Division of holomorphic functions and growth conditions

2012/07/06 by William Alexandre, Alexandre, William, Emmanuel Mazzilli +1
Mathematics · #32A22 #32A26 #32A27 #32A37 #32A40 #32A55 #Complex Variables (math.CV) #FOS: Mathematics #math.CV #msc:32A22 #msc:32A26 #msc:32A27 #msc:32A37 #msc:32A40 #msc:32A55

paper · pdf · doi:10.48550/arxiv.1207.1565

28 pages

arxiv created 2012/07/06 · arxiv updated 2012/07/09

Abstract

Let D be a strictly convex domain of Cⁿ, f₁ and f₂ be two holomorphic functions defined on a neighborhood of closure of D and set X_l=z, f_l(z)=0, l=1,2. Suppose that X_l∩ bD is transverse for l=1 and l=2, and that X₁∩ X₂ is a complete intersection. We give necessary conditions when n>1 and sufficient conditions when n=2 under which a function g to be written as g=g₁f₁+g₂f₂ with g₁ and g₂ in L^q(D), q∈ [1,+∞), or g₁ and g₂ in BMO(D). In order to prove the sufficient condition, we explicitly write down the functions g₁ and g₂ using integral representation formulas and new residue currents.

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