2020/05/16 by Б. Н. Хабибуллин, Khabibullin, Bulat N., Enzhe Menshikova +1
Mathematics · #30C85 #31A05 #31A15 #31B05 (Primary) 32A60 #31B15 #31B25 #31C05 #32E35 (Secondary) #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Geometry and complex manifolds #Holomorphic and Operator Theory #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.2005.09582
openalex publication_date 2020/05/16 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
Let X be a class of extended numerical functions on a domain D of d-dimensional Euclidean space \mathbb Rd, H⊂ X. Given u,M∈ X, we write u\precH M if there is a function h∈ H such that u+h≤ M on D. We consider this special preorder \precH for a pair of subharmonic unctions u, M on D in cases where H is the space of all harmonic functions on D or H is the convex cone of all subharmonic functions h \not≡ -∞ on D. Main results are dual equivalent forms for this preorder \precH in terms of balayage processes for Riesz measures of subharmonic functions u and M, for Jensen and Arens-Singer (representing) measures, for potentials of these measures, and for special test functions generated by subharmonic functions on complements D∖ S of non-empty precompact subsets S\Subset D. Applications to holomorphic functions f on a domain D⊂ \mathbb Cn relate to the distribution of zero sets of functions f under upper restrictions |f|≤ exp M on D. If a domain D⊂ \mathbb C is a finitely connected domain with non-empty exterior or a simply connected domain with two different points on the boundary of D, then our conditions for the distribution of zeros of f≠ 0 with |f|≤ exp M on D are both necessary and sufficient.