Some Estimates for the Norm of the Bergman Projection on Besov Spaces

In this paper we obtain an estimate of the norm of the Bergman projection from L ^p (D, dλ) onto the Besov space B _p , 1 < p < + ∞. The result is asymptotically sharp when p → + ∞. Further for the case P : L ¹(D, dλ) → B ₁, we consider some weak type inequalities with the corresponding spaces.     

The annihilator of Bergman space

LetD be a bounded plane domain (with some smoothness requirements on its boundary). LetB _p(D), 1≤p<∞, be the Bergmanp-space ofD. In a previous paper we showed that the “natural projection”P, involving the Bergman kernel forD, is a bounded projection fromL _p(D) ontoB _p(D), 1<p<∞. With this we have the decompositionL _p(D)=B _p(D)⊕B _q ^⊥ (D,p ^–1+q ^–=1, 1<p< ∞. Here, we show that the annihilatorB _q ^⊥ (D) is the space of allL _p-complex derivatives of functions belonging to Sobolev space and which vanish on the boundary ofD. This extends a result of Schiffer for the casep=2. We also study certain operators onL _p(D). Especially, we show that , whereI is the identity operator and ? is an operator involving the adjoint of the Bergman kernel. Other relationships relevant toB _q ^⊥ (D) are studied.     

Lp ‐estimates for the Bergman projection on strictly pseudoconvex nonsmooth domains

We consider the Bergman projection on Henkin–Leiterer domains, bounded strictly pseudoconvex domains which have defining functions whose gradient is allowed to vanish. Our result describes the mapping properties of the Bergman projection between weighted L^p spaces, with the weights being powers of the gradient of the defining function. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Projections for harmonic Bergman spaces and applications

On the setting of general bounded smooth domains in , we construct L¹-bounded nonorthogonal projections and obtain related reproducing formulas for the harmonic Bergman spaces. In addition, we show that those projections satisfy Sobolev L^p-estimates of any order even for p=1. Among applications are Gleason's problems for the harmonic Bergman-Sobolev and (little) Bloch functions on star-shaped domains with strong reference points.     

On the Libera operator

We show that the Libera operator, L, on some spaces of analytic functions is a continuous extension of the conjugate of the Cesàro operator. Results on L acting on various spaces are obtained. In particular, L maps the Bloch space into BMOA. We also prove some results on the best approximation by polynomials in Hardy and Bergman spaces.     

Admissible convergence for the Poisson-Szegö integrals

We prove almost everywhere semirestricted admissible convergence of the Poisson-Szegö integrals ofL ^p functions (1 <p ≤ ∞) on the Bergman-Shilov boundary of a Siegel domain. In the case of symmetric domains our theorem can be deduced from the results by Peter Sjögren on admissible convergence to the boundary of Poisson integrals on symmetric spaces, although semirestricted admissible convergence means here a more general approach to the boundary then originally defined for symmetric spaces.     

An Analogue of the Kerzman-Stein Formula for the Bergman and Szegö Projections

We show that the difference between the Bergman and Szegö projections on a smooth, bounded planar domain gains a derivative in the L ^p -Sobolev and Lipschitz spaces.     

L (α)-Conjugates on Parabolic Bergman Spaces

The parabolic Bergman space is the set of all L ^p -solutions of the parabolic operator L ^(α). In this paper, we define L ^(α)-conjugates by using fractional derivatives, which are the extension of harmonic conjugates. We study several properties of L ^(α)-conjugates on parabolic Bergman spaces.     

Boundedness of Bergman projections on tube domains over light cones

Let be the future light cone in , and be the associated tube domain. We prove that the weighted Bergman projection is bounded on for , where Q denotes the Lorentz quadratic form. This theorem extends previous results by Bekollé and Bonami [BB]. Our proof relies on the analysis of the projection on mixed norm spaces, which allows us to exploit the oscillation of the Bergman kernel using the Laplace-Fourier transform. Received October 8, 1999 / Published online February 5, 2001     

Asymptotic expansions of Eisenstein integrals and Fourier transform on symmetric spaces

This paper studies the asymptotic expansions of spherical functions on symmetric spaces and Fourier transform of rapidly decreasing functions of L^p type (0 < p ? 2) on these spaces.     

Fonctions lipschitziennes et espaces de Sobolev fractionnaires

We consider a semigroup of Markovian and symmetric operators to which we associate fractional Sobolev spaces D^α_p (0 < α < 1 and 1 < p < ∞) defined as domains of fractional powers (−A_p)^α/2, where A_p is the generator of the semigroup in L^p. We show under rather general assumptions that Lipschitz continuous functions operate by composition on D^α_p if p ≥ 2. This holds in particular in the case of the Ornstein-Uhlenbeck semigroup on an abstract Wiener space.     

Differential recursion relations for Laguerre functions on symmetric cones

Let Ω be a symmetric cone and V the corresponding simple Euclidean Jordan algebra. In our previous papers (some with G. Zhang) we considered the family of generalized Laguerre functions on Ω that generalize the classical Laguerre functions on R⁺. This family forms an orthogonal basis for the subspace of L-invariant functions in L²(Ω,dμν), where dμν is a certain measure on the cone and where L is the group of linear transformations on V that leave the cone Ω invariant and fix the identity in Ω. The space L²(Ω,dμν) supports a highest weight representation of the group G of holomorphic diffeomorphisms that act on the tube domain T(Ω)=Ω+iV. In this article we give an explicit formula for the action of the Lie algebra of G and via this action determine second order differential operators which give differential recursion relations for the generalized Laguerre functions generalizing the classical creation, preservation, and annihilation relations for the Laguerre functions on R⁺.     

Maximal operator on variable Lebesgue spaces for almost monotone radial exponent

We study general Lebesgue spaces with variable exponent p. It is known that the classes L and N of functions p are such that the Hardy-Littlewood maximal operator is bounded on them provided p∈L∩P. The class L governs local properties of p and N governs the behavior of p at infinity.In this paper we focus on the properties of p near infinity. We extend the class N to a collection D of functions p such that the Hardy-Littlewood maximal operator is bounded on the corresponding variable Lebesgue spaces provided p∈L∩D and the class D is essentially larger than N.Moreover, the condition p∈D is quite easily verifiable in the practice.     

L p,q -Boundedness of Bergman Projections in Homogeneous Siegel Domains of Type II

In this paper, we generalize to homogeneous Siegel domains of second kind the L ^p -continuity properties of the Bergman projection. Precisely, we give an improvement of the index p using Fourier analysis as in the case of convex homogeneous tube type domains (Nana and Trojan in Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) X:477–511, 2011).     

Operators on harmonic Bergman spaces

We study the commutator of the multiplication and harmonic Bergman projection, Hankel and Toeplitz operators on the harmonic Bergman spaces. The same type operators have been well studied on the analytic Bergman spaces. The main difficulty of this study is that the bounded harmonic function space is not an algebra! In this paper, we characterize theL ^p boundedness and compactness of these operators with harmonic symbols. Results about operators in Schatten classes, the cut-off phenomenon and general symbols are also included.Partially supported by a grant from the Research Grants Committee of the University of Alabama.     

Optimal Sobolev estimates for ¯∂ on convex domains of finite type

We prove that solutions for ¯ get 1/M-derivatives more than the data in L^p-Sobolev spaces on a bounded convex domain of finite type M by means of the integral kernel method. Also we prove that the Bergman projection is invariant under the L^p-Sobolev spaces of fractional orders by different methods from McNeal-Stein's. By using these results, we can get L^p-Sobolev estimates of order 1/M for the canonical solution for ¯. The author was supported by grant No. R01-2000-000-00001-0 from the Basic Research Program of the Korea Science&Engineering Foundation.     

Composition Operators on a Class of Analytic Function Spaces Related to Brennan’s Conjecture

Brennan’s conjecture in univalent function theory states that if τ is any analytic univalent transform of the open unit disk \mathbbD{\mathbb{D}} onto a simply connected domain G and −1/3 < p < 1, then 1/(τ′) ^p belongs to the Hilbert Bergman space of all analytic square integrable functions with respect to the area measure. We introduce a class of analytic function spaces L²_a(m_p){L^2_a(\mu _p)} on G and prove that Brennan’s conjecture is equivalent to the existence of compact composition operators on these spaces for every simply connected domain G and all p ? (-1/3,1){p\in(-1/3,1)}. Motivated by this result, we study the boundedness and compactness of composition operators in this setting.     

Sharp Estimates in Bergman and Besov Spaces on Bounded Symmetric Domains

Sharp estimates of the point-evaluation functional in weighted Bergman spaces L ^p _a (Ω, dν _α) and for the point-evaluation derivalive functional in Besov spaces B ^p (Ω) are obtained for bounded symmetric domains Ω in ℂ ⁿ . Received October 25, 1999, Accepted December 6, 2000     

Approximation of the Müntz-Szász type in weight spaces L p and zeroes of functions of Bergman classes in a half-plane

We study the completeness of the system of exponents exp(?λ _n t), Re λ _n > 0, in spaces L ^p with the power weight on the semiaxis ?₊. We prove a sufficient condition for the completeness; one can treat it as a modification of the well-known Szász condition. With p = 2 it is unimprovable (in a sense). The proof is based on the results (which are also obtained in this paper) on the distribution of zeroes of functions of the Bergman classes in a half-plane.