The Mihlin–Hörmander multiplier theorem

Let V be a real inner product space of finite dimension n ≥ 1 and F a complex Hilbert space. A symbol m : V → ℂ, smooth away from the origin, satisfies Mihlin's condition of order N with constant A if

‖D^k m ξ‖ ≤ A / ‖ξ‖ ^ k for every k ≤ N and every ξ ≠ 0.

The Mihlin–Hörmander multiplier theorem (TauCeti.exists_eLpNorm_fourierMultiplierCLM_le_of_norm_iteratedFDeriv_le) says that if 2N > n, then for every 1 < p < ∞ the L² Fourier multiplier T_m (MeasureTheory.Lp.fourierMultiplierCLM) is bounded in the Lᵖ norm, ‖T_m f‖_p ≤ C A ‖f‖_p, with C independent of m and A. For a symbol of temperate growth, the multiplier on tempered distributions therefore maps every f ∈ Lᵖ to an Lᵖ function with the same bound (TauCeti.exists_fourierMultiplierCLM_eq_and_norm_le_of_norm_iteratedFDeriv_le), by density of the Schwartz functions in Lᵖ. The symbols of the Riesz transforms and of ∂_v ∂_w Δ⁻¹, and more generally every symbol that is smooth away from the origin and homogeneous of degree 0, satisfy Mihlin's condition of every order (TauCeti.exists_norm_iteratedFDeriv_le_div_pow_of_smul_eq).

The argument

Cut m into the Littlewood–Paley pieces ψⱼ • m of a bump function supported in the unit ball (ContDiffBump.littlewoodPaleyPiece), and let σₙ = ∑_{-n < j ≤ n} ψⱼ • m be the truncations. These are Schwartz functions, bounded by ‖m‖_∞, and they converge to m away from the origin.

• The pieces are band-limited at the scales 2 ^ j and satisfy the derivative bounds ‖D^k (ψⱼ • m) ξ‖ ≤ C A / (2 ^ j) ^ k with C independent of j (ContDiffBump.exists_norm_iteratedFDeriv_littlewoodPaleyPiece_le), so the kernels 𝓕⁻ σₙ satisfy Hörmander's condition ∫_{‖x‖ > 2 ‖h‖} ‖𝓕⁻ σₙ (x - h) - 𝓕⁻ σₙ x‖ dx ≤ B A uniformly in n (SchwartzMap.exists_setIntegral_norm_fourierInv_sum_sub_sub_le).
• T_{σₙ} is convolution with 𝓕⁻ σₙ (SchwartzMap.coeFn_fourierMultiplierCLM_toLp), and its adjoint is the multiplier with the truncations of the conjugate symbol, which satisfies the same condition. The Calderón–Zygmund theorem under Hörmander's condition (TauCeti.exists_eLpNorm_le_of_hormander) then bounds T_{σₙ} in the Lᵖ norm uniformly in n.
• T_{σₙ} f → T_m f in L² (MeasureTheory.Lp.tendsto_fourierMultiplierCLM_apply), so almost everywhere along a subsequence, and Fatou's lemma passes the bound to T_m.

The proof first treats symbols with Mihlin constant 1 and then rescales.

The symbol is assumed smooth away from the origin, rather than only N times differentiable, so that its Littlewood–Paley pieces are Schwartz functions.

References

• S. G. Mihlin, On the multipliers of Fourier integrals, Dokl. Akad. Nauk SSSR 109 (1956), 701–703.
• L. Hörmander, Estimates for translation invariant operators in Lᵖ spaces, Acta Math. 104 (1960), 93–140.
• L. Grafakos, Classical Fourier Analysis, 3rd ed., Theorem 6.2.7.
• E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Chapter IV, §3.

theorem TauCeti.exists_eLpNorm_fourierMultiplierCLM_le_of_norm_iteratedFDeriv_le {V : Type u_1} {F : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [NormedAddCommGroup F] [InnerProductSpace ℂ F] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] [CompleteSpace F] [Nontrivial V] {N : ℕ} (hN : Module.finrank ℝ V < 2 * N) {p : ENNReal} (hp : 1 < p) (hp_top : p ≠ ⊤) : ∃ (C : NNReal), ∀ {m : V → ℂ}, ContDiffOn ℝ (↑⊤) m {0}ᶜ → ∀ {A : NNReal}, (∀ k ≤ N, ∀ (ξ : V), ξ ≠ 0 → ‖iteratedFDeriv ℝ k m ξ‖ ≤ ↑A / ‖ξ‖ ^ k) → ∀ (m' : ↥(MeasureTheory.Lp ℂ ⊤ MeasureTheory.volume)), ↑↑m' =ᵐ[MeasureTheory.volume] m → ∀ (f : ↥(MeasureTheory.Lp F 2 MeasureTheory.volume)), MeasureTheory.eLpNorm (↑↑((MeasureTheory.Lp.fourierMultiplierCLM F m') f)) p MeasureTheory.volume ≤ ↑C * ↑A * MeasureTheory.eLpNorm (↑↑f) p MeasureTheory.volume

The Mihlin–Hörmander multiplier theorem. Let V be a real inner product space of finite dimension n ≥ 1, let N be an integer with 2N > n, and let 1 < p < ∞. There is a constant C with the following property. Let m : V → ℂ be smooth away from the origin and satisfy Mihlin's condition of order N with constant A:

‖D^k m ξ‖ ≤ A / ‖ξ‖ ^ k for every k ≤ N and every ξ ≠ 0.

Then the L² Fourier multiplier T_m with symbol m is bounded in the Lᵖ norm: ‖T_m f‖_p ≤ C A ‖f‖_p for every f ∈ L²(V, F), with values in any complex Hilbert space F. The symbol enters as any element of L^∞ almost everywhere equal to m, and the constant C does not depend on m or A.

theorem TauCeti.exists_fourierMultiplierCLM_eq_and_norm_le_of_norm_iteratedFDeriv_le {V : Type u_1} {F : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [NormedAddCommGroup F] [InnerProductSpace ℂ F] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] [CompleteSpace F] [Nontrivial V] {N : ℕ} (hN : Module.finrank ℝ V < 2 * N) {p : ENNReal} [Fact (1 ≤ p)] (hp : 1 < p) (hp_top : p ≠ ⊤) : ∃ (C : NNReal), ∀ {m : V → ℂ}, Function.HasTemperateGrowth m → ∀ {A : NNReal}, (∀ k ≤ N, ∀ (ξ : V), ξ ≠ 0 → ‖iteratedFDeriv ℝ k m ξ‖ ≤ ↑A / ‖ξ‖ ^ k) → ∀ (f : ↥(MeasureTheory.Lp F p MeasureTheory.volume)), ∃ (g : ↥(MeasureTheory.Lp F p MeasureTheory.volume)), (TemperedDistribution.fourierMultiplierCLM F m) (MeasureTheory.Lp.toTemperedDistribution f) = MeasureTheory.Lp.toTemperedDistribution g ∧ ‖g‖ ≤ ↑C * ↑A * ‖f‖

The Mihlin–Hörmander multiplier theorem on Lᵖ. Let V be a real inner product space of finite dimension n ≥ 1, let 2N > n and 1 < p < ∞. There is a constant C such that for every symbol m of temperate growth satisfying Mihlin's condition of order N with constant A,

‖D^k m ξ‖ ≤ A / ‖ξ‖ ^ k for every k ≤ N and every ξ ≠ 0,

the Fourier multiplier 𝓕⁻ (m • 𝓕 f) of the tempered distribution of any f ∈ Lᵖ(V, F) is an Lᵖ function g with ‖g‖_p ≤ C A ‖f‖_p.