We describe the strong dual space $({\mathcal O} (D))^*$ for the space ${\mathcal O} (D)$ of holomorphic functions of several complex variables over a bounded Lipschitz domain $D$ with connected boundary $\partial D$ (as usual, ${\mathcal O} (D)$ is endowed with the topology of the uniform convergence on the compact subsets of $D$). We identify the dual space with a closed subspace of the space of harmonic functions on the closed set ${\mathbb C}^n\setminus D$, $n>1$, with elements vanishing at the infinity and satisfying the tangential Cauchy-Riemann equations on $\partial D$. In particular, we extend in a way the classical Grothendieck-K{ö}the-Sebastião e Silva duality for the space of holomorphic functions of one complex variable to the multi-dimensional situation. We use the Bochner-Martinelli kernel ${\mathfrak U}_n$ in ${\mathbb C}^n$, $n>1$, instead of the Cauchy kernel over the complex plane ${\mathbb C}$ and we prove that the duality holds true if and only if the space ${\mathcal O} (D)\cap H^1 (D)$ of the Sobolev holomorphic functions over $D$ is dense in ${\mathcal O} (D)$.
We solve the $\bar{\partial}$-equation for $(p,q)$-forms locally on any reduced pure-dimensional complex space and we prove an explicit version of Serre duality by introducing suitable concrete fine sheaves of certain $(p,q)$-currents. In particular this gives a precise condition for the $\bar{\partial}$-equation to be globally solvable. Our results extend results for $(0,q)$-forms and give information about holomorphic $p$-forms on singular spaces.
Let V and F be holomorphic bundles over a complex manifold M, and s be a holomorphic section of V. We study the cohomology associated to the Koszul complex induced by s, and prove a generalized Serre duality theorem for them.
We study the holomorphic tent spaces $\mathcal{HT}^p_{q,α}(\Bn)$, which are motivated by the area function description of the Hardy spaces on one hand, and the maximal function description of the Hardy spaces on the other. Characterizations for these spaces under general fractional differential operators are given. We describe the dual of $\mathcal{HT}^p_{q,α}(\Bn)$ for the full range $0<p,q<\infty$ and $α>-n-1$. Here the case $1<p<\infty$, $q\leq 1$ leads us to the Hardy-Bloch type spaces $\mathcal{BT}^{p}(\Bn)$ and $\mathcal{BT}^{p,0}(\Bn)$, the latter being the predual of $\mathcal{HT}^p_{1…
Let $n \geq 3$ and $Ω$ be a bounded domain in $\mathbb{C}^n$ with a smooth negative plurisubharmonic exhaustion function $φ$. As a generalization of Y. Tiba's result, we prove that any holomorphic function on a connected open neighborhood of the support of $(i\partial \bar \partial φ)^{n-2}$ in $Ω$ can be extended to the whole domain $Ω$. To prove it, we combine an $L^2$ version of Serre duality and Donnelly-Fefferman type estimates on $(n,n-1)$- and $(n,n)$- forms.
We study several connected problems of holomorphic function spaces on homogeneous Siegel domains. The main object of our study concerns weighted mixed norm Bergman spaces on homogeneous Siegel domains of type II. These problems include: sampling, atomic decomposition, duality, boundary values, boundedness of the Bergman projectors. Our analysis include the Hardy spaces, and suitable generalizations of the classical Bloch and Dirichlet spaces. One of the main novelties in this work is the generality of the domains under consideration, that is, homogeneous Siegel domains, extending many results …
We study codimension one holomorphic distributions on the projective three-space, analyzing the properties of their singular schemes and tangent sheaves. In particular, we provide a classification of codimension one distributions of degree at most 2 with locally free tangent sheaves, and show that codimension one distributions of arbitrary degree with only isolated singularities have stable tangent sheaves. Furthermore, we describe the moduli space of distributions in terms of Grothendieck's Quot-scheme for the tangent bundle. In certain cases, we show that the moduli space of codimension one …
Let $(M,h)$ be a Hermitian manifold and $ψ$ a smooth weight function on $M$. The $\partial$-complex on weighted Bergman spaces $A^2_{(p,0)}(M,h, e^{-ψ})$ of holomorphic $(p,0)$-forms was recently studied in [[10] and [9]. It was shown that if $h$ is Kähler and a suitable density condition holds, the $\partial$-complex exhibits an interesting holomorphicity/duality property when $(\bar\partialψ)^{\sharp}$ is holomorphic (i.e., when the real gradient field $\mathrm{grad}_hψ$ is a real holomorphic vector field). For general Hermitian metrics this property does not hold without the holomorphicity …
We consider versions of the local duality theorem in $\mathbb{C}^n$. We show that there exist canonical pairings in these versions of the duality theorem which can be expressed explicitly in terms of residues of Grothendieck, or in terms of residue currents of Coleff-Herrera and Andersson-Wulcan, and we give several different proofs of non-degeneracy of the pairings. One of the proofs of non-degeneracy uses the theory of linkage, and conversely, we can use the non-degeneracy to obtain results about linkage for modules. We also discuss a variant of such pairings based on residues considered by …
