Siegel Upper-half Space Research Articles

Carleson measures for Hardy-Sobolev spaces in the Siegel upper half-space

We give a capacitary type characterization of Carleson measures for a class of Hardy-Sobolev spaces (also known as weighted Dirichlet spaces) on the Siegel upper half-space, introduced by Arcozzi et al. in [7]. This answers in part a question raised by the same authors.

Fundamental properties of Cauchy–Szegő projection on quaternionic Siegel upper half space and applications

Abstract We investigate the Cauchy–Szegő projection for quaternionic Siegel upper half space to obtain the pointwise (higher order) regularity estimates for Cauchy–Szegő kernel and prove that the Cauchy–Szegő kernel is nonzero everywhere, which further yields a non-degenerated pointwise lower bound. As applications, we prove the uniform boundedness of Cauchy–Szegő projection on every atom on the quaternionic Heisenberg group, which is used to give an atomic decomposition of regular Hardy space H p {H^{p}} on quaternionic Siegel upper half space for 2 3 < p ≤ 1 {\frac{2}{3}<p\leq 1} . Moreover, we establish the characterisation of singular values of the commutator of Cauchy–Szegő projection based on the kernel estimates. The quaternionic structure (lack of commutativity) is encoded in the symmetry groups of regular functions and the associated partial differential equations.

Bounds for the Bergman Kernel and the Sup-Norm of Holomorphic Siegel Cusp Forms

Abstract We prove “polynomial in $k$” bounds on the size of the Bergman kernel for the space of holomorphic Siegel cusp forms of degree $n$ and weight $k$. When $n=1,2$ our bounds agree with the conjectural bounds, while the lower bounds match for all $n \ge 1$. For an $L^{2}$-normalized Siegel cusp form $F$ of degree $2$, our bound for its sup-norm is $O_{\epsilon } (k^{9/4+\epsilon })$. Further, we show that in any compact set $\Omega $ (which does not depend on $k$) contained in the Siegel fundamental domain of $\textrm {Sp}(2,\mathbb {Z})$ on the Siegel upper half space, the sup-norm of $F$ is $O_{\Omega }(k^{3/2 - \eta })$ for some $\eta>0$, going beyond the “generic” bound in this setting.

The Boundedness of Forelli–Rudin Type Operators on the Siegel Upper Half Space

The Boundedness of Forelli–Rudin Type Operators on the Siegel Upper Half Space

A uniqueness property for Bergman functions on the Siegel upper half-space

In this paper, we show that the Bergman functions on the Siegel upper half-space enjoy the following uniqueness property: if f ∈ A t p ( U ) f\in A_t^p(\mathcal {U}) and L α f ≡ 0 \mathcal {L}^{\alpha } f\equiv 0 for some nonnegative multi-index α \alpha , then f ≡ 0 f\equiv 0 , where L α ≔ ( L 1 ) α 1 ⋯ ( L n ) α n \mathcal {L}^{\alpha }≔(\mathcal {L}_1)^{\alpha _1} \cdots (\mathcal {L}_n)^{\alpha _n} with L j = ∂ ∂ z j + 2 i z ¯ j ∂ ∂ z n \mathcal {L}_j = \frac {\partial }{\partial z_j} + 2i \bar {z}_j \frac {\partial }{\partial z_n} for j = 1 , … , n − 1 j=1,\ldots , n-1 and L n = ∂ ∂ z n \mathcal {L}_n = \frac {\partial }{\partial z_n} . As a consequence, we obtain a new integral representation for the Bergman functions on the Siegel upper half-space. In the end, as an application, we derive a result that relates the Bergman norm to a “derivative norm”, which suggests an alternative definition of the Bloch space and a notion of the Besov spaces over the Siegel upper half-space.

Extension of Q‐type spaces related with weight functions via convolution operators on Heisenberg groups

In this paper, in the setting of Heisenberg groups , we introduce two classes of ‐type spaces related with weight functions denoted by . We investigate several basic properties of , including the average oscillation property, the John–Nirenberg‐type inequality and the relations with classical function spaces. By the family of convolution operators , we extend , to the function spaces on the Siegel upper half space . The wavelet characterizations of are also obtained.

Bergman kernel and period map for curves

As for any symmetric space the tangent space to Siegel upper-half space is endowed with an operation coming from the Lie bracket on the Lie algebra. We consider the pull-back of this operation to the moduli space of curves via the Torelli map. We characterize it in terms of the geometry of the curve, using the Bergman kernel form associated to the curve. It is known that the second fundamental form of the Torelli map outside the hyperelliptic locus can be seen as the multiplication by a certain meromorphic form. Our second result says that the Bergman kernel form is the harmonic representative—in a suitable sense—of this meromorphic form.

Integrality and cuspidality of pullbacks of nearly holomorphic Siegel Eisenstein series

We study nearly holomorphic Siegel Eisenstein series of general levels and characters on $\mathbb{H}_{2n}$, the Siegel upper half space of degree $2n$. We prove that the Fourier coefficients of these Eisenstein series (once suitably normalized) lie in the ring of integers of $\mathbb{Q}_p$ for all sufficiently large primes $p$. We also prove that the pullbacks of these Eisenstein series to $\mathbb{H}_n \times \mathbb{H}_n$ are cuspidal under certain assumptions.

A symplectic restriction problem

We investigate the norm of a degree 2 Siegel modular form of asymptotically large weight whose argument is restricted to the 3-dimensional subspace of its imaginary part. On average over Saito–Kurokawa lifts an asymptotic formula is established that is consistent with the mass equidistribution conjecture on the Siegel upper half space as well as the Lindelöf hypothesis for the corresponding Koecher–Maaß series. The ingredients include a new relative trace formula for pairs of Heegner periods.

Half line Titchmarsh–Weyl m functions of vector-valued discrete Schrödinger operators

We show that the half-line m functions associated with the vector-valued discrete Schrödinger operators are the elements in the Siegel upper half space. We introduce a metric on the space of m functions associated to these operators. Then, we show that the action of transfer matrices on these m functions is distance decreasing.

Schatten class positive Toeplitz operators on Bergman spaces of the Siegel upper half-space

We characterize Schatten class membership of positive Toeplitz operators defined on the Bergman spaces over the Siegel upper half-space in terms of averaging functions and Berezin transforms in the range of \(0<p<\infty \).

The Precise Norm of a Class of Forelli-Rudin Type Operators on the Siegel Upper Half Space

The Precise Norm of a Class of Forelli-Rudin Type Operators on the Siegel Upper Half Space

An explicit formula of Cauchy-Szego kernel for quaternionic Siegel upper half space and applications

In this paper we obtain an explicit formula of Cauchy--Szego kernel for quaternionic Siegel upper half space, and then based on this, we prove that the Cauchy--Szego projection on quaternionic Heisenberg group is a Calderon--Zygmund operator via verifying the size and regularity conditions for the kernel. Next, we also obtain a suitable version of pointwise lower bound for the kernel, which further implies the characterisations of the boundedness and compactness of commutator of the Cauchy--Szego operator via the BMO and VMO spaces on quaternionic Heisenberg group, respectively.

English

The aim of this paper is to characterize the Lp − Lq boundedness of two classes of integral operators from Lp( $${\cal U}$$ , dVα) to Lq( $${\cal U}$$ , dVβ) in terms of the parameters a, b, c, p, q and α, β, where $${\cal U}$$ is the Siegel upper half-space. The results in the presented paper generalize a corresponding result given in C. Liu, Y. Liu, P. Hu, L. Zhou (2019).

Dispersionless Hirota Equations and the Genus 3 Hyperelliptic Divisor

Equations of dispersionless Hirota type F(uxixj)=0\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} F(u_{x_ix_j})=0 \\end{aligned}$$\\end{document}have been thoroughly investigated in mathematical physics and differential geometry. It is known that the parameter space of integrable Hirota type equations in 3D is 21-dimensional, and that the action of the natural equivalence group {mathrm {Sp}}(6,{mathbb {R}}) on the parameter space has an open orbit. However the structure of the generic equation corresponding to the open orbit remained elusive. Here we prove that the generic 3D Hirota equation is given by the remarkable formula ϑm(τ)=0,τ=iHess(u)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\vartheta _m(\\tau )=0, \\qquad \\tau =i\\ \\text {Hess}(u) \\end{aligned}$$\\end{document}where vartheta _m is any genus 3 theta constant with even characteristics and text {Hess}(u) is the 3times 3 Hessian matrix of a (real-valued) function u(x_1, x_2, x_3). Thus, generic Hirota equation coincides with the equation of the genus 3 hyperelliptic divisor (to be precise, its intersection with the imaginary part of the Siegel upper half space {mathfrak {H}}_3). The rich geometry of integrable Hirota type equations sheds new light on local differential geometry of the genus 3 hyperelliptic divisor, in particular, the integrability conditions can be viewed as local differential-geometric constraints that characterise the hyperelliptic divisor uniquely modulo {mathrm {Sp}}(6,{mathbb {C}})-equivalence.

Positive Toeplitz Operators on the Bergman Spaces of the Siegel Upper Half-Space

We characterize bounded and compact positive Toeplitz operators defined on the Bergman spaces over the Siegel upper half-space.

One-line formula for automorphic differential operators on Siegel modular forms

We consider the Siegel upper half space $$H_{2m}$$ of degree 2m and a subset $$H_m\times H_m$$ of $$H_{2m}$$ consisting of two $$m\times m$$ diagonal block matrices. We consider two actions of $$Sp(m,{\mathbb R})\times Sp(m,{\mathbb R}) \subset Sp(2m,{\mathbb R})$$ , one is the action on holomorphic functions on $$H_{2m}$$ defined by the automorphy factor of weight k on $$H_{2m}$$ and the other is the action on vector valued holomorphic functions on $$H_m\times H_m$$ defined on each component by automorphy factors obtained by $$det^k \otimes \rho $$ , where $$\rho $$ is a polynomial representation of $$GL(n,{\mathbb C})$$ . We consider vector valued linear holomorphic differential operators with constant coefficients on holomorphic functions on $$H_{2m}$$ which give an equivariant map with respect to the above two actions under the restriction to $$H_m\times H_m$$ . In a previous paper, we have already shown that all such operators can be obtained either by a projection of the universal automorphic differential operator or alternatively by a vector of monomial basis corresponding to the partition $$2m=m+m$$ . Here in this paper, based on a completely different idea, we give much simpler looking one-line formula for such operators. This is obtained independently from our previous results. The proofs also provide more algorithmic approach to our operators.

Paley–Wiener Theorems on the Siegel Upper Half-Space

In this paper we study spaces of holomorphic functions on the Siegel upper half-space $\mathcal U$ and prove Paley-Wiener type theorems for such spaces. The boundary of $\mathcal U$ can be identified with the Heisenberg group $\mathbb H_n$. Using the group Fourier transform on $\mathbb H_n$, Ogden-Vagi proved a Paley-Wiener theorem for the Hardy space $H^2(\mathcal U)$. We consider a scale of Hilbert spaces on $\mathcal U$ that includes the Hardy space, the weighted Bergman spaces, the weighted Dirichlet spaces, and in particular the Drury-Arveson space, and the Dirichlet space $\mathcal D$. For each of these spaces, we prove a Paley-Wiener theorem, some structure theorems, and provide some applications. In particular we prove that the norm of the Dirichlet space modulo constants $\dot{\mathcal D}$ is the unique Hilbert space norm that is invariant under the action of the group of automorphisms of $\mathcal U$.

Modular invariants for genus 3 hyperelliptic curves

In this article we prove an analogue of a theorem of Lachaud, Ritzenthaler, and Zykin, which allows us to connect invariants of binary oc-tics to Siegel modular forms of genus 3. We use this connection to show that certain modular functions, when restricted to the hyperelliptic locus, assume values whose denominators are products of powers of primes of bad reduction for the associated hyperelliptic curves. We illustrate our theorem with explicit computations. This work is motivated by the study of the value of these modular functions at CM points of the Siegel upper-half space, which, if their denominators are known, can be used to effectively compute models of (hyperelliptic, in our case) curves with CM.

Titchmarsh–Weyl theory for vector-valued discrete Schrödinger operators

We develop the Titchmarsh–Weyl theory for vector-valued discrete Schrodinger operators. We show that the Weyl m functions associated with these operators are matrix valued Herglotz functions that map complex upper half plane to the Siegel upper half space. We discuss about the Weyl disk and Weyl circle corresponding to these operators by defining these functions on a bounded interval. We also discuss the geometric properties of Weyl disk and find the center and radius of the Weyl disk explicitly in terms of matrices.