1. Statement of result

Let R_{1}, …, R_{d} be the Riesz transforms of ℝ^{d} (thus R_{j} is a Fourier multiplier with symbol iξ_{j}/|ξ_{j}|). The famous Fefferman-Stein decomposition ^{\underline{fefferman-stein}} [1] asserts:

fs Theorem 1.1 (Fefferman-Stein theorem). Suppose thatf ∈ BMO(ℝ^{d}) goes to zero at infinity (thus lim_{x→∞} f(x) = 0). Then there exists g_{0}, g_{1}, …, g_{d} ∈ L^{∞}(ℝ^{d}) such that

f = g_{0} + R_{1}g_{1} + … + R_{d}g_{d}

and furthermore

∥g_{0}∥_{L^{∞}(ℝ^{d})}, …, ∥g_{d}∥_{L^{∞}(ℝ^{d})} ≲ ∥f∥_{BMO(ℝ^{d})};

here and in the sequel we allow implied constants to depend on d.

The proof in ^{\underline{fefferman-stein}} [1] was indirect, using the Hahn-Banach theorem followed by a harmonic majorisation argument. In ^{\underline{uchiyama}} [3] Uchiyama gave a constructive proof of Theorem \underline{fs} 1.1 avoiding both the Hahn-Banach theorem and harmonic majorisation, in particular extending the decomposition to more general Fourier multipliers than R_{1}, …, R_{d}. We present Uchiyama’s proof here (but not in full generality).

2. Preliminary reductions

We may take f (and the g_{j}) to be real-valued. Write R⃗ for the vector-valued operator R⃗ := (1, R_{1}, …, R_{d}), and g⃗ for g⃗ = (g_{0}, …, g_{d}). Thus we are asserting for each f ∈ BMO(ℝ^{d}) going to zero at infinity, there exists a solution to the vector-valued problem

\begin{matrix}R⃗ · g⃗ & = f \\ ∥g⃗∥_{L^{∞}(ℝ^{d}→ℝ^{d+1})} & ≲ ∥f∥_{BMO(ℝ^{d})}. \end{matrix}  (2.1) linprob

This is a linear problem, but interestingly it is not possible to express g⃗ as a linear operator of f; if we had g⃗ = T⃗f for some linear operator T⃗ : BMO(ℝ^{d}) → L^{∞}(ℝ^{d} → ℝ^{d+1}), then by averaging over translations one can make T⃗ translation invariant; similarly one can make it dilation invariant, and reflection and rotation invariant (rotating and reflecting the Riesz transform vector along with the underlying space). This makes T⃗ a combination of the Riesz transforms themselves, but these are not bounded from BMO to L^{∞}.

On the other hand, suppose f was localised in frequency to an annulus {ξ : |ξ| ∼ 2^{j}}, thus one has a reproducing formula f = P_{j}f for some Littlewood-Paley type operator P_{j}. Then we can write

f = ∑_{k=1}^{d} R_{k}(R_{k}P_{j}f).

It is easy to see that R_{k}P_{j} is bounded from BMO(ℝ^{d}) to L^{∞}(ℝ^{d}), and so we can solve the problem (^{\underline{linprob}} 2.1) by linear means at each individual frequency scale. The remaining problem is how to concatenate all the frequency scales together (in some nonlinear fashion) without losing control of the solution.

We first perform a standard reduction to an apparently weaker assertion in which a small error in the equation is permitted.

Theorem 2.1 (First reduction). Let 0 < ε ≪ 1 be small. Then for any f ∈ BMO(ℝ^{d}) going to zero at infinity with ∥f∥_{BMO(ℝ^{d})} ≤ ε there exists g⃗ ∈ L^{∞}(ℝ^{d} → ℝ^{d+1}) going to a constant at infinity with ∥g⃗∥_{L^{∞}(ℝ^{d}→ℝ^{d+1})} = 1 such that

∥f − R⃗ · g⃗∥_{BMO(ℝ^{d})} ≲ ε^{2} log \frac{1}{ε}.

Indeed, by taking ε sufficiently small we conclude

∥f − R⃗ · g⃗∥_{BMO(ℝ^{d})} ≤ ε/2

and f − R⃗ · g⃗ goes to zero at infinity, at which point one can easily iterate the theorem and use the completeness of BMO and L^{∞} to obtain the result.

Now for the nonlinear trick: the function g⃗ is not just bounded in L^{∞}, but will in fact take values in the unit sphere S^{d} := {v ∈ ℝ^{d+1} : |v| = 1}, thus g⃗ · g⃗ = 1. We are thus reduced to

Theorem 2.2 (Second reduction). Let 0 < ε ≪ 1 be small. Then for any f ∈ BMO(ℝ^{d}) going to zero at infinity with ∥f∥_{BMO(ℝ^{d})} ≤ ε there exists g⃗ ∈ L^{∞}(ℝ^{d} → ℝ^{d+1}) with g⃗ → e_{1} = (1, 0, …, 0) at infinity such that

g⃗ · g⃗ = 1

and

∥f − R⃗ · g⃗∥_{BMO(ℝ^{d})} ≲ ε^{2} log \frac{1}{ε}.

The key point will be that in the ε-neighbourhood of any w ∈ S^{d}, the sphere S^{d} is approximately equal to a hyperplane \{v : v·w = 1\}, up to errors of O(ε^{2} log \frac{1}{ε}). This will allow us to concatenate the various frequency scales together while only losing errors of O(ε^{2} log \frac{1}{ε}) in the final error.

3. The linearised problem

In order to solve this problem we will first need to solve the linearised problem

w⃗ · g⃗ = 0
R⃗ · g⃗ = f

for each unit vector w⃗ ∈ S^{d}. The key lemma here is that this is solvable (with good estimates) in every direction w⃗:

linear Lemma 3.1 (Linear solvability). For each w⃗ ∈ S^{d} there exists a vector-valued Hörmander-Mikhlin multiplier T⃗w⃗ (i.e. the symbol is a homogeneous symbol of order 0, uniformly in w⃗) such that

R⃗ · T⃗w⃗f = f  (3.1) row

and

w⃗ · T⃗w⃗f = 0  (3.2) wow

for all f. Also T⃗w⃗ maps real scalar functions to real vector-valued functions Proof. Taking Fourier transforms, the task is to obtain a vector-valued symbol m⃗w⃗ of order 0 such that

(1, \frac{iξ}{|ξ|}) · m⃗w⃗(ξ) = 0

for all non-zero ξ, and also

w⃗ · m⃗w⃗(ξ) = 0.

To map real scalar functions to real vector-valued functions, we also need m⃗w⃗(−ξ) = \overline{m⃗w⃗(ξ)}. If we write w⃗ = (w_{0}, w) and m⃗w⃗ = (m_{0}, m) then these equations become

m_{0}(ξ) = −i\frac{ξ}{|ξ|} · m(ξ)

and

w_{0}m_{0}(ξ) = −w · m(ξ)

Solving for m_{0} in terms of m, we just need to find a symbol m of order 0 which solves the equation

(w − w_{0} \frac{iξ}{|ξ|}) · m(ξ) = 0

and

m(−ξ) = \overline{m(ξ)}.

The explicit choice

m(ξ) := w + w_{0} \frac{iξ}{|ξ|}

will work here. □

4. The nonlinear problem

Now we return to the nonlinear problem. We use the standard Littlewood-Paley decomposition 1 = ∑_{j} P_{j}, where P_{j} is frequency-localised to the region {ξ : 2^{j−1} ≤ |ξ| ≤ 2^{j+1}}, and write P_{<j} := ∑_{k<j} P_{k} and P_{≥j} := ∑_{k≥j} P_{k}. Since f decays at infinity, we see that P_{≥j}f converges to f in BMO norm as j → −∞. Thus we have ∥f − P_{≥−M}f∥_{BMO(ℝ^{d})} ≲ ε^{2} for some sufficiently large M. By rescaling we may take M = 0. Our task is then to solve the problem

g⃗ · g⃗ = 1
∥P_{≥0}f − R⃗ · g⃗∥_{BMO(ℝ^{d})} ≲ ε^{2}.

It will in fact suffice to achieve the following recursive construction:

third Theorem 4.1 (Third reduction). Let 0 < ε ≪ 1 be small, and let f ∈ BMO(ℝ^{d}) be such that

∥f∥_{BMO(ℝ^{d})} ≲ ε  (4.1) fsmall

and such that f goes to zero at infinity. Then there exists a sequence g⃗_{<j}, h⃗_{j} for j ≥ 0 of functions obeying the conditions

g⃗_{<0} = e_{1}  (4.2) g1
g⃗_{<j} → e_{1} at infinity  (4.3) g1a
g⃗_{<j} · g⃗_{<j} = 1  (4.4) g2
R⃗ · h⃗_{j} = P_{j}f  (4.5) g3
∥ ∑_{0≤j≤J}(g⃗_{<j+1} − g⃗_{<j} − h⃗_{j})∥_{BMO(ℝ^{d})} ≲ ε^{2} log \frac{1}{ε} for all J ≥ 0  (4.6) g4

and such that g⃗_{<j} is locally convergent in L^{2}(ℝ^{d}).

If we can find such functions, then by setting g⃗ to be the limit of the g⃗_{<j} we easily conclude the claim (note that R⃗ is bounded on BMO(ℝ^{d})). Roughly speaking, the idea is to construct the sphere-valued function g⃗ one Littlewood-Paley component at a time, so that at the j^{th} frequency scale one creates an perturbation h⃗_{j} to capture the effect of P_{j}f, and then sets g⃗_{<j+1} equal to g⃗_{<j} + h⃗_{j} plus a small adjustment (required to keep g⃗_{<j+1} valued on the sphere). (By a strange co-incidence, I stumbled much later across a similar method for an unrelated problem ^{\underline{tao:wavemaps}} [2].)

5. Proof of Theorem ^{\underline{third}} 4.1

It remains to prove Theorem ^{\underline{third}} 4.1. We set g⃗_{<0} to be an arbitrary unit vector in S^{d}, say (1, 0, …, 0), thus establishing (^{\underline{g1}} 4.2). Now suppose inductively that j ≥ 0, and that g⃗_{<j} has already been chosen. To build h⃗_{j} and g⃗_{<j+1}, what we do first is we perform a wavelet decomposition

P_{j}f = ∑_{k∈ℤ^{d}} c_{j,k}ψ_{j,k}

where ψ_{j,k} is a standard wavelet-type object concentrated around the ball B(2^{−j}k, 2^{−j}) with L^{∞} normalisation (thus the height of this wavelet is O(1)), and c_{j,k} is a wavelet coefficient obeying a pointwise bound of the form

|c_{j,k}| ≲ MP_{j}f(2^{−j}k),  (5.1) cjk

where M is the Hardy-Littlewoodmaximal function; there are many ways to obtain such a decomposition, and the precise form is not important for us here. It is worth noting that the band-limited nature of P_{j}f (which in particular implies a reproducing formula of the form P_{j}f = P̃_{j}P_{j}f for some variant P̃_{j} of P_{j}) ensures that the maximal function MP_{j}f is essentially constant at scales 2^{−j} (thus MP_{j}f(x) ∼ MP_{j}f(y) whenever |x − y| ≲ 2^{−j}). We shall implicitly take advantage of this local constancy at a number of places in the sequel. It is slightly convenient to take ψ_{j,k} to be band-limited to the annulus {|ξ| ∼ 2^{j}}, and thus allow the wavelets to decay (rapidly) in space.

Let w⃗_{j,k} := g⃗_{<j}(2^{−j}k) ∈ S^{d}. We use Lemma ^{\underline{linear}} 3.1 and define

h⃗_{j} := ∑_{k∈ℤ^{d}} c_{j,k}Tw⃗_{j,k}ψ_{j,k},  (5.2) vhj

thus we invert the Riesz transform on each wavelet component of P_{j}f while trying to stay orthogonal to g⃗_{<j} at the centre 2^{−j}k of the support of that wavelet. The claim (^{\underline{g3}} 4.5) is then clear from (^{\underline{row}} 3.1). (Note that Tw⃗_{j,k}ψ_{j,k} is a “molecue” which is still bounded by O(1) and rapidly decreasing away from B(2^{−j}k, 2^{−j}), which makes it relatively easy to sum the series.)

From (^{\underline{fsmall}} 4.1) we have

∥P_{j}f∥_{L^{∞}} ≲ ε

and hence by (^{\underline{cjk}} 5.1)

|c_{j,k}| ≲ ε

and hence by (^{\underline{vhj}} 5.2)

∥h⃗_{j}∥_{L^{∞}} ≲ ε.

In particular, we see that g⃗_{<j} + h⃗_{j} takes values in the O(ε)-neighbourhood of the sphere S^{d}. We thus define g⃗_{<j+1} to be the radial projection of g⃗_{<j} + h⃗_{j} to the sphere, thus

g⃗_{<j+1} = \frac{g⃗_{<j} + h⃗_{j}}{|g⃗_{<j} + h⃗_{j}|}.

This gives (^{\underline{g2}} 4.4), and with a little more work (noting that P_{j}f goes to zero at infinity, hence h⃗_{j} does also) one also obtains (^{\underline{g1a}} 4.3).

It remains to show (^{\underline{g4}} 4.6) and the L^{2} convergence. The first step is to establish some more regularity on the approximate solutions g⃗_{<j}. It is convenient to introduce the local sizes

a_{j,k} := ∑_{m≥0}(2/3)^{m}MP_{j−m}f(2^{−j}k);

the BMO bound (^{\underline{fsmall}} 4.1) translates to a Carleson measure bound

∑_{j:2^{−j}≤r} 2^{−dj} ∑_{k:2^{−j}k∈B(x,r)} |a_{j,k}|^{2} ≲ ε^{2}r^{d}  (5.3) carl

for any ball B(x, r). (The exponential decay factor (2/3)^{m} ensures that the summation in m does not cause a difficulty.)

Lemma 5.1 (Lipschitz control on g⃗_{<j}). For any j ≥ 0, k ∈ ℤ^{d}, and x, y ∈ B(2^{−j}k, O(2^{−j})) we have

g⃗_{<j}(x) − g⃗_{<j}(y) = O(a_{j,k}2^{j}|x − y|)

and

h⃗_{j}(x) − h⃗_{j}(y) = O(a_{j,k}2^{j}|x − y|)

Proof. (Sketch) This is achieved by a straightforward induction, noting that h⃗_{j} = O(c_{j,k}) = O(a_{j,k}) and ∇h⃗_{j} = O(2^{j}a_{j,k}) on the ball B(2^{−j}k, O(2^{−j})), and also observing that radial projection to the sphere does not significantly increase the Lipschitz norm. The fact that 2/3 > 1/2 will allow one to keep the implied constants in the O() notation from blowing up. □

Let e⃗_{j} := g⃗_{<j+1} − g⃗_{<j} − h⃗_{j}. On the ball B(2^{−j}k, O(2^{−j})), g⃗_{<j} differs from w⃗_{j,k} by only O(a_{j,k}), while h⃗_{j} has magnitude O(a_{j,k}). Using (^{\underline{vhj}} 5.2) and (^{\underline{wow}} 3.2) one then sees that h⃗_{j} is mostly orthogonal to g⃗_{<j}, in that

h⃗_{j} · g⃗_{<j} = O(a_{j,k}^{2})

on B(2^{−j}k, 2^{−j}). The key here is the quadratic behaviour in a_{j,k}; the linear behaviour has been cancelled using (^{\underline{wow}} 3.2). As a consequence one concludes that the error function e_{j} is small:

e⃗_{j} = O(a_{j,k}^{2}).  (5.4) vej

Also, from the Lipschitz bounds on g⃗_{<j} and h⃗_{j} one can establish the relatively crude bound

e⃗_{j}(x) − e⃗_{j}(y) = O(2^{j}|x − y|).

Putting these together with the Carleson bounds (^{\underline{carl}} 5.3) one can quickly verify

∥ ∑_{0≤j≤J} e_{j}∥_{BMO(ℝ^{d})} ≲ ε^{2} log \frac{1}{ε}

for all J ≥ 0, which is (^{\underline{g4}} 4.6).

Finally, we show local L^{2} convergence. From (^{\underline{carl}} 5.3) and (^{\underline{vej}} 5.4), and dominated convergence, one can establish that ∑_{j=0}^{∞} e⃗_{j} is conditionally convergent in L_{loc}^{2}, while from the frequency support and magnitude bounds on h⃗_{j}, and (^{\underline{carl}} 5.3) again, one can also obtain conditional local convergence of ∑_{j=0}^{∞} h⃗_{j} in L_{loc}^{2}. The claim follows.

References

Math. Res. Not. 7 (2001), 299-328.

148 (1982), 215–241.

Department of Mathematics, UCLA, Los Angeles CA 90095-1555, USA.

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

fefferman-stein [1] C. Fefferman, E. M. Stein, H^{p} spaces of several variables, Acta Math. 129 (1972), 137–193

tao:wavemaps [2] T. Tao, Global regularity of wave maps I. Small critical Sobolev norm in high dimension, Internat.

uchiyama [3] A. Uchiyama, A constructive proof of the Fefferman-Stein decomposition of BMO(ℝ^{n}), Acta Math.