Singular integral operators of weak type (1, 1) #
Let T be a bounded linear operator on L²(ℝⁿ). The Calderón–Zygmund theorem says that T
is of weak type (1, 1) as soon as it satisfies a cancellation condition: there is a constant
B such that for every closed ball closedBall y r and every b ∈ L² vanishing off
that ball with integral zero,
∫_{ℝⁿ \ closedBall y (2r)} ‖T b‖ ≤ B ‖b‖₁.
The conclusion is that for every f ∈ L² and every t,
t · |{‖T f‖ > t}| ≤ (2ⁿ (4 ‖T‖² + 1) + 4 B) ‖f‖₁
(ContinuousLinearMap.mul_volume_lt_enorm_le_of_setLIntegral_compl_closedBall_le). The theorem
is stated more generally with closedBall y (c r) in place of the doubled ball, for any dilation
factor c, and then gives the constant 4 · 2ⁿ ‖T‖² + cⁿ + 4 B. Kernels satisfying only
Hörmander's condition in the Euclidean distance give the cancellation condition in the sup distance
with c = 2 √n.
The cancellation condition holds for an operator given off the support of b by a kernel,
T b x = ∫ K x y (b y) dy, satisfying Hörmander's kernel condition
∫_{dist x y' > 2 dist y y'} ‖K x y - K x y'‖ dx ≤ B for all y, y'
(TauCeti.setLIntegral_compl_closedBall_enorm_le_of_hormander). Since b has mean zero, K x y₀
can be subtracted from the kernel, and for y in the ball and x outside the doubled ball the
difference K x y - K x y₀ is controlled by Hörmander's condition. Combining the two gives the
classical form of the theorem (ContinuousLinearMap.mul_volume_lt_enorm_le_of_hormander). The
weak-type bound is the endpoint estimate that Marcinkiewicz interpolation against the L² bound
turns into Lᵖ bounds for 1 < p < 2.
The proof #
Decompose f = g + ∑_Q b_Q at height t (TauCeti.calderonZygmundGood,
TauCeti.calderonZygmundBad). The good part satisfies ‖g‖₂² ≤ 2ⁿ t ‖f‖₁, so Chebyshev's
inequality and the L² bound give t · |{‖T g‖ > t / 2}| ≤ 4 · 2ⁿ ‖T‖² ‖f‖₁. Each cube Q lies in
the closed sup-norm ball B_Q of radius half its side, and the union Ω of the dilated balls
c B_Q has t |Ω| ≤ cⁿ ‖f‖₁. Off Ω, the cancellation condition applied to each b_Q gives
∫_{ℝⁿ \ Ω} ‖T b‖ ≤ ∑_Q B ‖b_Q‖₁ ≤ 2 B ‖f‖₁, where ‖T b‖ ≤ ∑_Q ‖T b_Q‖ almost everywhere because
the b_Q sum to b in L² (MeasureTheory.Lp.ae_enorm_le_tsum_enorm_of_hasSum). A last
application of Chebyshev's inequality bounds t · |{‖T b‖ > t / 2} \ Ω| by 4 B ‖f‖₁.
Points of ℝⁿ are functions ι → ℝ, so distances and balls are taken in the sup norm.
Main declarations #
ContinuousLinearMap.mul_volume_lt_enorm_le_of_setLIntegral_compl_closedBall_le: anL²-bounded operator satisfying the cancellation condition is of weak type(1, 1).TauCeti.setLIntegral_compl_closedBall_enorm_le_of_hormander: Hörmander's kernel condition implies the cancellation condition, in any metric measure space.ContinuousLinearMap.setLIntegral_compl_closedBall_enorm_apply_le_of_hormander: the same for an operator onL²(ℝⁿ)given off balls by a kernel.ContinuousLinearMap.mul_volume_lt_enorm_le_of_hormander: anL²-bounded operator with a kernel satisfying Hörmander's condition is of weak type(1, 1).
References #
- A. P. Calderón and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (1952), 85–139.
- L. Hörmander, Estimates for translation invariant operators in Lᵖ spaces, Acta Math. 104 (1960), 93–140.
- E. Stein, Singular Integrals and Differentiability Properties of Functions, Chapter II, §2.
- L. Grafakos, Classical Fourier Analysis, Section 5.3.
Hörmander's kernel condition implies the cancellation condition. Let K be a kernel whose
differences satisfy ∫_{dist x y' > 2 dist y y'} ‖K x y - K x y'‖ dx ≤ B for all y, y'. If b
is integrable, vanishes off closedBall y₀ r and has integral zero, and h x = ∫ K x y (b y) dy
for almost every x off that ball, then the integral of ‖h‖ off closedBall y₀ (2 r) is at
most B ‖b‖₁.
The weak-type bound on a one-point space. For empty ι, the space ι → ℝ is a single
point carrying the Dirac measure, so t |{‖T f‖ > t}| ≤ ‖T‖ ‖f‖₁ for any bounded operator on
L².
The Calderón–Zygmund theorem, weak type (1, 1). Let T be a bounded linear operator on
L²(ℝⁿ) satisfying the cancellation condition with dilation factor c: for every b ∈ L²
vanishing off a closed ball closedBall y r with integral zero,
∫_{ℝⁿ \ closedBall y (c r)} ‖T b‖ ≤ B ‖b‖₁. Then for every f ∈ L² and every t,
t · |{‖T f‖ > t}| ≤ (4 · 2ⁿ ‖T‖² + cⁿ + 4 B) ‖f‖₁,
where cⁿ is read as 0 when c < 0 (it is ENNReal.ofReal c ^ n). The classical factor is
c = 2, for which the constant is 2ⁿ (4 ‖T‖² + 1) + 4 B. A larger factor makes the cancellation
condition weaker; it is needed when the condition comes from a kernel estimate in a distance other
than the sup distance. The bound is vacuous unless f is also integrable.
An operator on L²(ℝⁿ) given off balls by a kernel satisfying Hörmander's condition satisfies
the cancellation condition: if T b x = ∫ K x y (b y) dy for almost every x off any closed ball
outside which b vanishes, and ∫_{dist x y' > 2 dist y y'} ‖K x y - K x y'‖ dx ≤ B for all y,
y', then ∫_{ℝⁿ \ closedBall y (2r)} ‖T b‖ ≤ B ‖b‖₁ for every b ∈ L² vanishing off
closedBall y r with integral zero.
The Calderón–Zygmund theorem for an operator given by a kernel. Let T be a bounded linear
operator on L²(ℝⁿ) such that T b x = ∫ K x y (b y) dy for almost every x off any closed ball
outside which b vanishes, where the kernel K satisfies Hörmander's condition
∫_{dist x y' > 2 dist y y'} ‖K x y - K x y'‖ dx ≤ B for all y, y'. Then for every f ∈ L² and
every t,
t · |{‖T f‖ > t}| ≤ (2ⁿ (4 ‖T‖² + 1) + 4 B) ‖f‖₁.