The Calderón–Zygmund estimate for strong solutions in W^{2,p} of the whole space

For 1 < p < ∞, this file proves the W^{2,p} estimate of the Hessian on the whole space for constant-coefficient operators on ℝⁿ, and for the Laplacian on every finite-dimensional real inner product space V. Here D²u is the highest recorded weak derivative TauCeti.Wkp.iteratedGradient 1 u, the weak Hessian as an Lᵖ field of endomorphisms with the operator norm, so that ⟪D²u(x) eᵢ, eⱼ⟫ is the weak second derivative ∂ᵢ∂ⱼu.

• Constant coefficients (TauCeti.exists_norm_iteratedGradient_le_mul_norm_of_sum_mul_inner_eq). 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, every u ∈ W^{2,p}(ℝⁿ) with ∑ᵢⱼ Aᵢⱼ ∂ᵢ∂ⱼu = f ∈ Lᵖ(ℝⁿ) almost everywhere satisfies ‖D²u‖_{Lᵖ} ≤ (C / λ) ‖f‖_{Lᵖ}.
• The Laplacian. There is a constant C, depending only on V and p, such that ‖D²u‖_{Lᵖ} ≤ C ‖f‖_{Lᵖ} for every u ∈ W^{2,p}(V) and f ∈ Lᵖ(V) with Δu = f. The equation Δu = f is taken either strongly, the trace of the weak Hessian of u being f in Lᵖ (TauCeti.exists_norm_iteratedGradient_le_mul_norm_of_traceCLM_compLpL_eq), or against test functions, ∫ Δφ • u = ∫ φ • f for every φ ∈ C_c^∞(V) (TauCeti.exists_norm_iteratedGradient_le_mul_norm_of_integral_laplacian_smul_eq); the two agree by TauCeti.Wkp.forall_integral_laplacian_smul_value_eq_iff. Stating this on a general V rather than on ℝⁿ lets it apply on products such as ℝ × E, where half-spaces are described.

The argument

For Schwartz functions the constant-coefficient estimate is TauCeti.PDE.exists_eLpNorm_iteratedFDeriv_two_le_eLpNorm_sum_smul_fderiv_fderiv_real and the estimate for the Laplacian is TauCeti.exists_eLpNorm_iteratedFDeriv_two_le_eLpNorm_laplacian_real, the singular integral estimates for the Hessian, and test functions are Schwartz functions. Both sides of each estimate are continuous on W^{2,p}: D²u is a continuous projection, and D²u ↦ ∑ᵢⱼ Aᵢⱼ ⟪D²u eᵢ, eⱼ⟫ and D²u ↦ tr D²u are continuous linear functionals (Matrix.contraction A and traceCLM) applied pointwise. For a test function φ the trace of the Hessian is Δφ (TauCeti.Wkp.trace_iteratedGradient_ofTestFunctionₗ_ae_eq). Test functions are dense in W^{2,p} of the whole space for p < ∞ (TauCeti.Wkp.denseRange_ofTestFunctionₗ_top), so the estimates hold on all of W^{2,p}.

References

• D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Theorem 9.9 and §9.4.
• E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Chapter III, §1.3.

theorem TauCeti.exists_norm_iteratedGradient_le_mul_norm_of_sum_mul_inner_eq {n : ℕ} {p : ENNReal} [Fact (1 ≤ p)] (hp : 1 < p) (hp_top : p ≠ ⊤) : ∃ (C : NNReal), ∀ {A : Matrix (Fin n) (Fin n) ℝ} {lam : ℝ}, 0 < lam → (∀ (ξ : EuclideanSpace ℝ (Fin n)), lam * ‖ξ‖ ^ 2 ≤ A.toQuadraticForm' ξ.ofLp) → ∀ (u : Wkp MeasureTheory.volume ⊤ p 2) (f : ↥(MeasureTheory.Lp ℝ p (MeasureTheory.volume.restrict ↑⊤))), (∀ᵐ (x : EuclideanSpace ℝ (Fin n)), ∑ i : Fin n, ∑ j : Fin n, A i j * inner ℝ ((↑↑(Wkp.iteratedGradient 1 u) x) (EuclideanSpace.single i 1)) (EuclideanSpace.single j 1) = ↑↑f x) → ‖Wkp.iteratedGradient 1 u‖ ≤ ↑C / lam * ‖f‖

The Calderón–Zygmund estimate for constant coefficients in W^{2,p}(ℝⁿ). 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, every u ∈ W^{2,p}(ℝⁿ) solving ∑ᵢⱼ Aᵢⱼ ∂ᵢ∂ⱼu = f almost everywhere, with f ∈ Lᵖ(ℝⁿ), satisfies

‖D²u‖_{Lᵖ} ≤ (C / λ) ‖f‖_{Lᵖ},

where D²u is the weak Hessian TauCeti.Wkp.iteratedGradient 1 u, with the operator norm on endomorphisms of ℝⁿ.

theorem TauCeti.exists_norm_iteratedGradient_le_mul_norm_of_traceCLM_compLpL_eq {p : ENNReal} [Fact (1 ≤ p)] {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] (hp : 1 < p) (hp_top : p ≠ ⊤) : ∃ (C : NNReal), ∀ (u : Wkp MeasureTheory.volume ⊤ p 2) (f : ↥(MeasureTheory.Lp ℝ p (MeasureTheory.volume.restrict ↑⊤))), (ContinuousLinearMap.compLpL p (MeasureTheory.volume.restrict ↑⊤) (traceCLM ℝ V)) (Wkp.iteratedGradient 1 u) = f → ‖Wkp.iteratedGradient 1 u‖ ≤ ↑C * ‖f‖

The Calderón–Zygmund estimate for strong solutions in W^{2,p}. Let 1 < p < ∞, and let V be a finite-dimensional real inner product space with its volume measure. There is a constant C such that every u ∈ W^{2,p}(V) solving Δu = f with f ∈ Lᵖ(V) strongly, in the sense that the trace of the weak Hessian of u is f in Lᵖ, satisfies

‖D²u‖_{Lᵖ} ≤ C ‖f‖_{Lᵖ},

where D²u is the weak Hessian TauCeti.Wkp.iteratedGradient 1 u, with the operator norm on endomorphisms of V.

theorem TauCeti.exists_norm_iteratedGradient_le_mul_norm_of_integral_laplacian_smul_eq {p : ENNReal} [Fact (1 ≤ p)] {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] (hp : 1 < p) (hp_top : p ≠ ⊤) : ∃ (C : NNReal), ∀ (u : Wkp MeasureTheory.volume ⊤ p 2) (f : ↥(MeasureTheory.Lp ℝ p (MeasureTheory.volume.restrict ↑⊤))), (∀ (φ : TestFunction ⊤ ℝ ⊤), ∫ (x : V), Laplacian.laplacian (⇑φ) x • ↑↑(Wkp.value 2 u) x = ∫ (x : V), φ x • ↑↑f x) → ‖Wkp.iteratedGradient 1 u‖ ≤ ↑C * ‖f‖

The Calderón–Zygmund estimate in W^{2,p}, against test functions. Let 1 < p < ∞, and let V be a finite-dimensional real inner product space with its volume measure. There is a constant C such that every u ∈ W^{2,p}(V) solving Δu = f with f ∈ Lᵖ(V), in the sense that ∫ Δφ • u = ∫ φ • f for every test function φ, satisfies

‖D²u‖_{Lᵖ} ≤ C ‖f‖_{Lᵖ},

where D²u is the weak Hessian TauCeti.Wkp.iteratedGradient 1 u, with the operator norm on endomorphisms of V.