The Calderón–Zygmund estimate for constant-coefficient operators #
Let A be a constant n × n coefficient matrix whose quadratic form satisfies
λ ‖ξ‖² ≤ ξᵀ A ξ with 0 < λ, and write L_A u = ∑ᵢⱼ Aᵢⱼ ∂ᵢ∂ⱼu. For every dimension n and 1 < p < ∞,
this file proves
‖D²u‖_{Lᵖ} ≤ (C / λ) ‖L_A u‖_{Lᵖ}
for every Schwartz function u on ℝⁿ, real valued or with values in a fixed complex Hilbert
space F, with a constant C depending only on n and p, and on F in the Hilbert-space case
(TauCeti.PDE.exists_eLpNorm_iteratedFDeriv_two_le_eLpNorm_sum_smul_fderiv_fderiv and
TauCeti.PDE.exists_eLpNorm_iteratedFDeriv_two_le_eLpNorm_sum_smul_fderiv_fderiv_real). The
constant does not depend on A beyond the lower ellipticity constant λ; no upper bound on A
enters. This uniformity is what the W^{2,p} theory for continuous coefficients needs: the
coefficients are frozen at a point of each small ball, and the frozen matrices range over all
matrices with a common lower ellipticity constant.
The argument #
The case A = 1 is the Calderón–Zygmund estimate for the Laplacian
(TauCeti.exists_eLpNorm_iteratedFDeriv_two_le_eLpNorm_laplacian). For general A there is a
linear automorphism e with ‖e⁻¹‖ ≤ λ^(-1/2) such that Δ (u ∘ e) = (L_A u) ∘ e
(TauCeti.PDE.exists_continuousLinearEquiv_laplacian_comp_eq). The Schwartz function
v = u ∘ e then satisfies D²u (e y) = D²v y ∘ (e⁻¹ × e⁻¹), so ‖D²u ∘ e‖ ≤ λ⁻¹ ‖D²v‖
pointwise, while Δ v = (L_A u) ∘ e. Composing with e multiplies both Lᵖ norms by the same
factor |det e|^(-1 / p) (MeasureTheory.Measure.eLpNorm_comp_continuousLinearEquiv), which
cancels. This transfer is stated for any codomain and any exponent in which the estimate for the
Laplacian holds
(TauCeti.PDE.eLpNorm_iteratedFDeriv_two_le_eLpNorm_sum_smul_fderiv_fderiv_of_laplacian).
References #
- D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,
Section 6.1 (reduction of a constant-coefficient operator to the Laplacian by a linear change
of variables) and Section 9.5 (
W^{2,p}estimates by freezing the coefficients).
Transfer of the Calderón–Zygmund estimate from the Laplacian to constant coefficients.
Suppose that ‖D²v‖_p ≤ C ‖Δv‖_p for every Schwartz function v on ℝⁿ with values in G.
If λ ‖ξ‖² ≤ ξᵀ A ξ with 0 < λ, then every such Schwartz function u satisfies
‖D²u‖_p ≤ (C / λ) ‖∑ᵢⱼ Aᵢⱼ ∂ᵢ∂ⱼu‖_p.
The Calderón–Zygmund estimate for constant-coefficient operators. Let 1 < p < ∞, and let
F be a complex Hilbert space. There is a constant C, depending only on n,
p and F, such that for every matrix A with λ ‖ξ‖² ≤ ξᵀ A ξ and 0 < λ,
‖D²u‖_{Lᵖ} ≤ (C / λ) ‖∑ᵢⱼ Aᵢⱼ ∂ᵢ∂ⱼu‖_{Lᵖ}
for every Schwartz function u on ℝⁿ with values in F.
The Calderón–Zygmund estimate for constant-coefficient operators, real-valued form. Let
1 < p < ∞, in any dimension n. There is a constant C, depending only on n and p, such
that for every
matrix A with λ ‖ξ‖² ≤ ξᵀ A ξ and 0 < λ,
‖D²u‖_{Lᵖ} ≤ (C / λ) ‖∑ᵢⱼ Aᵢⱼ ∂ᵢ∂ⱼu‖_{Lᵖ}
for every real-valued Schwartz function u on ℝⁿ.