The Hadamard multiplication theorem in several complex variables

The $*$-product of domains in several complex variables

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In this article we continue the research, carried out in [25], on computing the * -product of domains in C N . Assuming that 0 ∈ G ⊂ C N is an arbitrary Runge domain and 0 ∈ D ⊂ C N is a bounded, smooth and linearly convex domain (or a non-decreasing union of such ones), we establish a geometric relation between D * G and another domain in C N which is 'extremal' (in an appropriate sense) with respect to a special coefficient multiplier dependent only on the dimension N .

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“…It was extensively studied in various aspects: as a bilinear form, as a linear operator with one factor fixed (see the survey [19] and, for instance, the papers [1], [2], [3], [4], [12], [14], [15], [16], [17], [18], [20], [21], [22], [25]), and also as a map acting on spaces of real analytic functions (see [6], [7], [8], [9], [10], [11]).…”

“…In [25] it was shown (see Proposition 4.1 there) that if at least one of domains 0 ∈ D, G ⊂ C N is Runge domain, then there exists the largest domain 0 ∈ Ω ⊂ C N having the property that the image of O 0,D ×O 0,G under the * -product lies in O 0,Ω (here, as well as in [25], we follow [13, Definition 2.7.1] and consider Runge domains to be pseudoconvex). This largest Ω, denoted as D * G, was the subject of research carried out in [25], which concluded in finding a description of D * G for domains of certain special class.…”

“…In [25] it was shown (see Proposition 4.1 there) that if at least one of domains 0 ∈ D, G ⊂ C N is Runge domain, then there exists the largest domain 0 ∈ Ω ⊂ C N having the property that the image of O 0,D ×O 0,G under the * -product lies in O 0,Ω (here, as well as in [25], we follow [13, Definition 2.7.1] and consider Runge domains to be pseudoconvex). This largest Ω, denoted as D * G, was the subject of research carried out in [25], which concluded in finding a description of D * G for domains of certain special class. The methodology employed in [25], as well as the results achieved there, was of symmetrical nature: both D and G were assumed to belong to the same family of domains and the fundamental integral formula for f * g relied on geometric properties of D to the same extent as on those of G.…”

“…This largest Ω, denoted as D * G, was the subject of research carried out in [25], which concluded in finding a description of D * G for domains of certain special class. The methodology employed in [25], as well as the results achieved there, was of symmetrical nature: both D and G were assumed to belong to the same family of domains and the fundamental integral formula for f * g relied on geometric properties of D to the same extent as on those of G.…”