We study smoothing properties of the Bergman projection and also of weighted Bergman projections. In particular, we relate these properties to the hyperconvexity index of a pseudoconvex domain in $\mathbb{C}^{n}$. The notion of a hyperconvexity index was first introduced by B.Y. Chen, which provides a flexible criterion for studying geometric properties of hyperconvex domains. We also obtain a new estimate of weighted Bergman projections, which improves a well-known estimate of Berndtsson and Charpentier. We give several applications of this estimate, including the study of smoothing properties of weighted Bergman projections.
Recently Herbig, McNeal, and Straube have showed that the Bergman projection of conjugate holomorphic functions is smooth up to the boundary on a class of pseudoconvex domains. 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.
It is proved that on any smoothly bounded domain $D$ in $\mathbb{R}^n$, $n>1$, the output of the harmonic Bergman projection belongs to the Sobolev space of order $k$ whenever all tangential derivatives of order up to k of the input function belong to $L^2(D)$.
Let B be the Bergman projection associated to a domain on which the dbar-Neumann operator is compact. We show that arbitrary L^2 derivatives of Bf are controlled by derivatives of f taken in a single, distinguished direction. As a consequence, functions that are not smooth up to the boundary but are mapped by B to functions which are smooth up to the boundary are explicitly described.
Let $Ω\subset\mathbb{C}^n$ be a bounded domain with smooth boundary, whose Bergman projection $B$ maps the Sobolev space $H^{k_{1}}(Ω)$ (continuously) into $H^{k_{2}}(Ω)$. We establish two smoothing results: (i) the full Sobolev norm $\|Bf\|_{k_{2}}$ is controlled by $L^2$ derivatives of $f$ taken along a single, distinguished direction (of order $\leq k_{1}$), and (ii) the projection of a conjugate holomorphic function in $L^{2}(Ω)$ is automatically in $H^{k_{2}}(Ω)$. There are obvious corollaries for when $B$ is globally regular.
We investigate $L^p$ regularity of weighted Bergman projections on the unit disc and $L^p$ regularity of ordinary Bergman projections in higher dimensions.
We investigate Lp regularity of weighted Bergman projections and zeros of weighted Bergman kernels for the weights that are radially symmetric and comparable to 1 on the unit disc.
In this paper we discuss the explicit solution of certain extremal problems in Bergman spaces. In order to do this, we develop methods to calculate the Bergman projections of various functions. As a special case, we deal with canonical divisors for certain values of $p$.
We apply the Bekollé-Bonami estimate for the (positive) Bergman projection on the weighted $L^p$ spaces on the unit disk. As the consequences, we obtain the boundedness of the Bergman projection on the weighted Sobolev space on the symmetrized bidisk. We also improve the boundedness result of the Bergman projection on the unweighted $L^p$ space on the symmetrized bidisk in \cite{CKY}.
Although the Bergman projection operator $\mathbf{B}_Ω$ is defined on $L^2(Ω)$, its behavior on other $L^p(Ω)$ spaces for $p\not =2$ is an active research area. We survey some of the recent results on $L^p$ estimates on the Bergman projection.
We estimate the norm of the harmonic Bergman projection in the context of harmonic Besov spaces. We obtain the two-side norm estimates in general Lp-case.
We study weighted Bergman projections on the Hartogs triangle in $\mathbb{C}^2$. We show that projections corresponding to exponentially vanishing weights have degenerate $L^p$ mapping properties.
It is known that the Bergman projection operator maps the space of essentially bounded functions in the unit ball in the d-dimensional complex vector space onto the Bloch space of the unit ball. This paper deals with the various semi-norms of the Bergman projection. We improve some recent results.
We bound integral means of the Bergman projection of a function in terms of integral means of the original function. As an application of these results, we bound certain weighted Bergman space norms of derivatives of Bergman projections in terms of weighted $L^p$ norms of certain derivatives of the original function in the $θ$ direction. These results easily imply the well known result that the Bergman projection is bounded from the Sobolev space $W^{k,p}$ into itself for $1 < p < \infty$. We also apply our results to derive certain regularity results involving extremal problems in Bergman spa…
We obtain $L^p$ regularity for the Bergman projection on some Reinhardt domains. We start with a bounded initial domain $Ω$ with some symmetry properties and generate successor domains in higher {dimensions}. We prove: If the Bergman kernel on $Ω$ satisfies appropriate estimates, then the Bergman projection on the successor is $L^p$ bounded. For example, the Bergman projection on successors of strictly pseudoconvex initial domains is bounded on $L^p$ for $1<p<\infty$. The successor domains need not have smooth boundary nor be strictly pseudoconvex.
We prove a general version of \cite[Theorem 4.1]{Boas84} to obtain Sobolev estimates for weighted Bergman projections on convex Reinhardt domains by using the Prékopa-Leindler inequality.
In this paper we investigate a two classes of domains in $\mathbb{C}^n$ generalizing the Hartogs triangle. We prove optimal estimates for the mapping properties of the Bergman projection on these domains.
We reveal a complex analogue to a result about polynomial solutions to the Dirichlet Problem on ellipsoids in $\mathbb{R}^n$ by showing that the Bergman projection on any ellipsoid in $\mathbb{C}^n$ is such that the projection of any polynomial function of degree at most $N$ is a holomorphic polynomial function of degree at most $N$. The discussion is motivated by a connection between the Bergman projection and the Khavinson-Shapiro conjecture in $\mathbb{C}$. We also relate the Khavinson-Shapiro conjecture to polyharmonic Bergman projections in $\mathbb{R}^n$ by showing that these projections…
We show that weighted Bergman projections, corresponding to weights of the form $M(z)(1-|z|^2)^α$ where $α>-1$ and $M(z)$ is a radially symmetric, strictly positive and at least $C^2$ function on the unit disc, are $L^p$ regular.
The main result of this paper is related to finding two-sided bounds of norm for the adjoint operator $P^{\ast}$ of the Bergman projection $P,$ where $P$ denotes the Bergman projection wich maps $L^{1}(D,dλ(z))$ onto the Besov space $B_{1}.$ Here $dλ(z)$ is the Möbius invariant measure ${(1-|z|^2)^{-2}}{dA(z)}$. It is shown that $2\leq\| P^{\ast}\|\leq 4.$
We show that on smooth complete Reinhardt domains, weighted Bergman projection operators corresponding to exponentially decaying weights are unbounded on $L^p$ spaces for all $p\not=2$. On the other hand, we also show that the exponentially weighted projection operators are bounded on Sobolev spaces on the unit ball.
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.
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 $1 <p< \infty$.
We give criteria for boundedness of the associated Bergman-type projections on Lp-spaces on Cn with respect to generalized Gaussian weights exp(-|z|^2m). Complete characterization is obtained for 0<m<=1 and 2n/(2n-1)<m<2; partial results are given for other values of m.
We construct higher-dimensional versions of the Diederich-Fornaess worm domains and show that the Bergman projection operators for these domains are not bounded on high-order $L^p$-Sobolev spaces for $1\leq p<\infty.$
We obtain new two-sided norm estimates for the family of Bergman-type projections arising from the standard weights $(1-|z|^2)^α$ where $α>-1$. As $α\to -1$, the lower bound is sharp in the sense that it asymptotically agrees with the norm of the Riesz projection. The upper bound is estimated in terms of the maximal Bergman projection, whose exact operator norm we calculate. The results provide evidence towards a conjecture that was posed very recently by the first author.
We obtain several estimates for the $L^p$ operator norms of the Bergman and Cauchy-Szegö projections over the the Siegel upper half-space. As a by-product, we also determine the precise value of the $L^p$ operator norm of a family of integral operators over the Siegel upper half-space.
A class of pseudoconvex domains in $\mathbb{C}^{n}$ generalizing the Hartogs triangle is considered. The $L^p$ boundedness of the Bergman projection associated to these domains is established, for a restricted range of $p$ depending on the "fatness" of domains. This range of $p$ is shown to be sharp.
If a bounded domain can be covered by the polydisk through a rational proper holomorphic map, then the Bergman projection is $L^p$-bounded for $p$ in a certain range depending on the ramified rational covering. This result can be applied to the symmetrized polydisk and to the Hartogs triangle with exponent $γ$.
For the Bergman projection operator $P$ we prove that $ \|P\|_{L^1(B,dλ)\rightarrow B_1}= \frac {(2n+1)!}{n!}.$ Here $λ$ stands for the invariant metric in the unit ball $B$ of $\mathbf{C}^n$, and $B_1$ denotes the Besov space with an adequate semi--norm. We also consider a generalization of this result. This generalizes some recent results due to Perälä.
We prove optimal estimates for the mapping properties of the Bergman projection on the Hartogs triangle in weighted $L^p$ spaces when $p>\frac{4}{3}$, where the weight is a power of the distance to the singular boundary point. For $1<p\leq\frac{4}{3}$ we show that no such weighted estimates are possible.
This paper deals with the the norm of the weighted Bergman projection operator $P_α:L^\infty\to \mathcal{B}$ where $α>-1$ and $\mathcal{B}$ is the Bloch space of the unit ball of the complex space $\mathbf{C}^n$. We consider two Bloch norms, the standard Bloch norm and invariant norm w.r.t. automorphisms of the unit ball. Our work contains as a special case the main result of the recent paper \cite{Perala}.
On a reasonable class of domains in $\CC^n$, we characterize those holomorphic functions which continue analytically past the boundary. Then we give some applications of this result to holomorphic mappings. In addition, some new results about the Lu Qi-Keng conjecture are treated.
Let D be a smoothly bounded domain in a complex vector space of dimension n. Suppose that D has a smooth defining function, such that the sum of any q eigenvalues of its complex Hessian are non-negative on the closure of D. We show that this implies global regularity of the Bergman projection on (0,j)-forms for j larger or equal to q-1.
In this paper we investigate a class of domains $Ω^{n+1}_k =\{(z,w)\in \mathbb{C}^n\times \mathbb{C}: |z|^k < |w| < 1\}$ for $k \in \mathbb{Z}^+$ which generalizes the Hartogs triangle. we first obtain the new explicit formulas for the Bergman kernel function on these domains and further give a range of $p$ values for which the $L^p$ boundedness of the Bergman projection holds. This range of $p$ is shown to be sharp.
In this paper we study the Bergman kernel and projection on the unbounded worm domain $$ \mathcal{W}_\infty = \big\{(z_1,z_2)\in\mathbb{C}^2 : \big|z_1-e^{i\log|z_2|^2}\big|^2<1, z_2\neq0\big\}. $$ We first show that the Bergman space of $\mathcal{W}_\infty$ is infinite dimensional. Then we study Bergman kernel $K$ and Bergman projection $\mathcal{P}$ for $\mathcal{W}_\infty$. We prove that $K(z,w)$ extends holomorphically in $z$ (and antiholomorphically in $w$) near each point of the boundary except for a specific subset that we study in detail. By means of an appropriate asymptotic expansion…
Sobolev irregularity of the Bergman projection on a family of domains containing the Hartogs triangle is shown. On the Hartogs triangle itself, a sub-Bergman projection is shown to satisfy better Sobolev norm estimates than its Bergman projection.
We study Bergman kernels $K_Π$ and projections $P_Π$ in unbounded planar domains $Π$, which are periodic in one dimension. In the case $Π$ is simply connected we write the kernel $K_Π$ in terms of a Riemann mapping $φ$ related to the bounded periodic cell $\varpi$ of the domain $Π$. We also introduce and adapt to the Bergman space setting the Floquet transform technique, which is a standard tool for elliptic spectral problems in periodic domains. We investigate the boundedness properties of the Floquet transform operators in Bergman spaces and derive a general formula connecting $P_Π$ to a pro…
We study estimates for Hardy space norms of analytic projections. We first find a sufficient condition for the Bergman projection of a function in the unit disc to belong to the Hardy space $H^p$ for $1 < p < \infty$. We apply the result to prove a converse to an extension of Ryabykh's theorem about Hardy space regularity for Bergman space extremal functions. We also prove that the $H^q$ norm of the Szegö projection of $F^{p/2} \overline{F}^{(p/2)-1}$ cannot be too small if $F$ is analytic, for certain values of $p$ and $q$. We apply this to show that the best analytic approximation in $L^p$ o…
In this paper we obtain sharp weighted estimates for solutions of the $\partial$-equation in a lineally convex domains of finite type. Precisely we obtain estimates in spaces of the form L p (Ω,$δ$ $γ$), $δ$ being the distance to the boundary, with gain on the index p and the exponent $γ$. These estimates allow us to extend the L p (Ω,$δ$ $γ$) and lipschitz regularity results for weighted Bergman projection obtained in [CDM14b] for convex domains to more general weights.
We consider weighted Bergman projection $P_α: L^{\infty}(\Bbb B) \rightarrow {\cal B} $ where $α>-1$ and $\cal B$ is the Bloch space of the unit ball $\Bbb B$ of the complex space $\Bbb C^n.$ We obtain the exact norm of the operator $P_α$ where the Bloch space is observed as a space with norm (and semi-norm) induced from the Besov space $B_{p},0<p<\infty,(B_{\infty}=\cal B).$ Our work contains, as a special case, the main results from \cite{Kalaj} and \cite{Ant}.
We apply integral representations for functions on non-smooth strictly pseudoconvex domains, the Henkin-Leiterer domains, to derive weighted $C^k$ estimates for the component of a given function, $f$, which is orthogonal to holomorphic functions in terms of $C^k$ norms of $\mdbar f$. The weights are powers of the gradient of the defining function of the domain.
In this note we show that the weighted $L^{2}$-Sobolev estimates obtained by P. Charpentier, Y. Dupain & M. Mounkaila for the weighted Bergman projection of the Hilbert space $L^{2}\left(Ω,dμ_{0}\right)$ where $Ω$ is a smoothly bounded pseudoconvex domain of finite type in $\mathbb{C}^{n}$ and $μ_{0}=\left(-ρ_{0}\right)^{r}dλ$, $λ$ being the Lebesgue measure, $r\in\mathbb{Q}_{+}$ and $ρ_{0}$ a special defining function of $Ω$, are still valid for the Bergman projection of $L^{2}\left(Ω,dμ\right)$ where $μ=\left(-ρ\right)^{r}dλ$, $ρ$ being any defining function of $Ω$. In fact a stronger direct…
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.
It is shown in quantitative terms that the maximal Bergman projection \begin{equation*} P^{+}_ω(f)(z)=\int_\mathbb{D} f(ζ)|B^ω_z(ζ)|ω(ζ)\,dA(ζ), \end{equation*} is bounded from $L^p_ν$ to $L^p_η$ if and only if \begin{equation*} \sup_{0<r<1}\left(\int_0^r\frac{η(s)}{\left(\int_{s}^1ω(t)\,dt\right)^p}\,ds\right)^{\frac{1}{p}} \left(\int_r^1\left(\frac{ω(s)}{ν(s)^\frac{1}{p}}\right)^{p'}ds\right)^{\frac{1}{p'}}<\infty, \end{equation*} provided $ω,ν,η$ are radial regular weights. A radial weight $σ$ is regular if it satisfies $σ(r)\asymp\int_{r}^1σ(t)\,dt/(1-r)$ for all $0\leq r<1$. It is also sh…
In this work we consider smooth unbounded worm domains $\mathcal Z_λ$ in $\mathbb C^2$ and show that the Bergman projection, densely defined on the Sobolev spaces $H^{s,p}(\mathcal Z_λ)$, $p\in(1,\infty)$, $s\ge0$, does not extend to a bounded operator $P_λ:H^{s,p}(\mathcal Z_λ)\to H^{s,p}(\mathcal Z_λ)$ when $s>0$ or $p\neq2$. The same irregularity was known in the case of the non-smooth unbounded worm. This improved result shows that the irregularity of the projection is not a consequence of the irregularity of the boundary but instead of the infinite windings of the worm domain.
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 $γ$. A surprising consequence of the analysis is that, whenever $γ$ is irrational, the Bergman projection is bounded only for $p=2$.
We consider the problem of $L^1$ (un)boundedness for a wide class of orthogonal projections, including Bergman projections on domains in complex manifolds and Szegö projections on abstract CR manifolds.
The Fock-Bargmann-Hartogs domain $D_{n, m}(μ)$ is defined by $$ D_{n, m}(μ):=\{(z, w)\in\mathbb{C}^{n}\times\mathbb{C}^m:\Vert w \Vert^2<e^{-μ\Vert z \Vert^2}\},$$ where $μ>0.$ The Fock-Bargmann-Hartogs domain $D_{n, m}(μ)$ is an unbounded strongly pseudoconvex domain with smooth real-analytic boundary. In this paper, we first compute the weighted Bergman kernel of $D_{n, m}(μ)$ with respect to the weight $(-ρ)^α$, where $ρ(z,w):=\|w\|^2-e^{-μ\|z\|^2}$ is a defining function for $D_{n, m}(μ)$ and $α>-1$. Then, for $p\in [1,\infty),$ we show that the corresponding weighted Bergman projection $P…
In this paper we investigate the regularity properties of weighted Bergman projections for smoothly bounded pseudo-convex domains of finite type in $\mathbb{C}^{n}$. The main result is obtained for weights equal to a non negative rational power of the absolute value of a special defining function $ρ$ of the domain: we prove (weighted) Sobolev-$L^{p}$ and Lipchitz estimates for domains in $\mathbb{C}^{2}$ (or, more generally, for domains having a Levi form of rank $\geq n-2$ and for "decoupled" domains) and for convex domains. In particular, for these defining functions, we generalize results o…
We prove sharp estimates for the Bergman projection in weighted Bergman spaces in terms of the Bekolle constant. Our main tools are a dyadic model dominating the operator and an adaptation of a method of Cruz-Uribe, Martell and Perez.
In this note we study some basic properties of general fractional derivatives induced by weighted Bergman kernels. As an application we demonstrate a method for generating pre-images of analytic functions under weighted Bergman projections. This approach is useful for proving the surjectivity of weighted Bergman projections in cases when the target space is not a subspace of the domain space (such situations arise often when dealing with Bloch and Besov spaces). We also discuss a fractional Littlewood-Paley formula.
