Holomorphic Spaces Research Articles

Existence and uniqueness of nonlinear Goursat problem in the class of Denjoy–Carleman

The problem of existence and uniqueness of nonlinear Goursat is studied by Claude Wagschal in different spaces: holomorphic spaces, partially holomorphic, continuous-Gevrey and Gevrey-holomorphic. In this paper, we replace the sequence (n!)d−1 in the case Gevery with suitable sequence (Mn) in the case Carleman. After that, we study the problem of existence and uniqueness of nonlinear Goursat problem in Denjoy–Carleman space.

On Immersions of Surfaces into SL(2, ℂ) and Geometric Consequences

Abstract We study immersions of smooth manifolds into holomorphic Riemannian space forms of constant sectional curvature -1, including $SL(2,\mathbb{C})$ and the space of geodesics of $\mathbb{H}^3$, and we prove a Gauss–Codazzi theorem in this setting. This approach has some interesting geometric consequences: (1) it provides a model for the transitioning of hypersurfaces among $\mathbb{H}^n$, ${\mathbb{A}}\textrm{d}{\mathbb{S}}^n$, $\textrm{d}{\mathbb{S}}^n$, and ${\mathbb{S}}^n$; (2) it provides an effective tool to construct holomorphic maps to the $\textrm{SO}(n,\mathbb{C})$-character variety, bringing to a simpler proof of the holomorphicity of the complex landslide; and (3) it leads to a correspondence, under certain hypotheses, between complex metrics on a surface (i.e., complex bilinear forms of its complexified tangent bundle) and pairs of projective structures with the same holonomy. Through Bers theorem, we prove a uniformization theorem for complex metrics.

On mixed norm holomorphic grand and small spaces

In this paper, we continue the study of new weighted spaces of holomorphic functions on the unit disc with the mixed norm defined in terms of conditions on Fourier coefficients of a function. Here we present the case in which the corresponding conditions are related to grand and small Lebesgues spaces, i.e. the Fourier coefficients as functions of radial variable belong to either grand or small space, and the norms of these coefficients taken in the corresponding space all together form a sequence. We provide the characterization of functions in such mixed norm spaces and study the boundedness of the holomorphic projection.

Integral Superposition-Type Operators on Some Analytic Function Spaces

All entire functions which transform a class of holomorphic Zygmund-type spaces into weighted analytic Bloch space using the so-called n -generalized superposition operator are characterized in this paper. Moreover, certain specific properties such as boundedness and compactness of the newly defined class of generalized integral superposition operators are discussed and established by using the concerned entire functions.

On some new sharp embedding theorems for multifunctional Herz-type and Bergman-type spaces in tubular domains over symmetric cones

We introduce new multifunctional mixed norm analytic Herz-type spaces in tubular domains over symmetric cones and provide new sharp embedding theorems for them. Some results are new even in case of onefunctional holomorphic spaces. Some new related sharp results for new multifunctional Bergman-type spaces will be also provided under one condition on Bergman kernel.

On Nontangential Limits and Shift Invariant Subspaces

In 1998, Conway and Yang wrote a paper (Holomorphic spaces, MSRI Publications, vol 33, pp 201–209, 1998) in which they posed a number of open questions regarding the shift on $$P^t(\mu )$$ spaces. A few of these have been completely resolved, while at least one remains wide open. In this paper, we review some of the solutions, mention some alternate approaches and discuss further the problem that remains unsolved.

On some new sharp embedding theorems for multifunctional Herz-type and Bergman-type spaces in pseudoconvex domains

We introduce new multifunctional mixed norm analytic Herz-type spaces in strongly pseudoconvex domains and provide new sharp embedding theorems for them. Some results are new even in case of onefunctional holomorphic spaces. Some new related sharp results for new multifunctional Bergman-type spaces will be also provided under one condition on Bergman kernel. Similar results with similar proofs in unbounded tubular domains over symmetric cones and bounded symmetric domains will be also shortly mentioned.

Weighted Inductive Limits of Holomorphic Functions with o-Growth Condition

We study topological properties of weighted (LB)-spaces of holomorphic functions on open sets with o-growth condition. We show that such spaces are semi-Montel if and only if they are (DFS). We also establish that, for a wide family of open sets, some important topological properties of such holomorphic spaces are always equivalent and necessary for a space to be equal algebraically to its projective hull. This means, in particular, that for such open sets these spaces behave in a very similar way as the corresponding spaces of continuous functions.

Berezin symbols on the unit sphere of ℂn

ABSTRACTWe describe the symbolic calculus of operators on the unit sphere in the complex n-space defined by the Berezin quantization. In particular, we derive an explicit formula for the composition of Berezin symbols and thus a non-commutative star product. In order to do so, it was necessary to introduce holomorphic spaces that admit a reproducing kernel in the form of generalized hypergeometric series.

Tent spaces and Littlewood-Paley g-functions associated with Bergman spaces in the unit ball of ℂn

ABSTRACTIn this paper, a family of holomorphic spaces of tent type in the unit ball of is introduced, which is closely related to maximal and area integral functions in terms of the Bergman metric. It is shown that these spaces coincide with Bergman spaces. Furthermore, Littlewood-Paley type g-functions for the Bergman spaces are introduced in terms of the radial derivative, the complex gradient and the invariant gradient. The corresponding characterizations for Bergman spaces are obtained as well. As an application, we obtain new maximal and area integral characterizations for Besov spaces.

Frames in Hermite-Bergman and special Hermite-Bergman spaces

It is well known that the image of \(L^2(\mathbb {R}^n)\) under the Hermite semigroup \(e^{-tH}\) is a Bergman space \(\mathcal {H}_t(\mathbb {C}^n)\) with the reproducing kernel \(L_t\), for \(t>0\). In this paper, it is shown that the discrete system \(\{e_{ap+ibq, t}:p,q\in \mathbb {Z}^n\}\), for \(a,b>0\), arising out of \(L_t\) turns out to be a frame for \(\mathcal {H}_t(\mathbb {C}^n)\) iff \(ab<\pi \sinh 4t\). A similar type of problem is also discussed for Bessel sequences and frames in holomorphic Sobolev spaces associated with the Hermite semigroup. An attempt is also made to study a similar type of problem for the Bergman space \(\mathcal {B}_t^*(\mathbb {C}^{2n})\), which is the image of \(L^2(\mathbb {C}^n)\) under the special Hermite semigroup.

On cotangent manifolds, complex structures and generalized geometry

We develop various properties of symmetric generalized complex structures (in connection with their holomorphic space and B-field transformations), which are analogous to the well-known results of Gualtieri on skew-symmetric generalized complex structures. Given a symmetric or skew-symmetric generalized complex structure \mathcal J and a connection D on a manifold M, we construct an almost complex structure J^{\mathcal J,D} on the cotangent manifold T^{*}M and we study its integrability. For \mathcal J skew-symmetric, we relate the Courant integrability of \mathcal J with the integrability of J^{\mathcal J, D}. We consider in detail the case when M=G is a Lie group and \mathcal J , D are left-invariant. We also show that our approach generalizes various well-known results from special complex geometry.

Fractional derivatives of pointwise multipliers between holomorphic spaces

We study the action of fractional differential type operators on the space of pointwise multipliers between holomorphic Triebel–Lizorkin spaces on the unit ball B of C. As an application, we obtain new characterizations and examples of multipliers on Hardy–Sobolev spaces, and we improve some well-known results about the integrals operators Ib and Jb.

Paragrassmann algebras as quantum spaces, Part II: Toeplitz Operators

This paper continues the study of paragrassmann algebras begun in Part I with the definition and analysis of Toeplitz operators in the associated holomorphic Segal-Bargmann space. These are defined in the usual way as multiplication by a symbol followed by the projection defined by the reproducing kernel. These are non-trivial examples of spaces with Toeplitz operators whose symbols are not functions and which themselves are not spaces of functions.

Holomorphic Spaces in the Unit Ball of

We introduce the and vector spaces of holomorphic functions defined in the unit ball of , generalizing previous work like Ouyang et al. (1998), Stroethoff (1989), and Choa et al. (1992). Likewise, we characterize those spaces in terms of harmonic majorants as a generalization of Arellano et al. (2000).

Erratum to: ``Painlevé null sets, dimension and compact embedding of weighted holomorphic spaces'' (Studia Math. 213 (2012), 169–187)

Erratum to: ``Painlevé null sets, dimension and compact embedding of weighted holomorphic spaces'' (Studia Math. 213 (2012), 169–187)

Continuous embeddings in harmonic mixed norm spaces on the unit ball in ℝ n

In this paper continuous embeddings in spaces of harmonic functions with mixed norm on the unit ball in ℝ n are established, generalizing some Hardy-Littlewood embeddings for similar spaces of holomorphic functions in the unit disc. Differences in indices between the spaces of harmonic and holomorphic spaces are revealed. As a consequence an analogue of classical Fejer-Riesz inequality is obtained. Embeddings in the special case of Riesz systems are also established.

Fock–Sobolev spaces and their Carleson measures

We study a class of holomorphic spaces Fp,m consisting of entire functions f on Cn such that ∂αf is in the Fock space Fp for all multi-indices α with |α|⩽m. We prove a useful Fourier characterization, namely, f∈Fp,m if and only if zαf(z) is in Fp for all α with |α|=m. We obtain duality and interpolation results for these spaces, including the interesting fact that, for 0<p⩽1, (Fp,m)⁎=F∞,m. We also characterize Carleson measures for Fp,m in terms of simple polynomial growth conditions.

Weighted composition operators into Lipschitz spaces

We study bounded and compact weighted composition operators that map into holomorphic Lipschitz spaces. Bibliography: 11 titles.

Painlevé null sets, dimension and compact embedding of weighted holomorphic spaces

We obtain, in terms of associated weights, natural criteria for compact embedding of weighted Banach spaces of holomorphic functions on a wide class of domains in the complex plane. Our study is based on a complete characterization of finite-dimensional w