The Cauchy transform with parameters

For a compactly supported C¹ map f : ℂ → F into a complex Banach space, the Cauchy transform T f w = π⁻¹ ∫ (w - z)⁻¹ • f z solves the inhomogeneous Cauchy–Riemann equation \bar∂ (T f) = f on the whole plane and is holomorphic off the support of f (TauCeti.Analysis.Complex.Pompeiu.Basic). This file studies the transform when f depends on parameters, as needed to solve \bar∂ u = f with smooth or holomorphic dependence on parameters. Normalizing constants play no role in regularity, so the statements are about the unnormalized integral ∫ (w - z)⁻¹ • f z.

• Let g : P → ℂ → F be jointly C^n on s ×ˢ univ, for an open set s of a real normed space P, with g p z = 0 for p ∈ s and z outside a fixed compact set k. The transform (p, w) ↦ ∫ (w - z)⁻¹ • g p z is jointly C^n on s ×ˢ univ: this is smoothness of the convolution of the locally integrable Cauchy kernel with a compactly supported family, after the substitution ζ = w - z. Its derivative in p is the transform of the derivative of g in p, so every parameter derivative commutes with the transform. By TauCeti.differentiableOn_integral_sub_inv_smul_compl, for every p ∈ s the transform of g p is holomorphic off k, a region that does not depend on p.
• When the parameter space Q is complex, the conjugate Wirtinger derivative in q commutes with the transform. In particular, if g q z is holomorphic in q for every z, so is the transform of g q at every point w.

Main results

• TauCeti.contDiffOn_integral_sub_inv_smul_with_param: joint smoothness in the parameter and the point.
• TauCeti.hasFDerivAt_integral_sub_inv_smul_with_param: parameter derivatives commute with the transform.
• TauCeti.conjWirtingerDeriv_integral_sub_inv_smul_with_param and TauCeti.differentiableAt_integral_sub_inv_smul_with_param: the conjugate Wirtinger derivative in a complex parameter commutes with the transform, which preserves holomorphy in parameters.

References

• L. Hörmander, An Introduction to Complex Analysis in Several Variables, 3rd ed., North-Holland, 1990, Theorem 1.2.2.
• P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley, 1978, Chapter 0, §§1–2 (the \bar∂-Poincaré lemma in one and in several variables, where the solution depends holomorphically on the remaining variables).

Smooth dependence on parameters

theorem TauCeti.contDiffOn_integral_sub_inv_smul_with_param {F : Type u_1} [NormedAddCommGroup F] [NormedSpace ℂ F] {P : Type u_2} [NormedAddCommGroup P] [NormedSpace ℝ P] {g : P → ℂ → F} {s : Set P} {k : Set ℂ} {n : ℕ∞} (hs : IsOpen s) (hk : IsCompact k) (hgs : ∀ (p : P) (x : ℂ), p ∈ s → x ∉ k → g p x = 0) (hg : ContDiffOn ℝ (↑n) (↿g) (s ×ˢ Set.univ)) : ContDiffOn ℝ (↑n) (fun (q : P × ℂ) => ∫ (z : ℂ), (q.2 - z)⁻¹ • g q.1 z) (s ×ˢ Set.univ)

Joint smoothness of the Cauchy transform in parameters. If g is C^n on s ×ˢ univ for an open set s of parameters, and g p vanishes outside a fixed compact set k for every p ∈ s, then (p, w) ↦ ∫ (w - z)⁻¹ • g p z is C^n on s ×ˢ univ.

theorem TauCeti.hasFDerivAt_integral_sub_inv_smul_with_param {F : Type u_1} [NormedAddCommGroup F] [NormedSpace ℂ F] {P : Type u_2} [NormedAddCommGroup P] [NormedSpace ℝ P] {g : P → ℂ → F} {s : Set P} {k : Set ℂ} [CompleteSpace F] (hs : IsOpen s) (hk : IsCompact k) (hgs : ∀ (p : P) (x : ℂ), p ∈ s → x ∉ k → g p x = 0) (hg : ContDiffOn ℝ 1 (↿g) (s ×ˢ Set.univ)) {p : P} (hp : p ∈ s) (w : ℂ) : HasFDerivAt (fun (p : P) => ∫ (z : ℂ), (w - z)⁻¹ • g p z) (∫ (z : ℂ), (w - z)⁻¹ • fderiv ℝ (fun (p : P) => g p z) p) p

Parameter derivatives commute with the Cauchy transform. Under the hypotheses of TauCeti.contDiffOn_integral_sub_inv_smul_with_param with n = 1, the derivative in p ∈ s of ∫ (w - z)⁻¹ • g p z is the transform ∫ (w - z)⁻¹ • D_p g p z of the derivative of g in p.

Holomorphic dependence on parameters

theorem TauCeti.conjWirtingerDeriv_integral_sub_inv_smul_with_param {F : Type u_1} [NormedAddCommGroup F] [NormedSpace ℂ F] [CompleteSpace F] {Q : Type u_2} [NormedAddCommGroup Q] [NormedSpace ℂ Q] {g : Q → ℂ → F} {s : Set Q} {k : Set ℂ} (hs : IsOpen s) (hk : IsCompact k) (hgs : ∀ (q : Q) (x : ℂ), q ∈ s → x ∉ k → g q x = 0) (hg : ContDiffOn ℝ 1 (↿g) (s ×ˢ Set.univ)) {q : Q} (hq : q ∈ s) (w : ℂ) (v : Q) : (conjWirtingerDeriv (fun (q : Q) => ∫ (z : ℂ), (w - z)⁻¹ • g q z) q) v = ∫ (z : ℂ), (w - z)⁻¹ • (conjWirtingerDeriv (fun (q : Q) => g q z) q) v

The conjugate Wirtinger derivative in a complex parameter commutes with the Cauchy transform: \bar∂_q ∫ (w - z)⁻¹ • g q z = ∫ (w - z)⁻¹ • \bar∂_q g q z, under the hypotheses of TauCeti.contDiffOn_integral_sub_inv_smul_with_param with n = 1.

theorem TauCeti.differentiableAt_integral_sub_inv_smul_with_param {F : Type u_1} [NormedAddCommGroup F] [NormedSpace ℂ F] [CompleteSpace F] {Q : Type u_2} [NormedAddCommGroup Q] [NormedSpace ℂ Q] {g : Q → ℂ → F} {s : Set Q} {k : Set ℂ} (hs : IsOpen s) (hk : IsCompact k) (hgs : ∀ (q : Q) (x : ℂ), q ∈ s → x ∉ k → g q x = 0) (hg : ContDiffOn ℝ 1 (↿g) (s ×ˢ Set.univ)) {q : Q} (hq : q ∈ s) (hgq : ∀ (z : ℂ), DifferentiableAt ℂ (fun (q : Q) => g q z) q) (w : ℂ) : DifferentiableAt ℂ (fun (q : Q) => ∫ (z : ℂ), (w - z)⁻¹ • g q z) q

The Cauchy transform preserves holomorphy in parameters. Under the hypotheses of TauCeti.contDiffOn_integral_sub_inv_smul_with_param with n = 1, if g q z is complex differentiable in q at a point q ∈ s for every z, then so is ∫ (w - z)⁻¹ • g q z, for every w.