Calderón–Zygmund operators on a finite-dimensional inner product space #
The Calderón–Zygmund theorems
(ContinuousLinearMap.mul_volume_lt_enorm_le_of_setLIntegral_compl_closedBall_le,
TauCeti.exists_eLpNorm_le_of_setLIntegral_compl_closedBall_le) are stated on
ℝⁿ = ι → ℝ with the sup distance, because the Calderón–Zygmund decomposition runs on dyadic
cubes. The singular integral kernels of analysis, such as the second derivatives D²Gₙ (x - y) of
the Newtonian kernel, are instead estimated in the Euclidean distance. This file states the
theorems on a finite-dimensional real inner product space V of dimension n ≥ 1, with its
Lebesgue measure volume.
Let T be a bounded linear operator on L²(V) given off the support of its argument by a kernel
K: if b ∈ L² vanishes off a compact set s, then T b x = ∫ K x y (b y) dy for almost every
x ∉ s. Let K be a standard kernel: for some δ > 0 it satisfies the size estimate and the
smoothness estimate
‖K x y‖ ≤ C₀ ‖x - y‖ ^ (-n) for x ≠ y, and
‖K x y - K x y'‖ ≤ C ‖y - y'‖ ^ δ ‖x - y'‖ ^ (-(n + δ)) whenever ‖x - y'‖ > 2 ‖y - y'‖.
Then T is of weak type (1, 1)
(ContinuousLinearMap.exists_mul_volume_lt_enorm_le_of_norm_sub_le_mul_rpow). If moreover the
adjoint T† has a standard kernel, then for every 1 < p < ∞ there is a constant C with
‖T f‖_p ≤ C ‖f‖_p for every f ∈ L²
(ContinuousLinearMap.exists_eLpNorm_le_of_norm_sub_le_mul_rpow).
An orthonormal basis b identifies ι → ℝ with V through x ↦ ∑ xᵢ bᵢ, preserving Lebesgue
measure, and precomposition with it conjugates T to an operator on L²(ι → ℝ) whose kernel is
K in coordinates. Since ‖x‖_∞ ≤ ‖∑ xᵢ bᵢ‖ ≤ √n ‖x‖_∞, the coordinate kernel satisfies the
smoothness estimate for the sup distance only when dist x y' > 2 √n dist y y', and the size
estimate extends it to dist x y' > 2 dist y y'
(TauCeti.norm_sub_le_mul_rpow_of_norm_le_mul_rpow). The kernel representation is assumed off
compact sets, rather than off balls of V, because a sup-distance ball of ι → ℝ is a cube of V.
Kernels that are not standard, such as the kernels of the Fourier multipliers of the Mihlin–Hörmander theorem, often satisfy only Hörmander's condition
∫_{‖x - y'‖ > 2 ‖y - y'‖} ‖K x y - K x y'‖ dx ≤ B for all y, y'.
Without a size estimate, the region ‖x - y'‖ > 2 ‖y - y'‖ cannot be enlarged, so this is used
with the cancellation condition in coordinates for the dilation factor 2 √n: the complement of
the sup-distance ball of radius 2 √n r around the centre of a cube of V lies in the region of
Hörmander's condition for every point of the cube. The resulting weak type (1, 1) and strong
type (p, p) bounds (TauCeti.exists_mul_volume_lt_enorm_le_of_hormander,
TauCeti.exists_eLpNorm_le_of_hormander) have constants depending only on n, p,
B and a bound for ‖T‖, so they hold uniformly over families of operators.
Main declarations #
ContinuousLinearMap.exists_mul_volume_lt_enorm_le_of_norm_sub_le_mul_rpow: an operator onL²(V)with a standard kernel is of weak type(1, 1).ContinuousLinearMap.exists_eLpNorm_le_of_norm_sub_le_mul_rpow: an operator onL²(V)such that both it and its adjoint have standard kernels is bounded in theLᵖnorm for1 < p < ∞.TauCeti.exists_mul_volume_lt_enorm_le_of_hormander: operators onL²(V)with kernels satisfying Hörmander's condition are of weak type(1, 1), uniformly.TauCeti.exists_eLpNorm_le_of_hormander: operators onL²(V)such that both they and their adjoints have kernels satisfying Hörmander's condition are bounded in theLᵖnorm for1 < p < ∞, uniformly.TauCeti.setLIntegral_enorm_sub_le_of_norm_sub_le: a kernel dominated by the differences of a translation-invariant kernelK (x - y)inherits Hörmander's condition fromK.
References #
- E. Stein, Singular Integrals and Differentiability Properties of Functions, Chapter II, §2.
- L. Grafakos, Classical Fourier Analysis, Sections 5.3 and 5.4.
- L. Hörmander, Estimates for translation invariant operators in Lᵖ spaces, Acta Math. 104 (1960), 93–140.
The Calderón–Zygmund theorem on an inner product space, weak type (1, 1). Let V be a
finite-dimensional real inner product space of dimension n ≥ 1, and let T be a bounded linear
operator on L²(V) such that T f x = ∫ K x y (f y) dy for almost every x off any compact set
outside which f vanishes, where for some δ > 0 the kernel K satisfies the size and smoothness
estimates
‖K x y‖ ≤ C₀ ‖x - y‖ ^ (-n) for x ≠ y, and
‖K x y - K x y'‖ ≤ C ‖y - y'‖ ^ δ ‖x - y'‖ ^ (-(n + δ)) whenever ‖x - y'‖ > 2 ‖y - y'‖.
Then there is a constant A with t · |{‖T f‖ > t}| ≤ A ‖f‖₁ for every f ∈ L² and every t.
The Calderón–Zygmund theorem on an inner product space under Hörmander's condition, weak
type (1, 1), with a constant depending only on the data. Let V be a finite-dimensional real
inner product space of dimension n ≥ 1, and let M and B be finite. There is a constant A
with the following property. Let T be a bounded linear operator on L²(V) with ‖T‖ ≤ M, such
that T f x = ∫ k x y (f y) dy for almost every x off any compact set outside which f vanishes,
where the kernel k satisfies Hörmander's condition
∫_{‖x - y'‖ > 2 ‖y - y'‖} ‖k x y - k x y'‖ dx ≤ B for all y, y'.
Then t · |{‖T f‖ > t}| ≤ A ‖f‖₁ for every f ∈ L² and every t. One can take
A = 4 · 2ⁿ M² + (2 √n)ⁿ + 4 B.
The Calderón–Zygmund theorem on an inner product space, strong type (p, p) for
1 < p < ∞. Let V be a finite-dimensional real inner product space of dimension n ≥ 1, and let
T be a bounded linear operator on L²(V) of Hilbert space valued functions. Suppose that T and
its adjoint T† are given off the support of their argument by kernels k and k': for every
compact set s and every f vanishing off s, T f x = ∫ k x y (f y) dy (respectively
T† f x = ∫ k' x y (f y) dy) for almost every x ∉ s. Suppose that, for some δ > 0, the
kernel k satisfies the size and smoothness estimates
‖k x y‖ ≤ C₀ ‖x - y‖ ^ (-n) for x ≠ y, and
‖k x y - k x y'‖ ≤ C ‖y - y'‖ ^ δ ‖x - y'‖ ^ (-(n + δ)) whenever ‖x - y'‖ > 2 ‖y - y'‖,
and that k' satisfies the same estimates with its own constants C₀', C' and exponent
δ' > 0.
Then for every 1 < p < ∞ there is a constant C with ‖T f‖_p ≤ C ‖f‖_p for every f ∈ L².
The Calderón–Zygmund theorem on an inner product space under Hörmander's condition, strong
type (p, p) for 1 < p < ∞, with a constant depending only on the data. Let V be a
finite-dimensional real inner product space of dimension n ≥ 1, let M, B and B' be finite,
and let 1 < p < ∞. There is a constant C with the following property. Let T be a bounded
linear operator on L²(V) of Hilbert space valued functions with ‖T‖ ≤ M. Suppose that T and
its adjoint T† are given off the support of their argument by kernels k and k': for every
compact set s and every f vanishing off s, T f x = ∫ k x y (f y) dy (respectively
T† f x = ∫ k' x y (f y) dy) for almost every x ∉ s. Suppose that both kernels satisfy
Hörmander's condition
∫_{‖x - y'‖ > 2 ‖y - y'‖} ‖k x y - k x y'‖ dx ≤ B and
∫_{‖x - y'‖ > 2 ‖y - y'‖} ‖k' x y - k' x y'‖ dx ≤ B' for all y, y'.
Then ‖T f‖_p ≤ C ‖f‖_p for every f ∈ L². Since C does not depend on T, the theorem applies
uniformly to families of operators, such as the truncations of a Fourier multiplier.
Hörmander's condition for a translation-invariant kernel. Let k be a kernel dominated by
the differences of an integrable function K, ‖k x y - k x y'‖ ≤ ‖K (x - y) - K (x - y')‖. If
K satisfies Hörmander's condition in the translation-invariant form
∫_{‖x‖ > 2 ‖h‖} ‖K (x - h) - K x‖ dx ≤ B, then k satisfies it in the two-variable form
∫_{‖x - y'‖ > 2 ‖y - y'‖} ‖k x y - k x y'‖ dx ≤ B of the Calderón–Zygmund theorems.