The Calderón–Zygmund inequality for the Cauchy–Riemann operator #

For 1 < p < ∞ there is a constant C with

‖Du‖_{Lᵖ} ≤ C ‖\bar∂u‖_{Lᵖ}

for every Schwartz function u : ℂ → F with values in a complex Hilbert space (TauCeti.exists_eLpNorm_fderiv_le_eLpNorm_conjWirtingerDeriv), and hence for every smooth compactly supported u (TauCeti.exists_eLpNorm_fderiv_le_eLpNorm_conjWirtingerDeriv_of_hasCompactSupport). Here Du is the real derivative of u and \bar∂u = (∂ₓu + I ∂ᵧu) / 2 is its conjugate Wirtinger derivative TauCeti.conjWirtingerDeriv u z 1. For F = ℂⁿ this is the inequality ‖∇u‖_{Lᵖ} ≤ C ‖∂ₛu + J₀ ∂ₜu‖_{Lᵖ} behind the elliptic estimates for J-holomorphic curves. It fails at the endpoints p = 1 and p = ∞.

The argument #

With Mathlib's normalization of the Fourier transform, 𝓕 (∂_v u) ξ = 2πi ⟪ξ, v⟫ 𝓕 u ξ and hence 𝓕 (\bar∂u) ξ = πi ξ 𝓕 u ξ, the real inner product on ℂ being ⟪ξ, v⟫ = re (conj ξ v). So on Schwartz functions ∂_v = T_{m_v} ∘ \bar∂, where T_{m_v} is the Fourier multiplier with symbol

m_v(ξ) = 2 ⟪ξ, v⟫ / ξ = conj v + v (conj ξ / ξ).

Up to the constant term, this is v times the symbol conj ξ / ξ of the Beurling transform. The symbol is smooth away from the origin and homogeneous of degree 0, so it satisfies Mihlin's condition (TauCeti.exists_norm_iteratedFDeriv_le_div_pow_of_smul_eq), and the Mihlin–Hörmander multiplier theorem (TauCeti.exists_eLpNorm_fourierMultiplierCLM_le_of_norm_iteratedFDeriv_le) bounds T_{m_v} on Lᵖ. Applying this to the directions v = 1 and v = I bounds the whole derivative.

References #

D. McDuff and D. Salamon, J-holomorphic Curves and Symplectic Topology, 2nd ed., AMS Colloquium Publications 52, 2012, Appendix B.2 (the Calderón–Zygmund inequality).
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Chapter IV, §3.

The symbol of `∂_v ∘ \bar∂⁻¹` #

The Cauchy–Riemann operator on the Fourier side #