Bergman Projection Research Articles

Weighted Sobolev regularity of the Bergman projection on the Hartogs triangle

We prove a weighted Sobolev estimate of the Bergman projection on the Hartogs triangle, where the weight is some power of the distance to the singularity at the boundary. This method also applies to the $n$-dimensional generalization of the Hartogs triangle.

Estimates of the $$\varvec{L^p}$$ L p Norms of the Bergman Projection on Strongly Pseudoconvex Domains

We give estimates of the \(L^p\) norm of the Bergman projection on a strongly pseudoconvex domain in \(\mathbb {C}^n\). We show that this norm is comparable to \(\frac{p^2}{p - 1}\) for \(1<p< \infty \).

Bergman Subspaces and Subkernels: Degenerate $$L^p$$ L p Mapping and Zeroes

Regularity and irregularity of the Bergman projection on $L^p$ spaces is established on a natural family of bounded, pseudoconvex domains. The family is parameterized by a real variable $\gamma$. A surprising consequence of the analysis is that, whenever $\gamma$ is irrational, the Bergman projection is bounded only for $p=2$.

Weighted Harmonic Bloch Spaces on the Ball

We study the family of weighted harmonic Bloch spaces \(b_\alpha , \alpha \in {\mathbb {R}}\), on the unit ball of \({\mathbb {R}}^n\). We provide characterizations in terms of partial and radial derivatives and certain radial differential operators that are more compatible with reproducing kernels of harmonic Bergman–Besov spaces. We consider a class of integral operators related to harmonic Bergman projection and determine precisely when they are bounded on \(L^\infty _\alpha \). We define projections from \(L^\infty _\alpha \) to \(b_\alpha \) and as a consequence obtain integral representations. We solve the Gleason problem and provide atomic decomposition for all \(b_\alpha , \alpha \in {\mathbb {R}}\). Finally we give an oscillatory characterization of \(b_\alpha \) when \(\alpha >-1\).

Norm Estimates for Weighted Bergman Projection on the Upper Half Plane

We obtain norm estimate for the weighted Bergman projection on the upper half plane by some integral formulas and properties of hypergeometric functions.

On the global Lipschitz continuity of the Bergman projection on a class of convex domains of infinite type in $\mathbb {C}^2$

On the global Lipschitz continuity of the Bergman projection on a class of convex domains of infinite type in $\mathbb {C}^2$

A curvature formula associated to a family of pseudoconvex domains

We shall give a definition of the curvature operator for a family of weighted Bergman spaces {H-t} associated to a smooth family of smoothly bounded strongly pseudoconvex domains {D-t}. In order to study the boundary term in the curvature operator, we shall introduce the notion of geodesic curvature for the associated family of boundaries {delta D-t} As an application, we get a variation formula for the norms of Bergman projections of currents with compact support. A flatness criterion for {H(t)1 and its applications to triviality of fibrations are also given in this paper.

Sobolev regularity of the Bergman projection on certain pseudoconvex domains

In this paper we study the Sobolev regularity of the Bergman projection B and the ∂¯-Neumann operator N on a certain pseudoconvex domain. We show that if Ω is a domain with Lipschitz boundary, which is relatively compact in an n-dimensional compact Kähler manifold and satisfies some “logδ-pseudoconvexity” condition, the operators B, N and ∂¯∗N are regular in the Sobolev spaces Wr,sk(Ω,E) for forms with values in a holomorphic vector bundle E and for any k<η/2, 0<η<1, 0≤r≤n, 0≤s≤n−1.

The Lp boundedness of the Bergman projection for a class of bounded Hartogs domains

We generalize the Hartogs triangle to a class of bounded Hartogs domains, and we prove that the corresponding Bergman projection is bounded on Lp if and only if p is in the range (2nn+1,2nn−1).

Weighted estimates for the Berezin transform and Bergman projection on the unit ball

Using modern techniques of dyadic harmonic analysis, we are able to prove sharp estimates for the Bergman projection and Berezin transform and more general operators in weighted Bergman spaces on the unit ball in $\mathbb{C}^n$. The estimates are in terms of the Bekolle-Bonami constant of the weight.

THE &lt;TEX&gt;${\bar{\partial}}$&lt;/TEX&gt;-PROBLEM WITH SUPPORT CONDITIONS AND PSEUDOCONVEXITY OF GENERAL ORDER IN KÄHLER MANIFOLDS

Let M be an n-dimensional <TEX>$K{\ddot{a}}hler$</TEX> manifold with positive holomorphic bisectional curvature and let <TEX>${\Omega}{\Subset}M$</TEX> be a pseudoconvex domain of order <TEX>$n-q$</TEX>, <TEX>$1{\leq}q{\leq}n$</TEX>, with <TEX>$C^2$</TEX> smooth boundary. Then, we study the (weighted) <TEX>$\bar{\partial}$</TEX>-equation with support conditions in <TEX>${\Omega}$</TEX> and the closed range property of <TEX>${\bar{\partial}}$</TEX> on <TEX>${\Omega}$</TEX>. Applications to the <TEX>${\bar{\partial}}$</TEX>-closed extensions from the boundary are given. In particular, for q = 1, we prove that there exists a number <TEX>${\ell}_0$</TEX> > 0 such that the <TEX>${\bar{\partial}}$</TEX>-Neumann problem and the Bergman projection are regular in the Sobolev space <TEX>$W^{\ell}({\Omega})$</TEX> for <TEX>${\ell}$</TEX> < <TEX>${\ell}_0$</TEX>.

A note on smoothing properties of the Bergman projection

Recently Herbig, McNeal, and Straube have showed that the Bergman projection of conjugate holomorphic functions is smooth up to the boundary on smoothly bounded domains that satisfy condition R. We show that a further smoothing property holds on a family of Reinhardt domains; namely, the Bergman projection of conjugate holomorphic functions is holomorphic past the boundary.

Weighted Bergman projections on the Hartogs triangle

We prove the Lp regularity of the weighted Bergman projections on the Hartogs triangle, where the weights are powers of the distance to the singularity at the boundary. The restricted range of p is proved to be sharp. By using a two-weight inequality on the upper half plane with Muckenhoupt weights, we can consider a slightly wider class of weights.

Harmonic Besov spaces on the ball

We initiate a detailed study of two-parameter Besov spaces on the unit ball of [Formula: see text] consisting of harmonic functions whose sufficiently high-order radial derivatives lie in harmonic Bergman spaces. We compute the reproducing kernels of those Besov spaces that are Hilbert spaces. The kernels are weighted infinite sums of zonal harmonics and natural radial fractional derivatives of the Poisson kernel. Estimates of the growth of kernels lead to characterization of integral transformations on Lebesgue classes. The transformations allow us to conclude that the order of the radial derivative is not a characteristic of a Besov space as long as it is above a certain threshold. Using kernels, we define generalized Bergman projections and characterize those that are bounded from Lebesgue classes onto Besov spaces. The projections provide integral representations for the functions in these spaces and also lead to characterizations of the functions in the spaces using partial derivatives. Several other applications follow from the integral representations such as atomic decomposition, growth at the boundary and of Fourier coefficients, inclusions among them, duality and interpolation relations, and a solution to the Gleason problem.

Sarason Conjecture on the Bergman Space

We provide a counterexample to the Sarason Conjecture for the Bergman space and present a characterisation of bounded Toeplitz products on the Bergman space in terms of test functions by means of a dyadic model approach. We also present some results about two-weighted estimates for the Bergman projection. Finally, we introduce the class B∞ and give sharp estimates for the one-weighted Bergman projection.

Mixed norm Bergman–Morrey-type spaces on the unit disc

We introduce and study the mixed-norm Bergman–Morrey space A q;p,λ $$(\mathbb{D})$$ , mixednorm Bergman–Morrey space of local type A loc ;,λ , and mixed-norm Bergman–Morrey space of complementary type C A q;p,λ $$(\mathbb{D})$$ on the unit disk D in the complex plane C. Themixed norm Lebesgue–Morrey space L q;p,λ $$(\mathbb{D})$$ is defined by the requirement that the sequence of Morrey L p,λ(I)-norms of the Fourier coefficients of a function f belongs to l q (I = (0, 1)). Then, A q;p,λ $$(\mathbb{D})$$ is defined as the subspace of analytic functions in L q;p,λ $$(\mathbb{D})$$ . Two other spaces A q;p,λ loc $$(\mathbb{D})$$ and C A q;p,λ $$(\mathbb{D})$$ are defined similarly by using the local Morrey L loc ,λ (I)-norm and the complementary Morrey C L p,λ(I)-norm respectively. The introduced spaces inherit features of both Bergman and Morrey spaces and, therefore, we call them Bergman–Morrey-type spaces. We prove the boundedness of the Bergman projection and reveal some facts on equivalent description of these spaces.

Endomorphisms of spaces of virtual vectors fixed by a discrete group

A study is made of unitary representations of a discrete group that are of type II when restricted to an almost-normal subgroup . The associated unitary representation of on the Hilbert space of `virtual' -invariant vectors is investigated, where runs over a suitable class of finite-index subgroups of . The unitary representation of is uniquely determined by the requirement that the Hecke operators for all are the `block-matrix coefficients' of . If is an integer multiple of the regular representation, then there is a subspace of the Hilbert space of that acts as a fundamental domain for . In this case the space of -invariant vectors is identified with . When is not an integer multiple of the regular representation (for example, if , is the modular group, belongs to the discrete series of representations of , and the -invariant vectors are cusp forms), is assumed to be the restriction to a subspace of a larger unitary representation having a subspace as above. The operator angle between the projection onto (typically, the characteristic function of the fundamental domain) and the projection onto the subspace (typically, a Bergman projection onto a space of analytic functions) is the analogue of the space of -invariant vectors. It is proved that the character of the unitary representation is uniquely determined by the character of the representation . Bibliography: 53 titles.

Mixed norm variable exponent Bergman space on the unit disc

We introduce and study the mixed norm variable order Bergman space on the unit disc in the complex plane. The mixed norm variable order Lebesgue-type space is defined by the requirement that the sequence of the variable exponent -norms of the Fourier coefficients of the function f belongs to Then is defined to be the subspace of which consists of analytic functions. We prove the boundedness of the Bergman projection and reveal the dependence of the nature of such spaces on possible growth of variable exponent p(r) when from inside the interval The situation is quite different in the cases and In the case we also characterize the introduced Bergman space as the space of Hadamards’s fractional derivatives of functions from the Hardy space The case is specially studied, and an open problem is formulated in this case. We also reveal the conditions on the rate of growth of p(r) when when isometrically, and when this is not longer true.

Mapping properties of weighted Bergman projection operators on Reinhardt domains

We show that on smooth complete Reinhardt domains, weighted Bergman projection operators corresponding to exponentially decaying weights are unbounded on L p L^p spaces for all p ≠ 2 p\not =2 . On the other hand, we also show that the exponentially weighted projection operators are bounded on Sobolev spaces on the unit ball.

On Hardy spaces on worm domains

AbstractIn this review article we present the problem of studying Hardy spaces and the related Szeg˝o projection on worm domains. We review the importance of the Diederich–Fornæss worm domain as a smooth bounded pseudoconvex domain whose Bergman projection does not preserve Sobolev spaces of sufficiently high order and we highlight which difficulties arise in studying the same problem for the Szeg˝o projection. Finally, we announce and discuss the results we have obtained so far in the setting of non-smooth worm domains.