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TauCeti.MeasureTheory.Integral.CalderonZygmund.StrongType

Singular integral operators of strong type (p, p) for 1 < p < ∞ #

Let T be a bounded linear operator on L²(ℝⁿ) satisfying the Calderón–Zygmund cancellation condition, for instance one given by a kernel satisfying Hörmander's condition. The weak type (1, 1) half of the Calderón–Zygmund theorem is ContinuousLinearMap.mul_volume_lt_enorm_le_of_setLIntegral_compl_closedBall_le. Marcinkiewicz interpolation between that bound and the L² bound (ContinuousLinearMap.eLpNorm_le_of_rpow_mul_meas_lt_le) gives the strong type (p, p) bound

‖T f‖_p ≤ (p / (p - 1) · 2 C + p / (2 - p) · 4 ‖T‖²) ^ (1 / p) ‖f‖_p

for every 1 < p < 2 and every f ∈ L², where C = 2ⁿ (4 ‖T‖² + 1) + 4 B is the weak type (1, 1) constant, or C = 4 · 2ⁿ ‖T‖² + cⁿ + 4 B for the cancellation condition with dilation factor c in place of 2. This constant blows up as p → 1, and also as p → 2 when ‖T‖ > 0. That is a limitation of the interpolation argument, not of every operator satisfying the hypotheses (the zero operator satisfies them), although classical singular integrals such as the Hilbert transform are indeed not bounded on L¹.

For Hilbert space valued functions, the range 2 < p < ∞ follows by duality (ContinuousLinearMap.eLpNorm_le_of_eLpNorm_adjoint_le): if the adjoint T† satisfies the cancellation condition, the bound for T† at the conjugate exponent p / (p - 1) ∈ (1, 2) gives

‖T f‖_p ≤ (p · 2 C + p / (p - 2) · 4 ‖T‖²) ^ (1 - 1 / p) ‖f‖_p

for every f ∈ L², with C = 2ⁿ (4 ‖T‖² + 1) + 4 B and B the cancellation constant of T†. When T and T† both satisfy the cancellation condition, T is therefore bounded in the Lᵖ norm for every 1 < p < ∞, with a constant depending only on n, p, the dilation factor, a bound for ‖T‖ and the cancellation constants.

Since T is only given on L², the estimates are stated for f ∈ L², and they are informative for f ∈ L² ∩ Lᵖ. Extending T to a bounded operator on all of Lᵖ is a separate density argument, not carried out here.

Points of ℝⁿ are functions ι → ℝ, so distances and balls are taken in the sup norm.

Main declarations #

References #

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theorem ContinuousLinearMap.eLpNorm_le_of_setLIntegral_compl_closedBall_le {ι : Type u_1} [Fintype ι] [Nonempty ι] {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] [NormedAddCommGroup F] [NormedSpace ℝ F] {p : ENNReal} (T : ↥(MeasureTheory.Lp E 2 MeasureTheory.volume) →L[ℝ] ↥(MeasureTheory.Lp F 2 MeasureTheory.volume)) {B : ENNReal} {c : ℝ} (hT : ∀ (b : ↥(MeasureTheory.Lp E 2 MeasureTheory.volume)) (y : ι → ℝ) (r : ℝ), (∀ᵐ (x : ι → ℝ), x ∉ Metric.closedBall y r → ↑↑b x = 0) → ∫ (x : ι → ℝ), ↑↑b x = 0 → ∫⁻ (x : ι → ℝ) in (Metric.closedBall y (c * r))ᶜ, ‖↑↑(T b) x‖ₑ ≤ B * ∫⁻ (x : ι → ℝ), ‖↑↑b x‖ₑ) (hp : 1 < p) (hp₂ : p < 2) (f : ↥(MeasureTheory.Lp E 2 MeasureTheory.volume)) :
MeasureTheory.eLpNorm (↑↑(T f)) p MeasureTheory.volume ≤ (ENNReal.ofReal (p.toReal / (p.toReal - 1)) * (2 * (2 ^ Fintype.card ι * (4 * ‖T‖ₑ ^ 2) + ENNReal.ofReal c ^ Fintype.card ι + 4 * B)) + ENNReal.ofReal (p.toReal / (2 - p.toReal)) * (4 * ‖T‖ₑ ^ 2)) ^ (1 / p.toReal) * MeasureTheory.eLpNorm (↑↑f) p MeasureTheory.volume

The Calderón–Zygmund theorem, strong type (p, p) for 1 < p < 2. 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 1 < p < 2 and every f ∈ L²,

‖T f‖_p ≤ (p / (p - 1) · 2 C + p / (2 - p) · 4 ‖T‖²) ^ (1 / p) ‖f‖_p,

where C = 4 · 2ⁿ ‖T‖² + cⁿ + 4 B is the weak type (1, 1) constant of ContinuousLinearMap.mul_volume_lt_enorm_le_of_setLIntegral_compl_closedBall_le. The bound is vacuous unless f also lies in Lᵖ.

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theorem ContinuousLinearMap.eLpNorm_le_of_hormander {ι : Type u_1} [Fintype ι] [Nonempty ι] {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] [NormedAddCommGroup F] [NormedSpace ℝ F] {p : ENNReal} [CompleteSpace F] (T : ↥(MeasureTheory.Lp E 2 MeasureTheory.volume) →L[ℝ] ↥(MeasureTheory.Lp F 2 MeasureTheory.volume)) {K : (ι → ℝ) → (ι → ℝ) → E →L[ℝ] F} (hK : MeasureTheory.StronglyMeasurable (Function.uncurry K)) {B : ENNReal} (hB : ∀ (y y' : ι → ℝ), ∫⁻ (x : ι → ℝ) in {x : ι → ℝ | 2 * dist y y' < dist x y'}, ‖K x y - K x y'‖ₑ ≤ B) (hrep : ∀ (b : ↥(MeasureTheory.Lp E 2 MeasureTheory.volume)) (y : ι → ℝ) (r : ℝ), (∀ᵐ (x : ι → ℝ), x ∉ Metric.closedBall y r → ↑↑b x = 0) → ∀ᵐ (x : ι → ℝ), x ∉ Metric.closedBall y r → ↑↑(T b) x = ∫ (z : ι → ℝ), (K x z) (↑↑b z)) (hp : 1 < p) (hp₂ : p < 2) (f : ↥(MeasureTheory.Lp E 2 MeasureTheory.volume)) :
MeasureTheory.eLpNorm (↑↑(T f)) p MeasureTheory.volume ≤ (ENNReal.ofReal (p.toReal / (p.toReal - 1)) * (2 * (2 ^ Fintype.card ι * (4 * ‖T‖ₑ ^ 2 + 1) + 4 * B)) + ENNReal.ofReal (p.toReal / (2 - p.toReal)) * (4 * ‖T‖ₑ ^ 2)) ^ (1 / p.toReal) * MeasureTheory.eLpNorm (↑↑f) p MeasureTheory.volume

The Calderón–Zygmund theorem for an operator given by a kernel, strong type (p, p) for 1 < p < 2. 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 1 < p < 2 and every f ∈ L²,

‖T f‖_p ≤ (p / (p - 1) · 2 (2ⁿ (4 ‖T‖² + 1) + 4 B) + p / (2 - p) · 4 ‖T‖²) ^ (1 / p) ‖f‖_p.

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theorem ContinuousLinearMap.eLpNorm_le_of_setLIntegral_compl_closedBall_adjoint_le {ι : Type u_1} [Fintype ι] [Nonempty ι] {p : ENNReal} {H : Type u_4} {K : Type u_5} [NormedAddCommGroup H] [InnerProductSpace ℝ H] [CompleteSpace H] [NormedAddCommGroup K] [InnerProductSpace ℝ K] [CompleteSpace K] (T : ↥(MeasureTheory.Lp H 2 MeasureTheory.volume) →L[ℝ] ↥(MeasureTheory.Lp K 2 MeasureTheory.volume)) {B : ENNReal} {c : ℝ} (hT : ∀ (b : ↥(MeasureTheory.Lp K 2 MeasureTheory.volume)) (y : ι → ℝ) (r : ℝ), (∀ᵐ (x : ι → ℝ), x ∉ Metric.closedBall y r → ↑↑b x = 0) → ∫ (x : ι → ℝ), ↑↑b x = 0 → ∫⁻ (x : ι → ℝ) in (Metric.closedBall y (c * r))ᶜ, ‖↑↑((adjoint T) b) x‖ₑ ≤ B * ∫⁻ (x : ι → ℝ), ‖↑↑b x‖ₑ) (hp : 2 < p) (hp_top : p ≠ ⊤) (f : ↥(MeasureTheory.Lp H 2 MeasureTheory.volume)) :
MeasureTheory.eLpNorm (↑↑(T f)) p MeasureTheory.volume ≤ (ENNReal.ofReal p.toReal * (2 * (2 ^ Fintype.card ι * (4 * ‖T‖ₑ ^ 2) + ENNReal.ofReal c ^ Fintype.card ι + 4 * B)) + ENNReal.ofReal (p.toReal / (p.toReal - 2)) * (4 * ‖T‖ₑ ^ 2)) ^ (1 - 1 / p.toReal) * MeasureTheory.eLpNorm (↑↑f) p MeasureTheory.volume

The Calderón–Zygmund theorem, strong type (p, p) for 2 < p < ∞. Let T be a bounded linear operator on L²(ℝⁿ) of Hilbert space valued functions whose adjoint T† satisfies 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 2 < p < ∞ and every f ∈ L²,

‖T f‖_p ≤ (p · 2 C + p / (p - 2) · 4 ‖T‖²) ^ (1 - 1 / p) ‖f‖_p,

where C = 4 · 2ⁿ ‖T‖² + cⁿ + 4 B. This is the bound of ContinuousLinearMap.eLpNorm_le_of_setLIntegral_compl_closedBall_le for T† at the conjugate exponent p / (p - 1) ∈ (1, 2), transferred to T by duality.

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theorem ContinuousLinearMap.eLpNorm_le_of_hormander_adjoint {ι : Type u_1} [Fintype ι] [Nonempty ι] {p : ENNReal} {H : Type u_4} {K : Type u_5} [NormedAddCommGroup H] [InnerProductSpace ℝ H] [CompleteSpace H] [NormedAddCommGroup K] [InnerProductSpace ℝ K] [CompleteSpace K] (T : ↥(MeasureTheory.Lp H 2 MeasureTheory.volume) →L[ℝ] ↥(MeasureTheory.Lp K 2 MeasureTheory.volume)) {k : (ι → ℝ) → (ι → ℝ) → K →L[ℝ] H} (hk : MeasureTheory.StronglyMeasurable (Function.uncurry k)) {B : ENNReal} (hB : ∀ (y y' : ι → ℝ), ∫⁻ (x : ι → ℝ) in {x : ι → ℝ | 2 * dist y y' < dist x y'}, ‖k x y - k x y'‖ₑ ≤ B) (hrep : ∀ (b : ↥(MeasureTheory.Lp K 2 MeasureTheory.volume)) (y : ι → ℝ) (r : ℝ), (∀ᵐ (x : ι → ℝ), x ∉ Metric.closedBall y r → ↑↑b x = 0) → ∀ᵐ (x : ι → ℝ), x ∉ Metric.closedBall y r → ↑↑((adjoint T) b) x = ∫ (z : ι → ℝ), (k x z) (↑↑b z)) (hp : 2 < p) (hp_top : p ≠ ⊤) (f : ↥(MeasureTheory.Lp H 2 MeasureTheory.volume)) :
MeasureTheory.eLpNorm (↑↑(T f)) p MeasureTheory.volume ≤ (ENNReal.ofReal p.toReal * (2 * (2 ^ Fintype.card ι * (4 * ‖T‖ₑ ^ 2 + 1) + 4 * B)) + ENNReal.ofReal (p.toReal / (p.toReal - 2)) * (4 * ‖T‖ₑ ^ 2)) ^ (1 - 1 / p.toReal) * MeasureTheory.eLpNorm (↑↑f) p MeasureTheory.volume

The Calderón–Zygmund theorem for an operator whose adjoint is given by a kernel, strong type (p, p) for 2 < p < ∞. Let T be a bounded linear operator on L²(ℝⁿ) of Hilbert space valued functions such that T† b x = ∫ k x y (b y) dy for almost every x off any closed ball outside which b vanishes, where 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 2 < p < ∞ and every f ∈ L²,

‖T f‖_p ≤ (p · 2 (2ⁿ (4 ‖T‖² + 1) + 4 B) + p / (p - 2) · 4 ‖T‖²) ^ (1 - 1 / p) ‖f‖_p.

Formally, if T has kernel K, then T† has kernel k x y = (K y x)†, and the hypothesis is Hörmander's condition for K with the roles of its two variables exchanged.

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theorem TauCeti.exists_eLpNorm_le_of_setLIntegral_compl_closedBall_le {ι : Type u_1} [Fintype ι] [Nonempty ι] {H : Type u_2} {K : Type u_3} [NormedAddCommGroup H] [InnerProductSpace ℝ H] [CompleteSpace H] [NormedAddCommGroup K] [InnerProductSpace ℝ K] [CompleteSpace K] {p : ENNReal} (c : ℝ) {M B B' : ENNReal} (hM : M ≠ ⊤) (hB : B ≠ ⊤) (hB' : B' ≠ ⊤) (hp : 1 < p) (hp_top : p ≠ ⊤) :
∃ (C : NNReal), ∀ (T : ↥(MeasureTheory.Lp H 2 MeasureTheory.volume) →L[ℝ] ↥(MeasureTheory.Lp K 2 MeasureTheory.volume)), ‖T‖ₑ ≤ M → (∀ (b : ↥(MeasureTheory.Lp H 2 MeasureTheory.volume)) (y : ι → ℝ) (r : ℝ), (∀ᵐ (x : ι → ℝ), x ∉ Metric.closedBall y r → ↑↑b x = 0) → ∫ (x : ι → ℝ), ↑↑b x = 0 → ∫⁻ (x : ι → ℝ) in (Metric.closedBall y (c * r))ᶜ, ‖↑↑(T b) x‖ₑ ≤ B * ∫⁻ (x : ι → ℝ), ‖↑↑b x‖ₑ) → (∀ (b : ↥(MeasureTheory.Lp K 2 MeasureTheory.volume)) (y : ι → ℝ) (r : ℝ), (∀ᵐ (x : ι → ℝ), x ∉ Metric.closedBall y r → ↑↑b x = 0) → ∫ (x : ι → ℝ), ↑↑b x = 0 → ∫⁻ (x : ι → ℝ) in (Metric.closedBall y (c * r))ᶜ, ‖↑↑((ContinuousLinearMap.adjoint T) b) x‖ₑ ≤ B' * ∫⁻ (x : ι → ℝ), ‖↑↑b x‖ₑ) → ∀ (f : ↥(MeasureTheory.Lp H 2 MeasureTheory.volume)), MeasureTheory.eLpNorm (↑↑(T f)) p MeasureTheory.volume ≤ ↑C * MeasureTheory.eLpNorm (↑↑f) p MeasureTheory.volume

The Calderón–Zygmund theorem on Lᵖ, 1 < p < ∞, with a constant depending only on the data. Let c be a dilation factor, M a bound on operator norms and B, B' cancellation constants, all finite. Then for every 1 < p < ∞ there is a constant C with the following property. Let T be a bounded linear operator on L²(ℝⁿ) of Hilbert space valued functions with ‖T‖ ≤ M, such that T and its adjoint satisfy the cancellation condition with dilation factor c and constants B and B'. Then ‖T f‖_p ≤ C ‖f‖_p for every f ∈ L².

Explicit constants are given by ContinuousLinearMap.eLpNorm_le_of_setLIntegral_compl_closedBall_le for p < 2 and by ContinuousLinearMap.eLpNorm_le_of_setLIntegral_compl_closedBall_adjoint_le for p > 2. That C does not depend on T is what makes the theorem apply uniformly to families of operators, such as the truncations of a singular integral.