2}(∑_{q} ∑_{r} |c_{q,r}|)∥Q_{j_{1}}^{+}ϕ∥_{V_{j_{1}}} ∥Q_{j_{2}}^{+}ψ∥_{V_{j_{2}}} .

Since j_{1} = O(1), j_{2} < −C, and c_{q,r} is rapidly decreasing, we thus obtain (11) as desired.

Department of Mathematics, UCLA, Los Angeles, CA 90024

   E-mail address: tao@math.ucla.edu