Time frequency analysis and the

bilinear Hilbert transform II

Christoph Thiele Nara, January 1999

The Walsh Functions

The Walsh functions W_{0}, W_{1}, W_{2}, …: R → R are defined by the recursive formulas:

W_{0} = 1_{[0,1)} ,

W_{2l} = W_{l}(2x) + W_{l}(2x − 1) ,

W_{2l+1} = W_{l}(2x) − W_{l}(2x − 1) .

A tile is a dyadic rectangle of area 1:

p = [2^{k}n, 2^{k}(n + 1)) × [2^{−k}l, 2^{−k}(l + 1))

with integers k, n, l and l ≥ 0. We write p = I + P × ω_{p}. The Walsh wave packet w_{p} is defined by:

w_{p}(x) = 2^{−k/2}W_{l}(2^{−k}x − n) .

The Quartile Operator

A quartile P as a dyadic rectangle of area 4:

P = [2^{k}n, 2^{k}(n + 1)) × [2^{2−k}l, 2^{2−k}(l + 1))

with integers k, n, l and l ≥ 0. The first interval is denoted by I_{P}. We divide a quartile P into four tiles:

p_{1} p_{2} p_{3} p_{4} .

We write w_{1,P} , w_{2,P} , w_{3,P} , and w_{4,P} for the corresponding Walsh wave packets. The bilinear quartile operator H(f_{1}, f_{2}) is:

_{P:quartile}^{∑} \frac{1}{√|I_{P} |} ⟨f_{1}, w_{1,P} ⟩ ⟨f_{2}, w_{2,P} ⟩ w_{3,P} .

The trilnear form Λ(f_{1}, f_{2}, f_{3}) is given by

_{P:quartile}^{∑} \frac{1}{√|I_{P} |} 3 ; ∏ ; j=1 ⟨f_{j}, w_{j,P} ⟩ .

The Main Estimate

Theorem C. T. 1995-98 If −1/2 < p_{j} < 1 for j = 1, 2, 3 and ∑ 1/p_{j} = 1, then the form Λ is of type (p_{1}, p_{2}, p_{3}).

Recall that for negative p_{j} the type is defined by an estimate for an appropriate dual bilinear operator of the trilinear form.

For the bilinear operator H this implies: If 1 < p_{1}, p_{2}, ≤ ∞, 2/3 < p < ∞, and 1/p = 1/p_{1} + 1/p_{2}, then

∥H(f_{1}, f_{2})∥_{p} ≤ C∥f_{1}∥_{p_{1}}∥f_{2}∥_{p_{2}} .

The Diagram of Exponents

The region of estimates of the main theorem is given by the convex hull of the three shaded triangles.

[figure: (0, 0, 1); (0, 1, 0); (1, 0, 0)]

One of the shaded triangles corresponds to the estimates

∥H(f_{1}, f_{2})∥_{p} ≤ C∥f_{1}∥_{p_{1}}∥f_{2}∥_{p_{2}}

for 1/p = 1/p_{1} + 1/p_{2}, 2/3 < p < 1, 1 < p_{1}, p_{2} ≤ 2.

Interpolation

Inside the shaded region we can interpret the points as estimates for one bilinear operator:

[figure: (0, 0, 1); (0, 1, 0); (1, 0, 0)]

The two extremal points are estimates

∥∑ H(f_{1}, f_{2})∥_{p} ≤ C∥f_{1}∥_{p_{1}} ∥f_{2}∥_{p_{2}} ,

∥∑ H(f_{1}, f_{2})∥_{q} ≤ C∥f_{1}∥_{q_{1}} ∥f_{2}∥_{q_{2}} .

By complex interpolation we then obtain the estimates on the line connecting the two extremal points.

A Bootstrapping Argument

[figure: (0, 0, 1); (0, 1, 0); (1, 0, 0); 1; 2; 3; 4]

We are given (first j = 1, second j = 2)

∥∑ ⟨f, w_{a_{P}} ⟩ ⟨g, w_{b_{P}} ⟩ w_{c_{P}} ∥_{p^{j}} ≤ C_{j}∥f∥_{p_{1}^{j}} ∥g∥_{p_{2}^{j}}

∥∑ ⟨f, w_{c_{P}} ⟩ ⟨g, w_{b_{P}} ⟩ w_{a_{P}} ∥_{q^{j}} ≤ C_{j}∥f∥_{q_{1}^{j}} ∥g∥_{q_{2}^{j}}

For nice characteristic functions g, these estimates also hold for j = 3, 4 for certain finite C_{3}, C_{4} depending on g. Then interpolation gives C_{3} ≤ min (C_{1}, C_{4}) , C_{4} ≤ min (C_{2}, C_{3}). Hence C_{3}, C_{4} are really independent of g. Real interpolation gives the estimates for arbitrary g.

Easy Reductions

Consider the range 1 < p_{1}, p_{2} < 2 and 1/p_{1} + 1/p_{2} < 3/2. By Marcinkiewicz interpolation, homogeneity, and linearity it suffices to prove for ∥f_{1}∥_{p_{1}} = ∥f_{2}∥_{p_{2}} = 1 that

|{x : H(f_{1}, f_{2})(x) > 1} | ≤ C .

Fix such f_{1}, f_{2} and define

E := {x : max (M_{p_{1}}f_{1}(x), M_{p_{2}}f_{2}(x)) ≥ 1} ,

where

M_{p}f(x) := (sup \frac{1}{|I|} ∫_{I} |f(x)|^{p} dx)^{1/p} .

By the maximal theorem |E| < C, hence it suffices to prove a bound on the set F:

\{x : _{P:I_{P} ⊄E}^{∑} |I_{P} |^{−\frac{1}{2}} ⟨f_{1}, w_{a_{P}} ⟩ ⟨f_{2}, w_{b_{P}} ⟩ w_{c_{P}} > 1\}

A Symmetric Statement

We can assume |F | > 1. Let f_{3} be the characteristic function of F normalized to ∥f_{3}∥_{2} = 1. Then we have

|F |^{\frac{1}{2}} ≤ _{P:I_{P} ⊄E}^{∑} |I_{P} |^{−\frac{1}{2}} 3 ; ∏ ; j=1 |⟨f_{j}, w_{j,_{P}} ⟩| .

Hence it remains to prove: Lemma Let 1 < p_{1}, p_{2}, p_{3} ≤ 2 with

1 < 1/p_{1} + 1/p_{2} + 1/p_{3} < 2

Then for all f_{1}, f_{2}, f_{3} with ∥f_{j}∥_{p_{j}} = 1 we have _{P:I_{P} ⊄E}^{∑} |I_{P} |^{−\frac{1}{2}} |⟨f_{1}, w_{a_{P}} ⟩ ⟨f_{2}, w_{b_{P}} ⟩ ⟨f_{3}, w_{c_{P}} ⟩| ≤ C , where

E = {x : max_{j=1,2,3} (M_{p_{j}}f_{j}(x) ≥ 1} .

Further Reductions

We will only prove this Lemma in the case p_{1} = p_{2} = p_{3} = 2, in order to make this exposition brief. Fix f_{1}, f_{2}, f_{3}. It suffices to consider the sum over any finite set P of quartiles with I_{P} ⊄ E for all P ∈ P. By splitting P into two subsets, we can assume that P consists only of even (odd) scales k. This has the effect that whenever two quartiles intersect, one of the quartiles intersects only one tile of the other quartile.

Trees

Define a partial ordering of tiles by p < q if I_{p} ⊂ I_{q} and p ∩ q ≠ ∅.

q  p I_{p}  .

A tree of type j, j = 1, 2, 3, is a set of quartiles such that the set of j-th tiles of these quartiles has a unique maximal element. This element is denoted p_{T} = I_{T} × ω_{T}. Trees of types 1 and 2:

I_{T} .

Disjointness Properties of Trees

If i ≠ j and T is a tree of type i, then the j-th tiles of the quartiles of T are pairwise disjoint.

ω_{1,P} ω_{2,P} ω_{1,Q} ω_{2,Q} Q  P  .

Proof: Type 2 implies ω_{2,Q} ⊂ ω_{2,P} , w.l.o.g ω_{2,Q} ≠ ω_{2,P} , and we conclude ω_{1,Q} ⊂ ω_{2,P} and ω_{1,Q} ∩ ω_{1,P} = ∅. Therefore we will for a tree T of type i consider the estimate:

_{P ∈T}^{∑} |I_{P} |^{−\frac{1}{2}} 3 ; ∏ ; j=1 |⟨f_{j}, w_{j,P} ⟩| ≤

(sup_{P ∈T} \frac{|⟨f_{i}, w_{i,P} ⟩|}{|I_{P} |^{\frac{1}{2}}}) _{j≠i}^{∏} (_{P ∈T}^{∑} |⟨f_{j}, w_{j,P} ⟩|^{2} )^{\frac{1}{2}} .

Size of Trees

A tree T satisfies a size condition (m, i, j), if

_{P ∈T}^{∑} |⟨f_{j}, w_{j,P} ⟩|^{2} ≥ 2^{2m+5} |I_{T} |, i ≠ j, or

inf_{P ∈T} |⟨f_{j}, w_{j,P} ⟩| ≥ 2^{m}|I_{P} |^{\frac{1}{2}} , i = j,

The size of a tree is the maximal m such that the tree satisfies some size condition (m, i, j). The size of any T ⊂ P is bounded by 0: For i ≠ j:

_{P ∈T}^{∑} |⟨f_{j}, w_{j,P} ⟩|^{2} |I_{T} |^{−1} ≤ inf_{x∈I_{T}} M_{2}f(x)^{2} ≤ 1 ,

because I_{T} ⊄ E, and for i = j and all P ∈ T:

|⟨f_{j}, w_{j,P} ⟩| |I_{P} |^{−1/2} ≤ inf_{x∈I_{T}} M_{2}f(x) ≤ 1 ,

because I_{P} ⊄ I_{T}.

Selection of the Trees

We partition the set P of quartiles inductively into trees T_{0}, …, T_{N}. We can assume

P_{ν} := P \ _{ν′<ν}^{⋃} T_{ν′} .

is not empty. Let m_{ν} be the maximal integer for which there is a tree T ⊂ P_{ν} satisfying some size condition (m, i, j). Let T_{ν} be the set of all such trees that are maximal with respect to set inclusion. If there is a tree in T_{ν} which satisfies a size condition (m, i, j) with i < j, let T_{ν} be such a tree such that the center of ω_{T_{ν+1}} is maximal. Else, let T_{ν} ∈ T_{ν} such that the center of ω_{T_{ν+1}} is minimal.

[figure: i; j; center(ω_{T})] .

Summing over the Trees

Let T_{m,i,j} be the set of trees selected due to size condition (m, i, j).

_{P ∈P}^{∑} \frac{1}{√|I_{P} |} | 3 ; ∏ ; j′=1 ⟨f_{j′}, w_{j′,P} ⟩|

= _{m≤0}^{∑} 3 ; ∑ ; ı,ȷ=1 _{T ∈T_{m,ı,ȷ}}^{∑} _{P ∈T}^{∑} \frac{1}{√|I_{P} |} | 3 ; ∏ ; j′=1 ⟨f′_{j}, w_{j′,P} ⟩|

≤ _{m,i,j,T}^{∑} sup_{P ∈T} \begin{matrix}| & ⟨f_{i}, w_{i,P} ⟩ & | \\ √ & |I_{P} | \end{matrix} _{j′≠ı}^{∏} (_{P ∈T}^{∑} |⟨f′_{j}, w_{j′,P} ⟩|^{2} )^{\frac{1}{2}}

≤ _{m≤0}^{∑} 3 ; ∑ ; ı,ȷ=1 _{T ∈T_{(m,ı,ȷ)}}^{∑}|I_{T} | 2^{3m+8} .

Hence it suffices to show that for each m, i, j _{T ∈T_{(m,ı,ȷ)}}^{∑}|I_{T} | ≤ C 2^{−2m}

Counting the Trees in Case i = j

Let T, T′ be two trees in T_{m,i,i} and let p_{T} , p_{T′} be the maximal tiles in these trees. Then p_{T} ∩ p_{T′} = ∅, because T ∪ T′ is not a tree, by maximality of T and T′.

[figure] . Hence we have:

_{T ∈T_{m,i,j}}^{∑} |I_{T} | ≤ C2^{−2m} _{T ∈T_{m,i,i}}^{∑} ∥⟨f_{i}, w_{p_{T}} ⟩∥^{2}

≤ C2^{−2m}∥f_{i}∥_{2}^{2} = C2^{−2m} .

Reduced Trees

For a tree T let T^{r} be the set of quartiles P ∈ T such that I_{P} is not minimal in {I_{P} : P ∈ T }.

[figure] .

Observe that the intervals I_{P} with P ∈ T \ T^{r} are pairwise disjoint.

Size of Reduced Trees

For T ∈ T_{m,i,j} we have

_{P ∈T^{r}}^{∑} |⟨f, w_{j,P} ⟩|^{2}

≥ _{P ∈T}^{∑} |⟨f, w_{j,P} ⟩|^{2} − _{P ∈T \T^{r}}^{∑} |⟨f, w_{j,P} ⟩|^{2}

≥ 2^{2m+5} |I_{T} | − _{P ∈T \T^{r}}^{∑} 2^{2m+2} |I_{P} | |⟨f, w_{j,P} ⟩|^{2} .

≥ 2^{2m+3} |I_{T} |

The last line follows because the I_{P} with P ∈ T \ T^{r} are pairwise disjoint.

Disjointness of Reduced Trees

Let T, T′ ∈ T_{m,i,j} with i ≠ j. let P ∈ T^{r}, P′ ∈ T′^{r}, then the tiles p_{j} and p′_{j} of P and P′ are disjoint. In the diagram below, which shows a proof by contradiction, Q and Q′ are the top elements in the trees T and T′, and R is an appropriate element in T \ T^{r}. Since T′ has been selected first, we reach a contradiction to the maximality of T′.

[figure: Q; p_{i}; p_{j}; r_{i}; r_{j}; ^{p′^{i}}; ^{p′^{j}}; Q′].

Counting Trees in Case i ≠ j

Fix (m, i, j) with i ≠ j. Since the reduced tress still have large size, and the tiles p_{j} with P ∈ T^{r}, T ∈ T_{m,i,j} are pairwise disjoint, we can argue similarly as in the case i = j:

_{T ∈T_{m,i,j}}^{∑} |I_{T} |

≤ _{T ∈T_{m,i,j}}^{∑} C2^{2m} _{P ∈T^{r}}^{∑} ∥⟨f_{j}, w_{j,P} ⟩∥^{2}

≤ C2^{2m}∥f_{j}∥_{2}^{2} = C2^{2m} .

This completes the proof of the Lemma in the case p_{1} = p_{2} = p_{3} = 2.

Two Remarks to min (p_{1}, p_{2}, p_{3}) < 2

1) If p_{j} < 2, then the j-size of trees of type i is measured with the aid of the square function

(_{P ∈T}^{∑} \frac{|⟨f_{j}, w_{j,P} ⟩|^{2}}{|I_{P} |} 1_{I_{P}} )^{\frac{1}{2}}

and its L^{q} and BMO norms. Its L^{2} norm is equal to the square sum of coefficients considered in the case p_{j} = 2.

2) If p is a set of pairwise disjoint tiles, then one considers the vector valued function

(\frac{⟨f, w_{p}⟩}{|I_{p}|^{1/2}}1_{I_{p}})_{p∈p}

and interpolates between estimates for its L^{2}(l^{2}) norm, obtained by Bessel’s inequality, and estimates for its L^{1+ϵ} (l^{∞}) norm, obtained by the maximal theorem.