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| 2. |
- Backlund, Ulf, et al.
(author)
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Semi-Bloch functions in several complex variables
- 2013
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Reports (other academic/artistic)abstract
- Let $M$ be an $n$-dimensional complex manifold. A holomorphic function $f:M\to \mathbb C$ is said to be semi-Bloch if for every $\lambda\in \mathbb C$ the function $g_\lambda=\exp(\lambda f(z))$ is normal on $M$. We characterise Semi-Bloch functions on infinitesimally Kobayashi non-degenerate $M$ in geometric as well as analytic terms. Moreover, we show that on such manifolds, semi-Bloch functions are normal.
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| 3. |
- Backlund, Ulf, et al.
(author)
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Semi-Bloch Functions in Several Complex Variables
- 2016
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In: Journal of Geometric Analysis. - : Springer Science and Business Media LLC. - 1050-6926 .- 1559-002X. ; 26:1, s. 463-473
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Journal article (peer-reviewed)abstract
- Let M be an n-dimensional complex manifold. A holomorphic function f : M -> C is said to be semi-Bloch if for every lambda is an element of C the function g(lambda) = exp(lambda f(z)) is normal on M. We characterize semi-Bloch functions on infinitesimally Kobayashi non-degenerate M in geometric as well as analytic terms. Moreover, we show that on such manifolds, semi-Bloch functions are normal.
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| 4. |
- Carlsson, Linus, 1972-, et al.
(author)
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A note on B-envelope of holomorphy and B-extendable domains
- 2008
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In: Complex Variables and Elliptic Equations. - Abingdon, Oxon, UK : Taylor & Francis. - 1747-6933 .- 1747-6941. ; 53:4, s. 307-309
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Journal article (peer-reviewed)abstract
- Let B subset of H(infinity)(X) be a Banach Algebra on a Riemann domain X over C(n). We show that under certain conditions on B and X, all functions in B can be extended to functions in B(E(B, X)) where E(B, X) is the B-envelope of holomorphy.
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| 5. |
- Carlsson, Linus, 1972-
(author)
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Ideals and boundaries in Algebras of Holomorphic functions
- 2006
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Doctoral thesis (other academic/artistic)abstract
- We investigate the spectrum of certain Banach algebras. Properties like generators of maximal ideals and generalized Shilov boundaries are studied. In particular we show that if the ∂-equation has solutions in the algebra of bounded functions or continuous functions up to the boundary of a domain D ⊂⊂ Cn then every maximal ideal over D is generated by the coordinate functions. This implies that the fibres over D in the spectrum are trivial and that the projection on Cn of the n − 1 order generalized Shilov boundary is contained in the boundary of D.For a domain D ⊂⊂ Cn where the boundary of the Nebenhülle coincide with the smooth strictly pseudoconvex boundary points of D we show that there always exist points p ∈ D such that D has the Gleason property at p.If the boundary of an open set U is smooth we show that there exist points in U such that the maximal ideals over those points are generated by the coordinate functions.An example is given of a Riemann domain, Ω, spread over Cn where the fibers over a point p ∈ Ω consist of m > n elements but the maximal ideal over p is generated by n functions.
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| 6. |
- Carlsson, Linus, 1972-, et al.
(author)
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Spectrum of certain Banach algebras and DBAR-problems
- 2007
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In: Annales Polonici Mathematici. - 0066-2216. ; 90:1, s. 51-58
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Journal article (peer-reviewed)abstract
- We study the spectrum of certain Banach algebras of holomorphic functions defined on a domain Ω where ∂−-problems with certain estimates can be solved. We show that the projection of the spectrum onto Cn equals Ω−− and that the fibers over Ω are trivial. This is used to solve a corona problem in the special case where all but one generator are continuous up to the boundary.
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| 7. |
- Carlsson, Linus, et al.
(author)
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Spectrum of certain Banach algebras and б-problems
- 2007
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In: Annales Polonici Mathematici. - Warsawa : Institute of Mathematics, Polish Academy of Sciences. - 0066-2216 .- 1730-6272. ; 90:1, s. 51-58
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Journal article (peer-reviewed)
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| 9. |
- Fällström, Anders, 1967-
(author)
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Algebras of bounded holomorphic functions
- 1994
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Doctoral thesis (other academic/artistic)abstract
- Some problems concerning the algebra of bounded holomorphic functions from bounded domains in Cn are solved. A bounded domain of holomorphy Q in C2 with nonschlicht i7°°- envelope of holomorphy is constructed and it is shown that there is a point in Q for which Gleason’s Problem for H°°(Q) cannot be solved.If A(f2) is the Banach algebra of functions holomorphic in the bounded domain Q in Cn and continuous on the boundary and if p is a point in Q, then the following problem is known as Gleason’s Problem for A(Q) :Is the maximal ideal in A(Q) consisting of functions vanishing at p generated by (Zl ~Pl) , ■■■ , (Zn - Pn) ?A sufficient condition for solving Gleason’s Problem for A(Q) for all points in Q is given. In particular, this condition is fulfilled by a convex domain Q with Lipi+£-boundary (0 < e < 1) and thus generalizes a theorem of S.L.Leibenzon. One of the ideas in the methods of proof is integration along specific polygonal lines.If Gleason’s Problem can be solved in a point it can be solved also in a neighbourhood of the point. It is shown, that the coefficients in this case depends holomorphically on the points.Defining a projection from the spectrum of a uniform algebra of holomorphic functions to Cn, one defines the fiber in the spectrum over a point as the elements in the spectrum that projects on that point. Defining a kind of maximum modulus property for domains in Cn, some problems concerning the fibers and the number of elements in the fibers in certain algebras of bounded holomorphic functions are solved. It is, for example,shown that the set of points, over which the fibers contain more than one element is closed. A consequence is also that a starshaped domain with the maximum modulus property has schlicht /y°°-envelope of holomorphy. These kind of problems are also connected with Gleason’s problem.A survey paper on general properties of algebras of bounded holomorphic functions of several variables is included. The paper, in particular, treats aspects connecting iy°°-envelopes of holomorphy and some areas in the theory of uniform algebras.
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