Fourier multipliers on L² #
Let E be a finite-dimensional real inner product space and F a complex Hilbert space. For an
essentially bounded symbol m ∈ L^∞(E), the Fourier multiplier with symbol m is the
operator
T_m f = 𝓕⁻ (m • 𝓕 f)
on L²(E, F), where 𝓕 = MeasureTheory.Lp.fourierTransformₗᵢ is the Fourier transform on L².
By Plancherel's theorem it is bounded, with ‖T_m‖ ≤ ‖m‖_∞.
Mathlib defines Fourier multipliers on Schwartz functions and on tempered distributions
(SchwartzMap.fourierMultiplierCLM, TemperedDistribution.fourierMultiplierCLM), but they are
meaningful only for symbols of temperate growth, which are in particular smooth. The symbols
of the classical singular integral operators are bounded but not smooth at the origin: the
Riesz transforms have symbols -i ξⱼ / ‖ξ‖, and ∂ⱼ∂ₖ Δ⁻¹ has symbol ξⱼ ξₖ / ‖ξ‖². The L²
multiplier accepts any essentially bounded symbol, and it is the L²-bounded operator that the
Calderón–Zygmund theory and the Mihlin–Hörmander multiplier theorem start from. Where both are
defined, it agrees with Mathlib's multipliers.
Main declarations #
MeasureTheory.Lp.fourierMultiplierCLM: the Fourier multiplier onL²with symbol inL^∞.MeasureTheory.Lp.fourier_fourierMultiplierCLM:𝓕 (T_m f) = m • 𝓕 f, andMeasureTheory.Lp.eq_fourierMultiplierCLM_iff, which characterizesT_mby it.MeasureTheory.Lp.norm_fourierMultiplierCLM_le: the bound‖T_m‖ ≤ ‖m‖_∞.MeasureTheory.Lp.fourierMultiplierCLM_compL_fourierMultiplierCLM:T_{m₁} ∘ T_{m₂} = T_{m₁ m₂}, with linearity in the symbol andT_c = c • idfor a constant symbol.MeasureTheory.Lp.adjoint_fourierMultiplierCLM: the adjoint ofT_misT_{star m}.SchwartzMap.toLp_fourierMultiplierCLM_eqandMeasureTheory.Lp.fourierMultiplierCLM_toTemperedDistribution_eq: for a bounded symbol of temperate growth,T_magrees with Mathlib's multipliers on Schwartz functions and, as a tempered distribution, on everyf ∈ L².SchwartzMap.toLp_eq_fourierMultiplierCLM_of_fourier_eq_smul: for any bounded symbolm, a Schwartz functiongwith𝓕 g = m • 𝓕 fisT_m finL².SchwartzMap.coeFn_fourierMultiplierCLM_toLp: for a Schwartz symbolσ,T_σis convolution with the Schwartz kernel𝓕⁻ σ, that isT_σ f x = ∫ 𝓕⁻ σ (x - y) • f yfor almost everyx. On Schwartz functions this isSchwartzMap.fourierMultiplierCLM_eq_convolution.MeasureTheory.Lp.tendsto_fourierMultiplierCLM_apply: if the symbolsm iare eventually uniformly bounded and converge tom₀almost everywhere along a countably generated filter, thenT_{m i} f → T_{m₀} finL².
Kernels of multipliers #
A bounded symbol m is in general not the Fourier transform of an integrable function, so T_m
is not a convolution with an integrable kernel. The Calderón–Zygmund theory instead asks for a
kernel representation off the support of the argument, and the proof of the Mihlin–Hörmander
multiplier theorem obtains it by approximation: m is cut into Schwartz symbols σ, for which
T_σ is convolution with 𝓕⁻ σ almost everywhere, the convolution integral converging at every
point, and T_m is recovered as the strong limit of the operators T_σ.
References #
- L. Grafakos, Classical Fourier Analysis, 3rd ed., Sections 2.5 and 6.2.2.
- E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Chapter IV, §3.
The Fourier multiplier on L²(E, F) with symbol m ∈ L^∞(E): the bounded operator
f ↦ 𝓕⁻ (m • 𝓕 f), where 𝓕 is the Fourier transform on L².
Instances For
On the Fourier side, the multiplier with symbol m is multiplication by m.
A bounded operator on L² is the multiplier with symbol m if and only if it acts as
multiplication by m on the Fourier side.
The multiplier with symbol m changes the L² norm by at most the factor ‖m‖_∞.
Plancherel bound for Fourier multipliers: ‖T_m‖ ≤ ‖m‖_∞.
Dependence on the symbol #
The multiplier of a finite sum of symbols is the sum of the multipliers.
A multiplier whose symbol is almost everywhere the constant c is multiplication by c.
The composition of two multipliers is the multiplier with the product symbol.
The adjoint of the multiplier with symbol m is the multiplier with the complex conjugate
symbol star m. In particular a real symbol gives a self-adjoint multiplier.
The pairing of g with the multiplier T_m f, computed on the Fourier side:
⟪g, T_m f⟫ = ∫ m ξ ⟪𝓕 g ξ, 𝓕 f ξ⟫ dξ.
Comparison with multipliers on Schwartz functions and tempered distributions #
On Schwartz functions, the L² multiplier with a bounded symbol g of temperate growth is
the Schwartz-space multiplier SchwartzMap.fourierMultiplierCLM F g.
A Schwartz function g whose Fourier transform is m • 𝓕 f is, in L², the multiplier with
symbol m applied to f. Here m need not have temperate growth: any bounded symbol works, so
this also covers symbols that are discontinuous at the origin.
For a bounded symbol g of temperate growth, the L² multiplier agrees with the multiplier
TemperedDistribution.fourierMultiplierCLM F g on tempered distributions: for every f ∈ L²,
both send f to the same tempered distribution.
Schwartz symbols and convolution kernels #
The L² multiplier with a Schwartz symbol σ is convolution with the Schwartz kernel
𝓕⁻ σ: for every f ∈ L², T_σ f x = ∫ 𝓕⁻ σ (x - y) • f y for almost every x. The
integral converges for every x, since y ↦ 𝓕⁻ σ (x - y) lies in L².
With k x y = 𝓕⁻ σ (x - y) • id, this is the kernel representation of T_σ that the
Calderón–Zygmund theorems take as hypothesis, valid at almost every point and not only off the
support of f.
If the symbols m i are eventually bounded in L^∞ uniformly in i along a countably
generated filter l and converge to m₀ almost everywhere along l, then
T_{m i} f → T_{m₀} f in L² along l for every f ∈ L².