Littlewood–Paley cutoffs #
Let φ : E → ℝ be a smooth bump function on a real normed space, equal to 1 on
closedBall 0 φ.rIn and supported in ball 0 φ.rOut. Its dyadic differences
ψⱼ ξ = φ (2^{-j} ξ) - φ (2^{1-j} ξ), j ∈ ℤ,
form a smooth Littlewood–Paley partition of unity of E \ {0}: each ψⱼ is supported in the
annulus 2^{j-1} φ.rIn ≤ ‖ξ‖ ≤ 2^j φ.rOut, the partial sums telescope,
∑_{a < j ≤ b} ψⱼ ξ = φ (2^{-b} ξ) - φ (2^{-a} ξ), and ∑ⱼ ψⱼ ξ = 1 for every ξ ≠ 0, with
only finitely many nonzero terms. In finite dimension, the k-th derivative of ψⱼ is
O(2^{-jk}) uniformly in j.
Keeping φ itself as a low-frequency piece and discarding the cutoffs with j ≤ 0 gives the
inhomogeneous partition ψ₀ = φ, ψ₁, ψ₂, …, which sums to 1 on all of E. Its partial sums
are ∑_{j ≤ N} ψⱼ = φ (2^{-N} ·), each ψ_{j+1} is the rescaling ψ₁ (2^{-j} ·), and pieces whose
indices differ by at least a fixed L have disjoint supports, so that ∑ⱼ ψⱼ ξ ² is bounded
independently of ξ. This is the partition used to decompose a tempered distribution into
frequency blocks for Besov and Bessel-potential spaces.
The sum Ψⱼ of the pieces with indices within L of j is then a smooth cutoff, equal to 1 on
the support of ψⱼ and supported in a slightly larger ball or annulus, with the same rescaled
derivative bounds; it localizes operators to the frequencies of a single block.
Main declarations #
ContDiffBump.littlewoodPaley: the dyadic cutoffψⱼ.ContDiffBump.sum_littlewoodPaley_Ioc: the telescoping partial sums.ContDiffBump.hasSum_littlewoodPaley:∑ⱼ ψⱼ ξ = 1forξ ≠ 0.ContDiffBump.exists_norm_iteratedFDeriv_littlewoodPaley_le:‖D^k ψⱼ‖ ≤ B 2^{-jk}.ContDiffBump.inhomogeneousLittlewoodPaley: the inhomogeneous partitionψ₀ = φ, ψ₁, ψ₂, ….ContDiffBump.hasSum_inhomogeneousLittlewoodPaley:∑ⱼ ψⱼ ξ = 1for everyξ.ContDiffBump.exists_inhomogeneousLittlewoodPaley_mul_eq_zero: finite overlap of the pieces.ContDiffBump.exists_sum_sq_inhomogeneousLittlewoodPaley_le:∑ⱼ ψⱼ ξ ²is bounded uniformly inξand in the set of indices.ContDiffBump.exists_one_add_norm_sq_rpow_le_of_mem_tsupport_inhomogeneousLittlewoodPaley: on the support ofψⱼ,(1 + ‖ξ‖ ^ 2) ^ (t / 2) ≤ c (2^j) ^ twithcindependent ofj.ContDiffBump.exists_norm_iteratedFDeriv_inhomogeneousLittlewoodPaley_le: the uniform rescaled derivative bounds for the inhomogeneous partition, andContDiffBump.exists_norm_iteratedFDeriv_inhomogeneousLittlewoodPaley_mul_le: their weighted form‖D^k ψⱼ ξ‖ (1 + ‖ξ‖) ^ k ≤ B, so the pieces form a bounded family of symbols of order zero;ContDiffBump.exists_norm_iteratedFDeriv_inhomogeneousLittlewoodPaley_mul_rpow_leis the same bound with the Japanese bracket weight(1 + ‖ξ‖ ^ 2) ^ (k / 2).ContDiffBump.enlargedLittlewoodPaley: the enlarged cutoffsΨⱼ, sums of the neighbouring pieces.ContDiffBump.enlargedLittlewoodPaley_mul_inhomogeneousLittlewoodPaley:Ψⱼ ψⱼ = ψⱼ.ContDiffBump.exists_norm_iteratedFDeriv_enlargedLittlewoodPaley_le: the uniform rescaled derivative bounds for the enlarged cutoffs.ContDiffBump.exists_one_add_norm_sq_le_of_mem_tsupport_enlargedLittlewoodPaley: on the support ofΨⱼ,1 + ‖ξ‖ ^ 2is comparable to4^juniformly inj.
References #
- L. Grafakos, Classical Fourier Analysis, 3rd ed., Section 6.2.2 (proof of Theorem 6.2.7).
- E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Chapter IV, Section 3.
- J. Bergh and J. Löfström, Interpolation Spaces: An Introduction, Section 6.2 (the inhomogeneous partition).
The Littlewood–Paley cutoff at scale 2^j built from the bump function φ:
ψⱼ ξ = φ (2^{-j} ξ) - φ (2^{1-j} ξ). It is supported in the annulus
2^{j-1} φ.rIn ≤ ‖ξ‖ ≤ 2^j φ.rOut, and these cutoffs sum to 1 on E \ {0}.
Instances For
Unfolding ContDiffBump.littlewoodPaley.
Each Littlewood–Paley cutoff is smooth.
The cutoff ψⱼ vanishes on the closed ball of radius 2^{j-1} φ.rIn, where both terms
are 1.
The cutoff ψⱼ vanishes outside the open ball of radius 2^j φ.rOut, where both terms
are 0.
The cutoff ψⱼ is supported in the closed annulus 2^{j-1} φ.rIn ≤ ‖ξ‖ ≤ 2^j φ.rOut.
The partial sums of the Littlewood–Paley cutoffs telescope:
∑_{a < j ≤ b} ψⱼ ξ = φ (2^{-b} ξ) - φ (2^{-a} ξ).
The Littlewood–Paley partition of unity. For ξ ≠ 0, ∑ⱼ ψⱼ ξ = 1, the sum having
only finitely many nonzero terms.
The inhomogeneous partition #
The inhomogeneous Littlewood–Paley partition built from the bump function φ: the
low-frequency piece ψ₀ = φ, and the dyadic cutoffs ψⱼ = φ (2^{-j} ·) - φ (2^{1-j} ·) for
j ≥ 1. The piece ψ₀ is supported in the ball of radius φ.rOut, each ψⱼ with j ≥ 1 in
the annulus 2^{j-1} φ.rIn ≤ ‖ξ‖ ≤ 2^j φ.rOut, and these functions sum to 1 on all of E.
Instances For
The low-frequency piece of the inhomogeneous partition is the bump function itself.
Away from the low-frequency piece, the inhomogeneous partition consists of the dyadic
cutoffs ψⱼ.
The pieces ψ_{j+1} of the inhomogeneous partition are the dyadic cutoffs.
The value of ψ_{j+1} in terms of the bump function, with natural-number powers.
Scaling of the dyadic pieces. Each ψ_{j+1} is the first dyadic piece rescaled to
frequency 2^j: ψ_{j+1} ξ = ψ₁ (2^{-j} ξ).
Each piece of the inhomogeneous partition is smooth.
Every piece ψⱼ vanishes outside the open ball of radius 2^j φ.rOut.
The piece ψ_{j+1} vanishes on the closed ball of radius 2^j φ.rIn.
The low-frequency piece ψ₀ is supported in the closed ball of radius φ.rOut.
The piece ψ_{j+1} is supported in the closed annulus
2^j φ.rIn ≤ ‖ξ‖ ≤ 2^{j+1} φ.rOut.
Every piece ψⱼ of the inhomogeneous partition is supported in the closed ball of radius
2^j φ.rOut.
On the support of ψⱼ, the Japanese bracket (1 + ‖ξ‖ ^ 2) ^ (1 / 2) is comparable to 2^j,
so every power (1 + ‖ξ‖ ^ 2) ^ (t / 2) is bounded by a constant independent of j times
(2^j) ^ t.
The partial sums of the inhomogeneous partition telescope:
∑_{j ≤ N} ψⱼ ξ = φ (2^{-N} ξ).
The inhomogeneous Littlewood–Paley partition of unity. For every ξ, ∑ⱼ ψⱼ ξ = 1, the
sum having only finitely many nonzero terms.
Finite overlap. There is L such that the pieces ψⱼ and ψₖ have disjoint supports
whenever j + L ≤ k.
The enlarged Littlewood–Paley cutoff Ψⱼ = ∑_{j + 1 - L ≤ k < j + L} ψₖ, the sum of the
pieces of the inhomogeneous partition whose indices are within L of j. It is supported in the
ball of radius 2^{j+L} φ.rOut, and for j ≥ L outside the ball of radius 2^{j-L} φ.rIn. When
the pieces whose indices differ by at least L have disjoint supports, as for the L of
exists_inhomogeneousLittlewoodPaley_mul_eq_zero, it equals 1 on the support of ψⱼ
(enlargedLittlewoodPaley_mul_inhomogeneousLittlewoodPaley), so that the Fourier multiplier
Δ'ⱼ by Ψⱼ satisfies Δ'ⱼ Δⱼ = Δⱼ.
Instances For
Unfolding ContDiffBump.enlargedLittlewoodPaley.
The enlarged cutoff is 1 on the support of ψⱼ. If the pieces of the inhomogeneous
partition whose indices differ by at least L have disjoint supports, then Ψⱼ ψⱼ = ψⱼ.
Each enlarged cutoff is smooth.
The enlarged cutoff Ψⱼ vanishes outside the open ball of radius 2^{j+L} φ.rOut.
For j ≥ L, the enlarged cutoff Ψⱼ vanishes on the closed ball of radius
2^{j-L} φ.rIn.
The enlarged cutoff Ψⱼ is supported in the closed ball of radius 2^{j+L} φ.rOut.
For j ≥ L, the enlarged cutoff Ψⱼ is supported where 2^{j-L} φ.rIn ≤ ‖ξ‖.
On the support of the enlarged cutoff Ψⱼ, the Japanese bracket is comparable to 2^j:
a 4^j ≤ 1 + ‖ξ‖ ^ 2 ≤ b 4^j with a > 0 and b independent of j.
Every piece of the inhomogeneous partition takes values in [-1, 1].
Bounded sum of squares. There is L, depending only on φ, with ∑ⱼ ψⱼ ξ ² ≤ L for every
ξ, over any finite set of indices: at each ξ at most L consecutive pieces are nonzero, and
each lies in [-1, 1].
Derivative bounds for the Littlewood–Paley cutoffs. The k-th derivative of ψⱼ is
bounded by B / 2^{jk}, with B independent of j.
Each piece of the inhomogeneous partition is compactly supported.
Each piece of the inhomogeneous partition has temperate growth, so it is a Fourier multiplier on Schwartz functions and on tempered distributions.
The pieces of the inhomogeneous partition, viewed as complex-valued symbols, have temperate growth, so they are Fourier multipliers on Schwartz functions and on tempered distributions.
Derivative bounds for the inhomogeneous partition. The k-th derivative of ψⱼ is
bounded by B / 2^{jk}, with B independent of j.
The cutoffs are a bounded family of symbols of order zero. The k-th derivative of ψⱼ
satisfies ‖D^k ψⱼ ξ‖ (1 + ‖ξ‖) ^ k ≤ B with B independent of j: the factor 2^{-jk} from the
derivative compensates 1 + ‖ξ‖ ≤ 2^j (1 + φ.rOut) on the support of ψⱼ.
The bound exists_norm_iteratedFDeriv_inhomogeneousLittlewoodPaley_mul_le with the Japanese
bracket weight: ‖D^k ψⱼ ξ‖ (1 + ‖ξ‖ ^ 2) ^ (k / 2) ≤ B with B independent of j.
Each enlarged cutoff is compactly supported.
Derivative bounds for the enlarged cutoffs. The k-th derivative of Ψⱼ is bounded by
B / 2^{jk}, with B independent of j.