Time frequency analysis and the

bilinear Hilbert transform I

Christoph Thiele Nara, January 1999

In memoriam Akihito Uchiyama

Outline

Part I: Introduction, bilinear Hilbert transform, phase plane analysis, Walsh analysis

Part II: Proof of the main theorem: boundedness of the bilinear Hilbert transform (or rather almost-proof in the easier Walsh case)

Part III: Connection to the classical Carleson theorem, recent results concerning uniform estimates and maximal operators.

The Hilbert Transform

For a Schwartz function f define

Hf(x) := p.v. ∫ f(x − t)\frac{dt}{t}

:= lim_{ϵ→0} ∫_{R\[−ϵ,ϵ]}f(x − t)\frac{dt}{t} .

For 0 < p ≤ ∞ consider the norm (quasi norm)

∥f∥_{p} := (∫ |f(x)|^{p} dx)^{1/p} .

Theorem A. Kolmogorov, 1925, M. Riesz 1926: For 1 < r < ∞ there is a constant C such that for all Schwartz functions f:

∥Hf∥_{r} ≤ C∥f∥_{r} .

Translation and Dilation Invariance

H is invariant under translation and dilation:

f_{a,b}(x) := f(ax + b) , a > 0 ,

H(f_{a,b})(x) = p.v. ∫ f(a(x − t) + b)\frac{dt}{t}

= p.v. ∫ f((ax + b) − t)\frac{dt}{t}

= (Hf)_{a,b}(x) .

Observe

∥f_{a,b}∥_{r} = a^{−1/r}∥f∥_{r} .

Hence the dilation invariance of H implies that equality of the exponents in the Riesz-Kolmogorov theorem is a necessary.

The Fourier Transform

For a Schwartz function f define the Fourier transform:

f̂(ξ) := ∫ f(x)e^{−2πixξ} dx .

Then f̂ is also a Schwartz function, and

f(x) = ∫ f̂(ξ)e^{2πixξ} dξ ,
∥f∥_{2} = ∥f̂∥_{2} .

Let A be any translation and dilation invariant operator bounded on L^{2}(R).

Then there is a function m, homogeneous of degree 0, such that for any Schwartz function f:

Af(x) = ∫ f̂(ξ)m(ξ)e^{iξx} dξ .

Littlewood Paley Decomposition

Consider the operator A with multiplier m = 1_{R_{+}} :

\overline{}ξ 0

We split the multiplier dyadically:

m(ξ) = _{k∈Z}^{∑} ψ̂_{k}(x) = _{k∈Z}^{∑} ψ̂(2^{k}x) .

\overline{}ξ 0

We split accordingly

A = ∑ A_{k} ,

Then A_{k} is convolution with ψ_{k}, and acts also (essentially) locally in time on intervals of length 2^{k}.

Wavelets

Each operator A_{k} acts locally in the following regions of the phase space

[figure: k = −1; k = 0; k = 1]

Pick an appropriate mean zero generating function ψ and define for all integers k, n the wavelets

ψ_{k,n}(x) := 2^{−\frac{k}{2}}ψ(2^{−k}x − n)

Then every singular integral operator is almost diagonal in the wavelet basis.

Bilinear singular integrals

We are looking for bilinear operators A which are invariant under simultaneous translations and dilations.

A(f_{a,b}, g_{a,b}) = A(f, g)_{a,b} ,
f_{a,b}(x) = f(ax + b) .

(E.g. pointwise product)

A formal calculation shows

A(e^{2πi . ξ}, e^{2πi . η}) = m(ξ, η)e^{2πi . (ξ+η)}

for some function m which is homogeneous of degree zero. Hence:

A(f, g)(x) = ∫ ∫ f̂(x)ĝ(η)m(ξ, η)e^{2πi(ξ+η)x} dξdη .

Trilinear forms, Adjoints

Sometimes a more convenient (symmetric) object is the corresponding trilinear form

Λ(f_{1}, f_{2}, f_{3}) = ∫ A(f_{1}, f_{2})(x)f_{3}(x) dx

= ∫_{ξ_{1}+ξ_{2}+ξ_{3}=0} m(ξ_{1}, ξ_{2}, ξ_{3}) 3 ; ∏ ; j=1 f_{j}(ξ_{j}) dμ .

For each permutation σ of three elements we have the adjoint

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

We say Λ is of type (p_{1}, p_{2}, p_{3}) if there is a σ and a constant C such that for all f_{1}, f_{2}:

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

Here 1/p′ + 1/p = 1. By dilation invariance we necessarily have ∑_{ȷ} p_{ȷ}^{−1} = 1. Observe that one of the p_{ȷ} is allowed to be negative.

Diagram of the plane ∑_{ȷ} p_{ȷ}^{−1} = 1

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

(Diagram for linear singular integrals)

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

The trilinear Hilbert forms

Consider the family of multipliers

m(ξ_{1}, ξ_{2}, ξ_{3}) = sign ((ξ_{1}, ξ_{2}, ξ_{3}) · u_{θ}) ,

where u is a unit vector in R^{3} perpendicular to (1, 1, 1).

[figure: ξ; ζ; η; θ; u]

The corresponding trilinear forms satisfy the modulation symmetries

Λ_{u}(f_{1}, f_{2}, f_{3}) = Λ_{u}(M_{v_{1}}f_{1}, M_{v_{2}}f_{2}, M_{v_{3}}f_{3}) ,

where v is perpendicular to u and (1, 1, 1), and

M_{ξ}f(x) = f(x)e^{2πiξx} .

The Bilinear Hilbert Transforms

The corresponding adjoint bilinear operators can be written

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

for varying α. The degenerate cases θ = 90^{o}, 210^{o}, 330^{o} correspond to

H(f) · g, f · H(g), H(f · g) .

Theorem M. Lacey, C. T. 1996/97 If −1/2 < 1/p_{j} < 1 for ȷ = 1, 2, 3, ∑ 1/p_{j} = 1, and u is non-degenerate, then Λ_{u} is of type (p_{1}, p_{2}, p_{3}).

Theorem C. T. 1998 The operator H_{α} is bounded from L^{2} × L^{∞} into L^{2} with bound independent of α when α varies in [−N, N] for large N.

Historical Motivation:

Interest in a bound for Calderon’s first commutator

T (f)(x) := p.v. ∫_{R} \frac{A(x) − A(y)}{(x − y)^{2}} f(y) dy

in L^{2} provided A is Lipschitz. Rewrite this as

p.v. ∫_{R} (∫_{0}^{1} A′(αy + (1 − α)x) dα) \frac{f(y)}{x − y} dy

= p.v. ∫_{R} (∫_{0}^{1} A′(x − αt) dα) f(x − t) \frac{dt}{t}

= ∫_{0}^{1} (p.v ∫_{R} f(x − t)A′(x − αt) \frac{dt}{t}) dα .

Hence we are looking for a uniform bound H_{α} : L^{2} × L^{∞} → L^{2} as α varies in (0, 1). Observe:

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

The Whitney Decomposition

[figure: v; u; ξ_{1}; ξ_{2}; ξ_{3}]

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

ξ^{k,l} = 2^{−k}(u + lv) .

Let ψ̂ be supported in [−1, 1] and define

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

ψ̂_{ȷ,k,l}(ξ_{ȷ}) := ψ̂(2^{k}(ξ_{ȷ} − ξ_{ȷ}^{k,l})) .

Phase Plane Localization

We have the following trilinear form, which is a variant of Λ:

_{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 .

For each pair (k, l) the corresponding summand comes with three frequency bands of the same width, whose centers are affine functions of each other.

[figure: ξ_{1}^{k,l}; ξ_{2}^{k,l}; ξ_{3}^{k,l}; ϕ_{1}^{k,l,n}; ϕ_{2}^{k,l,n}; ϕ_{3}^{k,l,n}]

As k, l, n vary, the rectangle ϕ_{ȷ}^{k,l,n} takes every possible position (area 1) in the plane.

Wave packets

[figure: x; ξ]

Fix a generating function ϕ. Then for all k, n, l ∈ Z we define the wave packet

ϕ_{k,n,l} = 2^{−k/2}e^{2πi2^{−k}xl} ϕ(2^{−k}x − n) .

In contrast to a wavelet the wavepacket can have arbitrary frequency relative to its scale.

A typical wavelet and wave packet:

\overline{}.

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) .

[figure: W_{0} :; W_{1} :; W_{2} :; W_{3} :; W_{4} :; W_{5} :; W_{6} :; W_{7} :; W_{8} :; 0; 1; …]

Tiles, Walsh Wave Packets

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 denote the first interval of p by I_{p}. The corresponding Walsh wave packet is the function w_{p} defined by:

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

[figure: p; w_{p}; 0; 1; 2; 3; 4; I_{p}]

The Recursion Relation

Let l, r, u, d (left, right, up, down) be tiles with l ∪ r = d ∪ u as below:

[figure: l; r; w_{l}; w_{r}; d; u; w_{d}; w_{u}]

Then we have the orthogonal transformation:

( \begin{matrix}w_{d} \\ w_{u} \end{matrix} ) = ( \begin{matrix}1/√2 & 1/√2 \\ 1/√2 & −1/√2 \end{matrix} ) ( \begin{matrix}w_{l} \\ w_{r} \end{matrix} ) .

Geometry of Dyadic Intervals

Dyadic intervals arise from successive subdivision of intervals.

Hence two dyadic intervals are either disjoint, or one is contained in the other

Two dyadic rectangles of area one always intersect like

.

They never intersect like

.

Covering and Linear Relations

Claim: If a tile p is covered by tiles p_{1}, …, p_{n}, then w_{p} is a linear combination of w_{p_{1}}, … w_{p_{n}}.

Proof: It suffices (symmetry!) to assume that already those p_{i} cover p, for which I_{p} ⊂ I_{p_{i}}.

.

Also it suffices to assume that the p_{i} are pairwise disjoint.

.

Then the longest tiles come in pairs, so one can successively use the recursion relation:

.

Orthogonality

The Walsh functions are pairwise orthogonal:

[figure: 0; 1; 0; 1; 0; 1] . The Walsh system is complete in L^{2}[0, 1]. If two tiles p_{1}, p_{2} are disjoint, then w_{p_{1}} and w_{p_{2}} are orthogonal:

.

If p_{1}, …, p_{n} is a set of pairwise disjoint tiles, then w_{p_{1}}, …, w_{p_{n}} is an orthonormal basis of a subspace of L^{2}(R). This subspace is an invariant of the set:

p_{1} ∪ … ∪ p_{n} .

The Quartile Operator

A quartile P is 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 as follows:

a_{P} b_{P} c_{P} d_{P} .

The bilinear quartile operator H is defined by:

H(f, g) = _{P:quartile}^{∑} |I_{P} |^{−1/2} ⟨f, w_{a_{P}} ⟩ ⟨g, w_{b_{P}} ⟩ w_{c_{P}} =

_{k,n,l∈Z,l≥0}^{∑} |I_{P} |^{−1/2} ⟨f, w_{k,n,4l}⟩ ⟨g, w_{k,n,4l+1} ⟩ w_{k,n,4l+2} .

Theorem C. T. 1995 If 1 < p_{1}, p_{2} ≤ ∞, 2/3 < p < ∞, 1/p = 1/p_{1} + 1/p_{2}, then there is a C such that for all f_{1}, f_{2}:

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