The kernel of the Hessian Fourier multiplier

For v, w ∈ ℝⁿ, the Fourier multiplier T_{m_{v,w}} on L²(ℝⁿ, F) with symbol m_{v,w}(ξ) = ⟪ξ, v⟫ ⟪ξ, w⟫ / ‖ξ‖² (TauCeti.hessianSymbolLp) is the operator ∂_v ∂_w Δ⁻¹ (SchwartzMap.toLp_lineDerivOp_lineDerivOp_eq_fourierMultiplierCLM). This file shows that for n ≥ 3 it is a singular integral operator whose kernel is the Hessian of the Newtonian kernel Gₙ = TauCeti.newtonianKernel n: if f ∈ L² vanishes off a compact set s, then for almost every x ∉ s

T_{m_{v,w}} f (x) = -∫ D²Gₙ(x - y)(v, w) f(y) dy

(TauCeti.ae_fourierMultiplierCLM_hessianSymbolLp_eq_neg_integral). Together with the size and smoothness estimates of D²Gₙ (TauCeti.norm_iteratedFDeriv_two_newtonianKernel_le, TauCeti.norm_iteratedFDeriv_two_newtonianKernel_sub_le), this is the kernel representation that the Calderón–Zygmund theory asks of an L²-bounded operator before it bounds the operator on Lᵖ, 1 < p < ∞; for T_{m_{v,w}} that bound is the estimate ‖∂_v ∂_w u‖_p ≤ C ‖Δu‖_p.

The argument

For φ smooth with compact support, the Newtonian potential u = Gₙ ⋆ φ solves -Δu = φ, so T_{m_{v,w}} φ should be -∂_v ∂_w u. The identity ∂_v ∂_w = T_{m_{v,w}} ∘ Δ holds on Schwartz functions, and u is not one, but its truncations χ_R u are: χ_R is a cutoff equal to 1 on ball 0 R with ‖Dχ_R‖ = O(1/R) and |Δχ_R| = O(1/R²) (TauCeti.exists_forall_contDiff_cutoff_ball_norm_laplacian_le). Since u = O(‖x‖ ^ (2 - n)) and Du = O(‖x‖ ^ (1 - n)) (TauCeti.norm_newtonianKernel_convolution_le, TauCeti.norm_fderiv_newtonianKernel_convolution_le), the error Δ(χ_R u) + φ = 2 Du(∇χ_R) + (Δχ_R) u lives on the annulus R ≤ ‖x‖ ≤ 2R, where it is O(R ^ (1 - n)), so its L² norm is O(R ^ (1 - n/2)), which tends to zero exactly because n ≥ 3. Hence T_{m_{v,w}} φ = -∂_v ∂_w u almost everywhere (TauCeti.coeFn_fourierMultiplierCLM_hessianSymbolLp_toLp).

Off the support of φ, integrating by parts twice moves the derivatives from φ onto the kernel: D²u(x)(v, w) = ∫ D²Gₙ(x - y)(v, w) φ(y) dy (TauCeti.iteratedFDeriv_two_newtonianKernel_convolution_apply_of_notMem_tsupport). A general f ∈ L² vanishing off s is the L² limit of smooth functions supported in shrinking thickenings of s (MeasureTheory.MemLp.exist_contDiff_tsupport_subset_eLpNorm_sub_le). For x ∉ s the kernel is bounded on a fixed thickening of s, so the kernel integrals converge (TauCeti.tendsto_integral_smul_of_tendsto_eLpNorm_sub), while T_{m_{v,w}} of the approximations converges almost everywhere along a subsequence.

Main declarations

• TauCeti.iteratedFDeriv_two_newtonianKernel_convolution_apply_of_notMem_tsupport: the second derivatives of the Newtonian potential off the support of the density.
• TauCeti.coeFn_fourierMultiplierCLM_hessianSymbolLp_toLp: T_{m_{v,w}} φ = -D²(Gₙ ⋆ φ)(v, w) for φ smooth with compact support.
• TauCeti.ae_fourierMultiplierCLM_hessianSymbolLp_eq_neg_integral: the kernel representation of T_{m_{v,w}} off the support of its argument.

References

• E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Chapter III, §1.
• D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Lemma 4.4 and Section 9.4.

theorem TauCeti.coeFn_fourierMultiplierCLM_hessianSymbolLp_toLp {n : ℕ} {F : Type u_2} [NormedAddCommGroup F] [InnerProductSpace ℂ F] [CompleteSpace F] {φ : EuclideanSpace ℝ (Fin n) → F} (hn : 3 ≤ n) (hφ : ContDiff ℝ (↑⊤) φ) (hc : HasCompactSupport φ) (v w : EuclideanSpace ℝ (Fin n)) : ↑↑((MeasureTheory.Lp.fourierMultiplierCLM F (hessianSymbolLp v w)) (MeasureTheory.MemLp.toLp φ ⋯)) =ᵐ[MeasureTheory.volume] fun (x : EuclideanSpace ℝ (Fin n)) => -(iteratedFDeriv ℝ 2 (MeasureTheory.convolution (newtonianKernel n) φ (ContinuousLinearMap.lsmul ℝ ℝ) MeasureTheory.volume) x) ![v, w]

The Hessian multiplier on smooth functions with compact support. For n ≥ 3 and φ smooth with compact support on ℝⁿ, the Fourier multiplier T_{m_{v,w}} with symbol ⟪ξ, v⟫ ⟪ξ, w⟫ / ‖ξ‖² sends φ to -D²(Gₙ ⋆ φ)(v, w). Since -Δ (Gₙ ⋆ φ) = φ, this says that T_{m_{v,w}} = ∂_v ∂_w Δ⁻¹ on such functions, with Δ⁻¹ given by the Newtonian potential.

The Newtonian potential u = Gₙ ⋆ φ is not a Schwartz function, so the identity ∂_v ∂_w = T_{m_{v,w}} ∘ Δ on Schwartz functions does not apply to it directly. It applies to truncations χ_R u, which agree with u on ball 0 R and whose Laplacians tend to -φ in L² because u decays like ‖x‖ ^ (2 - n) and n ≥ 3.

theorem TauCeti.ae_fourierMultiplierCLM_hessianSymbolLp_eq_neg_integral {n : ℕ} {F : Type u_2} [NormedAddCommGroup F] [InnerProductSpace ℂ F] [CompleteSpace F] (hn : 3 ≤ n) (v w : EuclideanSpace ℝ (Fin n)) (f : ↥(MeasureTheory.Lp F 2 MeasureTheory.volume)) {s : Set (EuclideanSpace ℝ (Fin n))} (hs : IsCompact s) (hf : ∀ᵐ (x : EuclideanSpace ℝ (Fin n)), x ∉ s → ↑↑f x = 0) : ∀ᵐ (x : EuclideanSpace ℝ (Fin n)), x ∉ s → ↑↑((MeasureTheory.Lp.fourierMultiplierCLM F (hessianSymbolLp v w)) f) x = -∫ (y : EuclideanSpace ℝ (Fin n)), (iteratedFDeriv ℝ 2 (newtonianKernel n) (x - y)) ![v, w] • ↑↑f y

The kernel of the Hessian multiplier. Let n ≥ 3 and v, w ∈ ℝⁿ, and let T_{m_{v,w}} be the Fourier multiplier with symbol ⟪ξ, v⟫ ⟪ξ, w⟫ / ‖ξ‖², that is ∂_v ∂_w Δ⁻¹. If f ∈ L² vanishes almost everywhere off a compact set s, then for almost every x ∉ s

T_{m_{v,w}} f (x) = -∫ D²Gₙ(x - y)(v, w) f(y) dy.

So off the support of its argument, T_{m_{v,w}} is the singular integral operator with kernel -D²Gₙ(x - y)(v, w), the Hessian of the Newtonian kernel; the integral converges because the kernel is bounded for y ∈ s and x away from s.