Cauchy-Riemann ̄∂-equations with some applications

Abstract

Abstract This paper shows that given 0 < p < 3 and a complex Borel measure µ on the unit disk 𝔻 the inhomogeneous Cauchy-Riemann ̄∂-equation ∂ z ¯ u ( z ) = d μ ( z ) ( 2 π i ) - 1 d z ¯ ∧ d z {\partial _{\bar z}}u\left( z \right) = {{d\mu \left( z \right)} \over {{{\left( {2\pi i} \right)}^{ - 1}}d\bar z \wedge dz}} − a complex Gauss curvature of the weighted disk (𝔻, µ) ᗄ z ∈ 𝔻, has a distributional solution (initially defined on ̄𝔻 = 𝔻 ∪ 𝕋) u ∈ ℒ2, p (𝕋) (formed of: (i) Morrey’s space M 2,0< p <1(𝕋); (ii) John-Nirenberg’s space BMO(𝕋) = 𝒧2,1(𝕋); (iii) Hölder-Lipschitz’s space C C 0 < p - 1 2 < 1 {C^{0 < {{p - 1} \over 2} < 1}} (𝕋)), if and only if 𝔻 ¯ ∋ z ↦ ∫ 𝔻 ( 1 - z w ¯ ) - 1 d μ ¯ ( w ) \mathbb{D} z \mapsto \int\limits_\mathbb{D} {{{\left( {1 - z\bar w} \right)}^{ - 1}}d\bar \mu } \left( w \right) belongs to the analytic Campanato space ϱ𝒜 p (𝔻), thereby not only extending Carleson’s corona & Wolff’s ideal theorems to the algebra M ϱ𝒜 p (𝔻) of all analytic pointwise multiplications of ϱ𝒜 p (𝔻), but quadratically generalizing Brownawell’s result on Hilbert’s Nullstellensatz for the analytic polynomial class 𝒫(ℂ).