In this paper we obtain a necessary and sufficient condition for the canonical solution operator to $\overline \partial $ restricted to radial symmetric Bergman spaces to be a Hilbert-Schmidt operator. We also discuss compactness of the solution operator in spaces of entire functions in one variable. In the sequel we consider several examples and also treat the case of weighted spaces of entire functions in several variables.
In this paper, we investigate a restricted version of Bergman kernels for high powers of a big line bundle over a smooth projective variety. The geometric meaning of the leading term is specified. As a byproduct, we derive some integral representations for the restricted volume.
We show that the exponential map of the Bochner connection on the restricted holomorphic tangent bundle of a complex manifold admitting the positive-definite Bergman metric coincides with the inverse of Bergman's representative map. We also present a generalization of the Lu theorem, as an application.
对于一个光滑有界的严格伪凸域，我们描述的边界奇异性加权Bergman内核相对于权力的定义函数的权力（可能是分数）的行为，更一般地说，任何实阶的全纯函数的索伯列夫空间的再生核。这推广了经典的结果February man为unweighted Bergman核。最后，我们还展示了一个全纯延续的内核相对于Sobolev参数的整个复平面。我们的主要工具是Boutet de Monvel和Guillemin的广义Toeplitz算子。
Given a holomorphic vector bundle $E$ over a compact Riemann surface $M$, and an open set $D$ in $M$, we prove that the Bergman space of holomorphic sections of the restriction of $E$ to $D$ must either coincide with the space of global holomorphic sections of $E$, or be infinite dimensional. Moreover, we characterize the latter entirely in terms of potential-theoretic properties of $D$.
The usual examples of Bergman spaces consist of the closure of an algebra of holomorphic functions on a domain. One can also take the real part of such functions, but essentially one is looking at the same object. In this paper the author shows that the properties of real analyticity and bounded point evaluation can be preserved under closure in a weighted $L^p$ space, if one takes the algebra generated by $z$ and $(\bar z)g(z)$, where $g$ is an entire function satisfying some condition on the distribution of zeros near $\infty$. The condition is not difficult to satisfy with simple examples. …
The asymptotic analysis of Bergman kernels with respect to exponentially varying measures near emergent interfaces has attracted recent attention. Such interfaces typically occur when the associated limiting Bergman density function vanishes on a portion of the plane, the off-spectral region. This type of behaviour is observed when the metric is negatively curved somewhere, or when we study partial Bergman kernels in the context of positively curved metrics. In this work, we cover these two situations in a unified way, for exponentially varying planar measures on the complex plane. We obtain u…
We obtain Littlewood-Paley formulas for Fock spaces $\mathcal{F}^q_{β,ω}$ induced by weights $ω\in A^{restricted}_\infty=\cup_{1\le p<\infty}A^{restricted}_{p}$, where $A^{restricted}_{p}$ is the class of weights such that the Bergman projection $P_α$, on the classical Fock space $\mathcal{F}^2_α$, is bounded on $$\mathcal{L}^p_{α,ω}:=\left\{f:\, \int_{\mathbb{C}}|f(z)|^pe^{-p\fracα{2}|z|^2}\,ω(z)dA(z)<\infty \right\}. $$ Using these equivalent norms for $\mathcal{F}^q_{β,ω}$ we characterize the Carleson measures for weighted Fock-Sobolev spaces $\mathcal{F}^{q,n}_{β,ω}$.
We prove that for a pseudoconvex domain of the form $\mathfrak{A} = \{(z, w) \in \mathbb C^2 : v > F(z, u)\}$, where $w = u + iv$ and F is a continuous function on ${\mathbb C}_z \times {\mathbb R}_u$, the following conditions are equivalent: (1) The domain $\mathfrak{A}$ is Kobayashi hyperbolic. (2) The domain $\mathfrak{A}$ is Brody hyperbolic. (3) The domain $\mathfrak{A}$ possesses a Bergman metric. (4) The domain $\mathfrak{A}$ possesses a bounded smooth strictly plurisubharmonic function, i.e. the core $\mathfrak{c}(\mathfrak{A})$ of $\mathfrak{A}$ is empty. (5) The graph $Γ(F)$ of $F$ c…
We give a general method to obtain from the integral restrictions of functions sharp pointwise and uniform estimates of these functions. This scheme is illustrated by the examples for Fock\,--\,Bargmann spaces of entire functions of several variables and Bergman spaces in balls and polydiscs.
We consider Fock spaces $F^{p,\ell}_α$ of entire functions on ${\mathbb C}$ associated to the weights $e^{-α|z|^{2\ell}}$, where $α>0$ and $\ell$ is a positive integer. We compute explicitly the corresponding Bergman kernel associated to $F^{2,\ell}_α$ and, using an adequate factorization of this kernel, we characterize the boundedness and the compactness of the small Hankel operator $\mathfrak{h}^{\ell}_{b,α}$ on $F^{p,\ell}_α$. Moreover, we also determine when $\mathfrak{h}^{\ell}_{b,α}$ is a Hilbert-Schmidt operator on $F^{2,\ell}_α$.
We study two-dimensional eigenvalue ensembles close to certain types of singular points in the bulk of the droplet. We prove existence of a microscopic density which quickly approaches the classical equilibrium density, as the distance from the singularity increases beyond the microscopic scale. As a consequence we obtain asymptotics for the Bergman function of certain Fock-Sobolev spaces of entire functions.
We derive a necessary condition for compactness of the weighted $\overline\partial$-Neumann operator on the space $L^2(\mathbb C^n,e^{-φ})$, under the assumption that the corresponding weighted Bergman space of entire functions has infinite dimension. Moreover, we compute the essential spectrum of the complex Laplacian for decoupled weights, $φ(z) = φ_1(z_1) + \dotsb + φ_n(z_n)$, and investigate (non-) compactness of the $\overline\partial$-Neumann operator in this case. More can be said if every $Δφ_j$ defines a nontrivial doubling measure.
We obtain sufficient conditions for an exponential type entire function not to have zeros in the open lower half-plane. An exact inequality containing the real and imaginary parts of such functions and their derivatives restricted to the real axis is deduced. A connection is established to the positive definite functions.
Self-commutators of Toeplitz operators and isoperimetric inequalities
对于一个亚正规算子，C. R. Putnam不等式给出了其自换位子范数的一个上界。在定义域的Smirnov空间中的带解析符号的Toeplitz算子的特殊情况下，也存在D. Khavinson（1985）指出，当与Putnam不等式相结合时，蕴涵经典的等周不等式。对于一个非平凡的域，我们比较这些估计精确的结果。然后，我们考虑这样的运营商的Bergman空间的域，我们得到的下界，也反映了几何的域。当与普特南不等式结合时，它们产生了域的基频的Faber-Krahn不等式和扭转刚度的圣维南不等式（但具有非尖锐常数）。我们猜想一个改进版本的普特南不等式在此限制设置。
We consider and provide an accurate study for the fractional Zernike functions on the punctured unit disc, generalizing the classical Zernike polynomials and their associated $β$-restricted Zernike functions. Mainly, we give the spectral realization of the latter ones and show that they are orthogonal $L^2$-eigenfunctions for certain perturbed magnetic (hyperbolic) Laplacian. The algebraic and analytic properties for the fractional Zernike functions to be established include the connection to special functions, their zeros, their orthogonality property, as well as the differential equations, r…
We use the folding theorem of Bishop to construct an entire function $f$ in class $B$ and a wandering domain $U$ of $f$ such that $f$ restricted to $f^n(U)$ is univalent, for all $n \geq 0$. The components of the wandering orbit are bounded and surrounded by the postcritical set.
