Bergman Projection Research Articles

On Grand and Small Bergman Spaces

Grand and small Bergman spaces of functions holomorphic in the unit disc are introduced. The boundedness of the Bergman projection operator on grand Bergman spaces is proved. The main result consists of estimates for functions in grand and small Bergman spaces near the boundary, which differ from those in the case of the classical Bergman space by a logarithmic multiplier with positive (for grand spaces) or negative (for small spaces) exponent.

The Bergman kernel on forms: General theory

The goal of this note is to explore the Bergman projection on forms. In particular, we show that some of most basic facts used to construct the Bergman kernel on functions, such as pointwise evaluation in $L^2_{0,q}(\Omega)\cap\ker\bar\partial_q$, fail for $(0,q)$-forms, $q \geq 1$. We do, however, provide a careful construction of the Bergman kernel and explicitly compute the Bergman kernel on $(0,n-1)$-forms. For the ball in $\mathbb{C}^2$, we also show that the size of the Bergman kernel on $(0,1)$-forms is not governed by the control metric, in stark contrast to Bergman kernel on functions.

Analytic Bergman operators in the semiclassical limit

Transposing the Berezin quantization into the setting of analytic microlocal analysis, we construct approximate semiclassical Bergman projections on weighted $L^2$ spaces with analytic weights, and show that their kernel functions admit an asymptotic expansion in the class of analytic symbols. As a corollary, we obtain new estimates for asymptotic expansions of the Bergman kernel on $\mathbb{C}^n$ and for high powers of ample holomorphic line bundles over compact complex manifolds.

The Harmonic Dirichlet–Besov Space and the Optimal Norm for the Bergman Projection

We found the optimal operator-norm constant for the harmonic Bergman projection in the context of harmonic Dirichlet–Besov space regarding the natural semi-norm defined on it. Also, we give the two-side norm estimation when it comes to the complete norm.

Sharp Weighted Bounds for Multilinear Fractional Type Operators Associated with Bergman Projection

We first introduce the multiple weights which are suitable for the study of Bergman type operators. Then, we give the sharp weighted estimates for multilinear fractional Bergman operators and fractional maximal function.

Bergman spaces with exponential weights

In this paper we introduce a kind of Bergman space Aφp on the unit disk D with exponential weights, which cover those defined by Borichev et al. (2007) [6]. We obtain upper and lower bound estimates on the Bergman kernel. As an application, we discuss the Bergman projection and duality.

On mixed norm Bergman–Orlicz–Morrey spaces

Abstract Following the ideas of our previous research, in this paper we continue the study of new Bergman-type spaces on the unit disc with mixed norm in terms of Fourier coefficients. Here we deal with the case where the sequence of norms of Fourier coefficients in the Orlicz–Morrey space in radial variable belongs to l q {l^{q}} . We study the boundedness of the Bergman projection and provide a description of functions in these spaces via the behavior of their Taylor coefficients.

Spectra of Generalized Ces\xe0ro Operators Acting on Growth Spaces

We study the spectrum of generalized Cesàro operators T_g acting on the class of growth spaces A^{-alpha }. We show how the problem of determining the spectrum is related to boundedness of standard weighted Bergman projections on weighted L^infty -spaces. Using this relation we establish some general spectral properties of these operators, and explicitly compute the spectrum for a large class of symbols g.

Boundedness of the Bergman Projection and Some Properties of Bergman Type Spaces

We give a simple proof of the boundedness of Bergman projection in various Banach spaces of functions on the unit disc in the complex plain. The approach of the paper is based on the idea of Zaharyuta and Yudovich (Uspekhi Mat Nauk 19(2):139–142, 1964) where the boundedness of the Bergman projection in Lebesgue spaces was proved using Calderon–Zygmund operators. We exploit this approach and treat the cases of variable exponent Lebesgue space, Orlicz space and variable exponent generalized Morrey spaces. In the case of variable exponent Lebesgue space the boundedness result is known, so in that case we provide a simpler proof, whereas the other cases are new. The major idea of this paper is to show that the approach can be applied to a wide range of function spaces. We also study the rate of growth of functions near the boundary in spaces under consideration and their approximation by mollified dilations.

English

We present a proof of the weighted estimate of the Bergman projection that does not use extrapolation results. This estimate is extended to product domains using an adapted definition of Bekolle-Bonami weights in this setting. An application to bounded Toeplitz products is also given.

$L^p$ estimates for the Bergman projection on some Reinhardt domains

$L^p$ estimates for the Bergman projection on some Reinhardt domains

Weighted and boundary $$L^{p}$$ L p estimates for solutions of the $$\overline{\partial }$$ ∂ ¯ -equation on lineally convex domains of finite type and applications

We obtain sharp weighted estimates for solutions of the equation ∂ u = f in a lineally convex domain of finite type. Precisely we obtain estimates in the spaces L p (Ω,δ γ), δ being the distance to the boundary, with two different types of hypothesis on the form f : first, if the data f belongs to L p Ω,δ γ Ω , γ > −1, we have a mixed gain on the index p and the exponent γ; secondly we obtain a similar estimate when the data f satisfies an apropriate anisotropic L p estimate with weight δ γ+1 Ω. Moreover we extend those results to γ = −1 and obtain L p (∂ Ω) and BMO(∂ Ω) estimates. These results allow us to extend the L p (Ω,δ γ)-regularity results for weighted Bergman projection obtained in [CDM14b] for convex domains to more general weights.

Bergman projections on weighted Fock spaces in several complex variables

Let ϕ be a real-valued plurisubharmonic function on {mathbb {C}}^{n} whose complex Hessian has uniformly comparable eigenvalues, and let mathcal{F}^{p}(phi) be the Fock space induced by ϕ. In this paper, we conclude that the Bergman projection is bounded from the pth Lebesgue space L^{p}(phi ) to mathcal{F}^{p}(phi) for 1leq p leqinfty. As a remark, we claim that Bergman projections are also well defined and bounded on Fock spaces mathcal{F}^{p}(phi) with 0< p<1. We also obtain the estimates for the distance induced by ϕ and the L^{p}(phi)-norm of Bergman kernel for mathcal{F}^{2}(phi).

Mixed norm spaces of analytic functions as spaces of generalized fractional derivatives of functions in Hardy type spaces

Abstract The aim of the paper is twofold. First, we present a new general approach to the definition of a class of mixed norm spaces of analytic functions 𝓐 q;X (𝔻), 1 ⩽ q &lt; ∞ on the unit disc 𝔻. We study a problem of boundedness of Bergman projection in this general setting. Second, we apply this general approach for the new concrete cases when X is either Orlicz space or generalized Morrey space, or generalized complementary Morrey space. In general, such introduced spaces are the spaces of functions which are in a sense the generalized Hadamard type derivatives of analytic functions having lq summable Taylor coefficients.

On Boundedness of Bergman Projection Operators in Banach Spaces of Holomorphic Functions in Half-Plane and Harmonic Functions in Half-Space

We present a simple proof of the boundedness of holomorphic and harmonic Bergman projection operators on a half-plane and a half-space respectively on the Orlicz space, the variable exponent Lebesgue space, and the variable exponent generalized Morrey space. The approach is based on an idea due to V. P. Zaharyuta and V. I. Yudovich (1962) to use Calderon–Zygmund operators for proving the boundedness of the Bergman projection in Lebesgue spaces on the unit disc. We also study the rate of growth of functions near the boundary in the spaces under consideration.

$$\varvec{L^p}$$ L p -norm estimate of the Bergman projection on Hartogs triangle

The purpose of this paper is to give an estimate of the $$L^p$$ -norm of the Bergman projection on the Hartogs triangle.

An application of the Prékopa–Leindler inequality and Sobolev regularity of weighted Bergman projections

An application of the Prékopa–Leindler inequality and Sobolev regularity of weighted Bergman projections

The hyper-singular cousin of the Bergman projection

The hyper-singular cousin of the Bergman projection

Bloch functions and asymptotic tail variance

Let P denote the Bergman projection on the unit disk D,Pμ(z):=∫Dμ(w)(1−zw¯)2dA(w),z∈D, where dA is normalized area measure. We prove that if |μ(z)|≤1 on D, then the integralIμ(a,r):=∫02πexp⁡{ar4|Pμ(reiθ)|2log⁡11−r2}dθ2π,0<r<1, has the bound Iμ(a,r)≤C(a):=10(1−a)−3/2 for 0<a<1, irrespective of the choice of the function μ. Moreover, for a>1, no such uniform bound is possible. We interpret the theorem in terms the asymptotic tail variance of such a Bergman projection Pμ (by the way, the asymptotic tail variance induces a seminorm on the Bloch space). This improves upon earlier work of Makarov, which covers the range 0<a<π264=0.1542… . We then apply the theorem to obtain an estimate of the universal integral means spectrum for conformal mappings with a k-quasiconformal extension, for 0<k<1. The estimate reads, for t∈C and 0<k<1,B(k,t)≤{14k2|t|2(1+7k)2,for|t|≤2k(1+7k)2,k|t|−1(1+7k)2,for|t|≥2k(1+7k)2, which should be compared with the conjecture by Prause and Smirnov to the effect that for real t with |t|≤2/k, we should have B(k,t)=14k2t2.

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.