Time frequency analysis and the

bilinear Hilbert transform III

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) .

Convergence of Walsh Series

Theorem P. Billard 1967 If f ∈ L^{2}[0, 1], then for almost every x ∈ [0, 1]:

f(x) = lim_{N→∞} N ; ∑ ; l=0 ⟨f, W_{l}⟩ W_{l}(x) .

If N : [0, 1] → R_{+} is a measurable function, define the linearized maximal partial sum operator S_{N} by:

S_{N}f(x) = _{l≤N(x)}^{∑} ⟨f, W_{l}⟩ W_{l}(x) .

Theorem Carleson, Hunt, Billard, Sjölin For each p > 1 there is a constant C_{p} such that for all f ∈ L^{p}[0, 1] and N : [0, 1] → R_{+} :

∥S_{N}f∥_{p} ≤ C_{p}∥f∥_{p} .

Rewriting the Partial Sum Operator I

Fix x ∈ [0, 1]. Let P_{x} be the set of all tiles I × ω such that I ⊂ [0, 1] and ω is a maximal dyadic interval in [0, N(x) + 1].

[figure: 0; N(x); 0; 1]

This set of tiles partitions the same rectangle as the set of tiles [0, 1)×[l, l+1) with l ≤ N(x).

[figure: 0; N(x); 0; 1]

Hence we have

_{l≤N(x)}^{∑} ⟨f, W_{l}⟩ W_{l} = _{p∈P_{x}}^{∑} ⟨f, w_{p}⟩ w_{p} .

Rewriting the Partial Sum Operator II

Evaluating at x we can simplify this expression:

S_{N}f(x) = _{p∈P_{x}:x∈I}^{∑} ⟨f, w_{p}⟩ w_{p}(x) .

[figure: 0; N(x); 0; 1; x]

A tile appears on the right hand side, if the tile above it contains the point (x, N(x)) and the union of the two tiles is a dyadic rectangle of area 2 (bitile). Hence:

S_{N}f = _{bitiles : u∪d}^{∑} ⟨f, w_{d}⟩ w_{d} 1_{{y:(y,N(y))∈u}} .

[figure: 0; N(x); 0; 1; x; N; d; u]

Idea of the Proof

Now the partial sum operator has form which has very similar geometric-combinatorial structure as the quartile operator. E.g., we encounter again the concept of trees. As before the Bessel inequality for the lower tiles d enters for a set d of pairwise disjoint lower tiles:

_{d∈d}^{∑} |⟨f, w_{d}⟩|^{2} ≤ ∥f∥_{2}^{2} .

For the upper tiles we have a formally similar inequality: If u is a set of pairwise disjoint upper tiles, then

_{u∈u}^{∑} |{y : (y, N(y) ∈ u} | ≤ 1 .

[figure: 0; 1; N; E_{u}; u]

Uniform Estimates

The bilinear Hilbert transform with parameter α ∈ R is defined by

H_{α}(f, g)(x) := p.v. ∫ f(x − t)g(x − αt)\frac{dt}{t} .

Theorem (M. Lacey, C. T., 1997) For α ∈ R \ \{0, 1\}, 1 < r, s ≤ ∞, \frac{2}{3} < \frac{rs}{r+s} < ∞ there is a constant C(α, r, s) such that for all f, g:

∥H_{α}(f, g)∥_{\frac{rs}{r+s}} ≤ C(α, r, s)∥f∥_{r}∥g∥_{s} .

If H is the linear Hilbert transform, we have

H_{0}(f, g) = H(f) · g ,
H_{1}(f, g) = H(f · g) .

Both operators satisfy trivially the above estimate for r = 2, s = ∞. The following is less trivial:

Theorem (C.T., 1998) The constant C(α, 2, ∞) in the above theorem can be chosen independently of α as long as |α| < N for large N > 0.

The Symmetric Approach

Consider the general trilinear form Λ = Λ_{β}:

Λ(f_{1}, f_{2}, f_{3}) = ∫ p.v. ∫ 3 ; ∏ ; ȷ=1 f_{ȷ}(x − β_{ȷ}t)\frac{dt}{t} dx .

We can assume that ∥β∥ = 1 and β · γ = 0 where γ = (1, 1, 1) Degeneration occurs when β_{ȷ} = β_{ȷ′} for some ȷ ≠ ȷ′. Then we have up to symmetry:

Λ(f_{1}, f_{2}, f_{3}) = ± ∫ H(f_{1})f_{2}f_{3} dx .

For any permutation σ of three elements we have the adjoint H_{σ} defined by

∫ H_{σ}(f_{σ(1)} , f_{σ(2)} )f_{σ(3)} dx = Λ(f_{1}, f_{2}, f_{3}) .

We say Λ is of type (p_{1}, p_{2}, p_{3}) if

|Λ(f_{1}, f_{2}, f_{3})| ≤ C∥f_{1}∥_{p_{1}}∥f_{2}∥_{p_{2}}∥f_{3}∥_{p_{3}} ,

or, if one of the p_{ȷ} is negative, there is a σ and a constant C such that for all f_{1}, f_{2}:

∥H_{σ}(f_{1}, f_{2})∥_{p_{σ(3)} _{′}} ≤ C ∥f_{1}∥_{σ(1)} ∥f_{2}∥_{σ(2)} .

Diagram of Exponents

The homogeneity of Λ implies that any type (p_{1}, p_{2}, p_{3}) of Λ satisfies ∑_{ȷ} p_{ȷ}^{−1} = 1. The shaded regions show the known types of the nondegenerate Λ and the degenerate Λ (one instance), and the uniform estimates.

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

The Whitney Decomposition

Assume β is away from the degenerate case. Up to a scalar multiple, Λ_{β} is given by:

∫_{ξ·γ=0} ( 3 ; ∏ ; j=1 f̂_{j}(ξ_{j})) sgn (β · ξ) dξ .

We take 1_{R^{+}} instead of sign and decompose the discontinuous multiplier into smooth multipliers living near the circles in the following figure:

[figure: α; β; ξ_{1}; ξ_{2}; ξ_{3}]

The Model Form

The centers of the circles are for k, l ∈ Z:

ξ^{k,l} = 2^{−k}(β + lα) .

Let ψ̂ be supported in [−1, 1] and define ψ̂_{ȷ,k,l}(ξ_{ȷ}) := ψ̂(2^{k}(ξ_{ȷ} − ξ_{ȷ}^{k,l})) ,

m_{k,l}(ξ_{1}, ξ_{2}, ξ_{3}) := 3 ; ∏ ; ȷ=1 ψ̂_{ȷ,k,l}(ξ_{ȷ}) .

The form Λ is then essentially:

_{k,l}^{∑} ∫_{ξ·γ=0} 3 ; ∏ ; ȷ=1 f̂_{ȷ}(ξ_{ȷ})ψ̂_{ȷ,k,l}(ξ_{ȷ}) dσ

= _{k,l}^{∑} ∫_{R} 3 ; ∏ ; ȷ=1 (f_{ȷ} ∗ ψ_{ȷ,k,l})(x) dx .

Trees

A tree of type j is a set T of pairs (k, l) such that ξ is contained in the support of ψ̂_{j,k,l}.

[figure: ξ_{3}; ξ_{2}; ξ_{1}]

We investigate the partial sum

_{(k,l)∈T}^{∑} ∫_{R} 3 ; ∏ ; ȷ=1 (f_{ȷ} ∗ ψ_{ȷ,k,l})(x) dx .

If j ≠ i, then the supports of ψ̂_{ȷ,k,l} for (k, l) ∈ T are pairwise disjoint and lacunary intervals.

Paraproducts

A model for a tree of type 1 is

_{k}^{∑} ∫ (f_{1} ∗ φ_{k})(f_{2} ∗ ψ_{k})(f_{3} ∗ ψ_{k}) dx ,

where φ,ψ are Schwartz functions with

\overline{}0φ̂  ψ̂ .

Define φ_{k} = 2^{−k}φ(2^{−k}x), and ψ_{k} likewise. This paraproduct is estimated by

∫ sup_{k}|f_{1} ∗ φ_{k}| 3 ; ∏ ; ȷ=2 (_{k}^{∑} |f_{ȷ} ∗ ψ_{k}|^{2} )^{1/2} dx

≤ ∥sup_{k}|f_{1} ∗ φ_{k}|∥_{p_{1}} 2 ; ∏ ; ȷ=1 ∥(_{k}^{∑} |f_{ȷ} ∗ ψ_{k}|^{2} )^{1/2} ∥_{p_{ȷ}}

≤ ∥Mf_{1}∥_{p_{1}} ∥Gf_{2}∥_{p_{2}} ∥Gf_{3}∥_{p_{3}} .

Here M is the Hardy Littlewood maximal operator and G is the square function.

Near the Degenerate Case

As β tends towards the degenerate case, one needs ellipses of excenticity 2^{κ}. The projections of the ellipses of fixed scale onto any axis are still a partition of the axis.

[figure: ξ_{3}; ξ_{2}; ξ_{1}]

For the tree of type 3 indicated above we have that the supports of ψ̂_{3,k,l} are large intervals which overlap, the supports of ψ̂_{2,k,l} are also large intervals which overlap, wheras only the small supports of ψ̂_{1,k,l} are disjoint and lacunary.

Coifman, McIntosh, Meyer Theory

Hence near the degenerate case one has to study model sums of the type

_{k}^{∑} ∫ (f_{1} ∗ ψ_{k})(f_{2} ∗ φ_{k−κ})(f_{3} ∗ φ_{k−κ}) dx

for some large κ. Such expressions have been studied by Coifman, Macintosh, Meyer. Observe that as κ tends to ∞, this degenerates to

_{k}^{∑} ∫ (f_{1} ∗ ψ_{k})f_{2}f_{3} dx ,

which is essentially

∫ H(f_{1})f_{2}f_{3} dx .

Telescopic Series

We have to estimate the model sums uniformly in κ. We write for the model sum

_{k′,k′′≥0}^{∑} _{k}^{∑} ∫ (f_{1}∗ψ_{k+κ} )(f_{2}∗ψ_{k+k′} )(f_{3}∗ψ_{k+k′′} ) dx

For each summand we can write

(f̂_{1}ψ̂_{k+κ} ) ∗ (f̂_{2}ψ̂_{k+k′} ) ∗ (f̂_{3}ψ̂_{k+k′′} )(0) .

This is nonzero only if the two smallest numbers of k + κ, k + k′, k + k′′ (largest frequency support) differ by at most one.

[figure: 0] . For simplicity we assume that the term is nonzero only if the two smallest indices are equal.

Recovering the Paraproducts

Summing over those terms for which k + k′ = k + k′′, is the minimal index we have

= _{k}^{∑} κ ; ∑ ; k′=0 ∫ (f_{1} ∗ ψ_{k+k′} )(f_{2} ∗ ψ_{k})(f_{3} ∗ ψ_{k}) dx

= _{k}^{∑} ∫ (f_{1} ∗ φ_{k})(f_{2} ∗ ψ_{k})(f_{3} ∗ ψ_{k}) dx

− _{k}^{∑} ∫ (f_{1} ∗ φ_{k+κ+1} )(f_{2} ∗ ψ_{k})(f_{3} ∗ ψ_{k}) dx

Those two expressions are again paraproducts as in the nondegenerate case. Similarly we obtain these paraproducts, if we consider those terms for which k + κ = k + k′ is the (strictly) minimal index or for which k + κ = k + k′′ is the (strictly) minimal index.

With these paraproducts one proceeds as in the nondegenerate case.

The Maximal Bilinear Hilbert Transform

Let α ∈ R \ {0, 1}. The maximal trunction of the bilinear Hilbert transform is

H^{max} (f_{1}, f_{2})(x) :=

sup_{ϵ>0} ∫_{R\[−ϵ,ϵ]}f_{1}(x − t)f_{2}(x − αt) \frac{dt}{t}

The linearized version of this operator is

H^{ϵ}(f_{1}, f_{2})(x) :=

∫_{R\[−ϵ(x),ϵ(x)]} f_{1}(x − t)f_{2}(x − αt) \frac{dt}{t}

for an arbitrary measurable function ϵ.

Theorem (M. Lacey, 1997) The trilinear form Λ^{ϵ} corresponding to H^{ϵ} satisfies the same type estimates as λ, independently of ϵ.

The Bi(sub)linear Maximal Operator

Let φ be a bump function

\overline{}0φ .

Define φ_{k} = 2^{−k}φ(2^{−k}x). A variant of the maximal bilinear Hilbert transform is given by

sup_{k∈Z} ∫ f_{1}(x − t)f_{2}(x − αt) (i ∗ φ_{k}) (t)dt ,

where i(x) = 1/x. Replacing i by δ_{0} gives

sup_{k∈Z} ∫ f_{1}(x − t)f_{2}(x − αt) (δ_{0} ∗ φ) (t)dt ,

which is equivalent to the maximal operator

M(f_{1}, f_{2})(x) = sup_{ϵ>0} \frac{1}{ϵ} ∫_{−ϵ}^{ϵ} f_{1}(x−t)f_{2}(x−αt) dt .

By similarity: M is bounded from L^{p_{1}} × L^{p_{2}} into L^{p} if 2/3 < p ≤ 1, 1 < p_{1}, p_{2} ≤ ∞, 1/p = 1/p_{1} + 1/p_{2}.

Almost Orthogonality Results

The linearized maximal quartile operator is given by

_{P:|I_{P} |>2^{k}(x)}^{∑} \frac{1}{√|I_{P} |} ⟨f_{1}, w_{1,P} ⟩ ⟨f_{2}, w_{2,P} ⟩ w_{3,P}

for some measurable function k. Define

v_{3,P} (x) = w_{3,P} (x) if |I_{P} | > 2^{k(x)}
v_{3,P} (x) = 0 if |I_{P} | ≤ 2^{k(x)}

Then the associated trilinear form is given by

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

The proof of boundedness proceeds formally as before. If P is a set of pairwise disjoint quartiles, one has to find good replacements for the (false) inequality

_{P ∈P}^{∑} |⟨f, v_{3,P} ⟩|^{2} ≤ C∥f∥_{2}^{2} .

Here the Rademacher Menshov theorem and some probabilistic Lemma by Bourgain comes into play.

Open Problems

1) The remaining triangle: Is there any universal estimate

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

for the nondegenerate bilinear Hilbert transform H_{α} with

1/2 < p < 2/3 ?

2) Is there any universal estimate

∥H_{α_{1},α_{2}}(f_{1}, f_{2}, f_{3})∥_{p} ≤ C∥f_{1}∥_{p_{1}}∥f_{2}∥_{p_{2}}∥f_{3}∥_{p_{3}}

for the (nondegenerate) trilinear Hilbert transform

H(f_{1}, f_{2}, f_{3})(x) :=

∫ f_{1}(x − t)f_{2}(x − α_{1}t)f_{3}(x − α_{2}t) \frac{dt}{t} ?

3) How about multidimensional results?