Second derivatives as a Fourier multiplier of the Laplacian #
On the Fourier side, ∂_v ∂_w is multiplication by -(2π)² ⟪ξ, v⟫ ⟪ξ, w⟫ and the Laplacian Δ is
multiplication by -(2π)² ‖ξ‖². The operator ∂_v ∂_w Δ⁻¹ is therefore the Fourier multiplier
with symbol
m_{v,w}(ξ) = ⟪ξ, v⟫ ⟪ξ, w⟫ / ‖ξ‖² (TauCeti.hessianSymbol),
which is bounded by ‖v‖ ‖w‖ but, for nonzero v and w, not continuous at the origin. It lies
in L^∞ (TauCeti.hessianSymbolLp), so the L² multiplier
MeasureTheory.Lp.fourierMultiplierCLM defines T_{m_{v,w}} as a bounded operator on L²(E, F),
and for every Schwartz function u
∂_v ∂_w u = T_{m_{v,w}} (Δ u) in L²
(SchwartzMap.toLp_lineDerivOp_lineDerivOp_eq_fourierMultiplierCLM).
In particular ‖∂_v ∂_w u‖_{L²} ≤ ‖v‖ ‖w‖ ‖Δ u‖_{L²}.
This realizes D² Δ⁻¹ as an L²-bounded operator, which is the starting point of the
Calderón–Zygmund theory: combined with the standard bounds on its kernel, the Hessian of the
Newtonian kernel, that theory upgrades the L² bound to ‖∂_v ∂_w u‖_{Lᵖ} ≤ C ‖Δ u‖_{Lᵖ} for
1 < p < ∞.
Main declarations #
TauCeti.hessianSymbol: the symbolξ ↦ ⟪ξ, v⟫ ⟪ξ, w⟫ / ‖ξ‖²of∂_v ∂_w Δ⁻¹, with the boundTauCeti.norm_hessianSymbol_le.TauCeti.hessianSymbolLp: the same symbol as an element ofL^∞.SchwartzMap.fourier_lineDerivOp_lineDerivOp_apply_eq_hessianSymbol_smul: pointwise on the Fourier side,𝓕 (∂_v ∂_w u) = m_{v,w} • 𝓕 (Δ u).SchwartzMap.toLp_lineDerivOp_lineDerivOp_eq_fourierMultiplierCLM:∂_v ∂_w u = T_{m_{v,w}} (Δ u)inL².SchwartzMap.norm_toLp_lineDerivOp_lineDerivOp_le:‖∂_v ∂_w u‖_{L²} ≤ ‖v‖ ‖w‖ ‖Δ u‖_{L²}.
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, Section 9.4.
The symbol ξ ↦ ⟪ξ, v⟫ ⟪ξ, w⟫ / ‖ξ‖² of the operator ∂_v ∂_w Δ⁻¹. At the origin it takes
the value 0.
Instances For
Multiplying the symbol of ∂_v ∂_w Δ⁻¹ by the symbol ‖ξ‖² of -(2π)⁻² Δ gives the symbol
⟪ξ, v⟫ ⟪ξ, w⟫ of -(2π)⁻² ∂_v ∂_w, at every point including the origin.
The symbol of ∂_v ∂_w Δ⁻¹ is bounded by ‖v‖ ‖w‖, by the Cauchy–Schwarz inequality.
For an orthonormal basis eⱼ, the symbols of ∂ⱼ ∂ⱼ Δ⁻¹ sum to 1 away from the origin:
the trace of D² Δ⁻¹ is the identity.
The symbol of ∂_v ∂_w Δ⁻¹ is essentially bounded for every measure.
The symbol ξ ↦ ⟪ξ, v⟫ ⟪ξ, w⟫ / ‖ξ‖² of ∂_v ∂_w Δ⁻¹, as an element of L^∞.
Instances For
The Hessian symbol is real, hence fixed by complex conjugation.
The L^∞ norm of the symbol of ∂_v ∂_w Δ⁻¹ is at most ‖v‖ ‖w‖.
On the Fourier side, the second directional derivative ∂_v ∂_w u of a Schwartz function is
the symbol ⟪ξ, v⟫ ⟪ξ, w⟫ / ‖ξ‖² times the Fourier transform of Δ u.
In L², the second directional derivative ∂_v ∂_w u of a Schwartz function is the Fourier
multiplier with symbol ⟪ξ, v⟫ ⟪ξ, w⟫ / ‖ξ‖² applied to Δ u: on Schwartz functions,
∂_v ∂_w = T_{m_{v,w}} ∘ Δ.
The Laplacian bounds every second directional derivative in L²:
‖∂_v ∂_w u‖_{L²} ≤ ‖v‖ ‖w‖ ‖Δ u‖_{L²} for a Schwartz function u.