The $L^p$- regularity problem for the Bergman projection of two-dimensional Rudin ball quotients
We solve the $L^p$-regularity problem of the Bergman projection of two-dimensional Rudin ball quotients.
math.CV↗
arXiv subjects
Debraj Chakrabarti
Publications and source records attributed to Debraj Chakrabarti.
Volume of the domain bounded by a Hermitian quadric in complex projective space
We compute explicitly the Riemannian volume, with respect to the Fubini-Study metric, of a domain bounded by a Hermitian quadric in complex projective space. The volume is a rational function of the eigenvalues of the defining quadratic form.
math.MG↗
A Hermitian metric on hyperbolic complex manifolds
We describe a method of defining a Hermitian metric on Kobayashi hyperbolic manifolds. The metric is distance decreasing under holomorphic mappings, up to a multiplicative constant. This method is distinct from the classical construction of Wu, and yields a metric which is expected to have superior regularity properties.
math.CV↗
Restricted type estimates on the Bergman projection of some singular domains
We obtain (weighted) restricted type estimates for the Bergman projection operator on monomial polyhedra, a class of domains generalizing the Hartogs triangle. From these estimates, we recapture $L^p$ boundedness results of the Bergman projection on these domains. On some monomial polyhedra, we also discover that the Bergman projection could fail to be of weak type $(q_*,q_*)$ where $q_*$ is the right endpoint of the interval of $L^p$-regularity of the domain.
math.CV↗
Projections onto $L^p$-Bergman spaces of Reinhardt Domains
For $1<p<\infty$, we emulate the Bergman projection on Reinhardt domains by using a Banach-space basis of $L^p$-Bergman space. The construction gives an integral kernel generalizing the ($L^2$) Bergman kernel. The operator defined by the kernel is shown to be absolutely bounded projection on the $L^p$-Bergman space on a class of domains where the $L^p$-boundedness of the Bergman projection fails for certain $p \neq 2$. As an application, we identify the duals of these $L^p$-Bergman spaces with weighted Bergman spaces.
math.CV↗
The $\bar\partial$-problem on $Z(q)$-domains
Given a complex manifold containing a relatively compact $Z(q)$ domain, we give sufficient geometric conditions on the domain so that its $L^2$-cohomology in degree $(p,q)$ (known to be finite dimensional) vanishes. The condition consists of the existence of a smooth weight function in a neighborhood of the closure of the domain, where the complex Hessian of the weight has a prescribed number of eigenvalues of a particular sign, along with good interaction at the boundary of the Levi form with the complex Hessian, encoded in a subbundle of common positive directions for the two Hermitian forms.
math.CV↗
Bergman kernels of Monomial Polyhedra
The Bergman kernels of monomial polyhedra are explicitly computed. Monomial polyhedra are a class of bounded pseudoconvex Reinhardt domains defined as sublevel sets of Laurent monomials. Their kernels are rational functions and are obtained by an application of Bell's transformation formula.
math.CV↗
Polarization Identities
We prove a generalization of the polarization identity of linear algebra expressing the inner product of a complex inner product space in terms of the norm, where the field of scalars is extended to an associative algebra equipped with an involution, and polarization is viewed as an averaging operation over a compact multiplicative subgroup of the scalars. Using this we prove a general form of the Jordan-von Neumann theorem on characterizing inner product spaces among normed linear spaces, when the scalars are taken in an associative algebra.
math.RA↗
Power series as Fourier series
An abstract theory of Fourier series in locally convex topological vector spaces is developed. An analog of Fejér's theorem is proved for these series. The theory is applied to distributional solutions of Cauchy-Riemann equations to recover basic results of complex analysis. Some classical results of function theory are also shown to be consequences of the series expansion.
math.CV↗
$L^p$-regularity of the Bergman projection on quotient domains
We obtain sharp ranges of $L^p$-boundedness for domains in a wide class of Reinhardt domains representable as sub-level sets of monomials, by expressing them as quotients of simpler domains. We prove a general transformation law relating $L^p$-boundedness on a domain and its quotient by a finite group. The range of $p$ for which the Bergman projection is $L^p$-bounded on our class of Reinhardt domains is found to shrink as the complexity of the domain increases.
math.CV↗
Exact sequences and estimates for the $\overline{\partial}$-problem
We study Sobolev estimates for solutions of the inhomogenous Cauchy-Riemann equations on annuli in $\cx^n$, by constructing exact sequences relating the Dolbeault cohomology of the annulus with respect to Sobolev spaces of forms with those of the envelope and the hole. We also obtain solutions with prescibed support and estimates in Sobolev spaces using our method.
math.CV↗
Bergman Kernels of Elementary Reinhardt Domains
We study the Bergman kernel of certain domains in $\mathbb{C}^n$, called elementary Reinhardt domains, generalizing the classical Hartogs triangle. For some elementary Reinhardt domains, we explicitly compute the kernel, which is a rational function of the coordinates. For some other such domains, we show that the kernel is not a rational function. For a general elementary Reinhardt domain, we obtain a representation of the kernel as an infinite series.
math.CV↗
Automorphism Groups of nilpotent Lie algebras associated to certain graphs
We consider a family of 2-step nilpotent Lie algebras associated to uniform complete graphs on odd number of vertices. We prove that the symmetry group of such a graph is the holomorph of the additive cyclic group $\Z_n$. Moreover, we prove that the (Lie) automorphism group of the corresponding nilpotent Lie algebra contains the dihedral group of order $2n$ as a subgroup.
math.DG↗
A Modified Morrey-Kohn-Hörmander Identity and Applications
We prove a modified form of the classical Morrey-Kohn-Hörmander identity, adapted to pseudoconcave boundaries. Applying this result to an annulus between two bounded pseudoconvex domains in $\mathbb{C}^n$, where the inner domain has $\mathcal{C}^{1,1}$ boundary, we show that the $L^2$ Dolbeault cohomology group in bidegree $(p,q)$ vanishes if $1\leq q\leq n-2$ and is Hausdorff and infinite-dimensional if $q=n-1$, so that the Cauchy-Riemann operator has closed range in each bidegree. As a dual result, we prove that the Cauchy-Riemann operator is solvable in the $L^2$ Sobolev space $W^1$ on any pseudoconvex domain with $\mathcal{C}^{1,1}$ boundary. We also generalize our results to annuli between domains which are weakly $q$-convex in the sense of Ho for appropriate values of $q$.
math.CV↗
On an observation of Sibony
It is shown that if the boundary of a Reinhardt domain in $\mathbb{C}^n$ contains the origin, each holomorphic function on the domain which is infinitely many times differentiable up to the boundary extends holomorphically to a neighborhood of the origin.
math.CV↗
The restriction operator on Bergman spaces
We study the restriction operator from the Bergman space of a domain in $\mathbb{C}^n$ to the Bergman space of a non-empty open subset of the domain. We relate the restriction operator to the Toeplitz operator on the Bergman space of the domain whose symbol is the characteristic function of the subset. Using the biholomorphic invariance of the spectrum of the associated Toeplitz operator, we study the restriction operator from the Bergman space of the unit disc to the Bergman space of subdomains with large symmetry groups, such as horodiscs and subdomains bounded by hypercycles. Furthermore, we prove a sharp estimate of the norm of the restriction operator in case the domain and the subdomain are balls. We also study various operator theoretic properties of the restriction operator such as compactness and essential norm estimates.
math.CV↗
Fourier Representations in Bergman Spaces
We consider a class of domains, generalizing the upper half-plane, and admitting rotational, translational and scaling symmetries, analogous to the half-plane. We prove Paley-Wiener type representations of functions in Bergman spaces of such domains with respect to each of these three groups of symmetries. The Fourier series, Fourier integral and Mellin integral representations so obtained may be used to give representations of the Bergman kernels of these domains.
math.CV↗
Proper holomorphic self-maps of symmetric powers of balls
We show that each proper holomorphic self map of a symmetric power of the unit ball is an automorphism naturally induced by an automorphism of the unit ball, provided the ball is of dimension at least two.
math.CV↗