Tube Domains Research Articles

Besov spaces and the boundedness of weighted Bergman projections over symmetric tube domains

We extend the analysis of weighted Bergman spaces $\mathcal{A}^{p,q}_{\mathbf{s}}$ on symmetric tube domains, contained in [2], to the case where the weights are positive powers $\Delta_{\mathbf{s}}\doteq\Delta_1^{s_1-s_2} \cdot\dotsc\cdot \Delta_{r-1}^{s_{r-1}-s_r} \Delta_r^{s_r}$ of the principal minors $\Delta_1,\dotsc,\Delta_r$ on the symmetric cone $\Omega$. We discuss the realization of the boundary distributions of functions in $\mathcal{A}^{p,q}_{\mathbf{s}}$ in terms of Besov-type spaces $B^{p,q}_{\mathbf{s}}$ adapted to the structure of the cone. We give a necessary and a sufficient condition on the values of $p$, $q$ and $\mathbf{s}$ for which this identification between $\mathcal{A}^{p,q}_{\mathbf{s}}$ and $B^{p,q}_{\mathbf{s}}$ holds. We also present a continuous version of these latter spaces which is new even for the case $s_1 =\dots =s_r$ considered in [2]. We use these results to discuss multipliers between Besov spaces and the boundedness of the weighted Bergman projection $P_{\mathbf{s}}\colon L^{p,q}_{\mathbf{s}}\rightarrow\mathcal{A}^{p,q}_{\mathbf{s}}$. The situation in the rank two case is specifically dealt with.

HODOGRAPH TRANSFORMATIONS AND THE CHERN-MOSER INVARIANTS ON THE BOUNDARY OF TUBE DOMAINS

By using a hodograph transformation, we calculate a part of the Chern-Moser invariants on the boundary of tube domains. As an application of this result, we give another proof of a result of Yang on the characterization of tube domains with a spherical boundary.

Polyhedra Dual to the Weyl Chamber Decomposition: A Précis

Let V_ℝ be a real vector space with an irreducible action of a finite reflection group W . We study the semi-algebraic geometry of the W -quotient affine variety V//W with the discriminant divisor D_W in it and the τ -quotient affine variety V//W//τ with the bifurcation set BW in it, where τ is the \mathbb G_a -action on V//W obtained by the integration of the primitive vector field D on V//W and B_W is the discriminant divisor of the induced projection : D_W → V//W//τ . Our goal is the construction of a one-parameter family of the semi-algebraic polyhedra K_W(λ) in V_ℝ which are dual to the Weyl chamber decomposition of V_ℝ . As an application, we obtain two geometric descriptions of generators for π_1((V//W)_ℂ^{\text{reg}}) , satisfying the Artin braid relations . The key of the construction of the polyhedra K_W(λ) is a theorem on a linearization of the tube domain in (V//W)_ℝ over the simplicial cone E_W in T_{W,ℝ} .

Fourier expansion of holomorphic modular forms on classical lie groups of tube type along the minimal parabolic subgroup

For holomorphic modular forms on tube domains, there are two types of known Fourier expansions, i.e. the classical Fourier expansion and the Fourier-Jacobi expansion. Either of them is along a maximal parabolic subgroup. In this paper, we discuss Fourier expansion of holomorphic modular forms on tube domains of classical type along the minimal parabolic subgroup. We also relate our Fourier expansion to the two known ones in terms of Fourier coefficients and theta series appearing in these expansions.

Towards a Classification of Homogeneous Tube Domains in ℂ4

We classify the tube domains in ℂ4 with affinely homogeneous base whose boundary contains a non-degenerate affinely homogeneous hypersurface. It follows that these domains are holomorphically homogeneous and amongst them, there are four new examples of unbounded homogeneous domains (that do not have bounded realisations). These domains lie to either side of a pair of Levi-indefinite hypersurfaces. Using the geometry of these two hypersurfaces, we find the automorphism groups of the domains.

Littlewood–Paley decompositions related to symmetric cones and Bergman projections in tube domains

Starting from a Whitney decomposition of a symmetric cone $\Omega$, analogous to the dyadic partition $[2^j, 2^{ j + 1})$ of the positive real line, in this paper we develop an adapted Littlewood?Paley theory for functions with spectrum in $\Omega$. In particular, we define a natural class of Besov spaces of such functions, $B^{p, q}_\nu$, where the role of the usual derivation is now played by the generalized wave operator of the cone $\Delta(\frac{\partial}{\partial x})$. We show that $B^{p, q}_\nu$ consists precisely of the distributional boundary values of holomorphic functions in the Bergman space $A^{p, q}_\nu (T_\Omega)$, at least in a 'good range' of indices $1 \leq q 2$. Moreover, we show the equivalence of this problem with the boundedness of Bergman projectors $P_\nu \colon L^{p, q}_\nu \to A^{p, q}_\nu$, for which our result implies a positive answer when $q_{\nu, p}' < q < q_{\nu, p}$. This extends, to general cones, previous work of the authors on the light-cone. Finally, we focus on light-cones and introduce new necessary and sufficient conditions for our conjecture to hold in terms of inequalities related to the cone multiplier problem. In particular, using recent work by Laba and Wolff, we establish the validity of our conjecture for light-cones when $p$ is sufficiently large.

Hox/Pbx and Brn binding sites mediate Pax3 expression in vitro and in vivo

Pax3 is a paired-homeodomain class transcription factor that serves a role in dorsal–ventral and medial–lateral patterning during vertebrate embryogenesis. Its expression is localized to dorsal domains within the developing neural tube and lateral domains within the developing somite. Additionally, modulation of its expression occurs along the rostral-caudal axis. Previous studies [Development 124 (1997) 617] have localized sequence elements required for expression of Pax3 in the neural tube and neural crest to a 1.6 kbp promoter fragment. In the present study, four discrete DNA elements within the 1.6 kbp promoter fragment are shown by electrophoretic mobility shift assays (EMSA) to exhibit sequence specific interactions with proteins present in nuclear extracts from P19 EC cells induced to express Pax3 by treatment with retinoic acid (RA). Proteins interacting at each of these elements are identified based on biochemical purification using DNA affinity chromatography or a candidate approach. These identifications were confirmed by the ability of specific antibodies to super-shift DNA-protein complexes in EMSA. Two of the four DNA sequence elements are shown to interact with the neural specific Pou-domain class III transcription factors Brn1 and Brn2. The remaining sites contain either consensus binding elements for heterodimers of Pbx and an anterior set of Hox family members, from paralogous groups 1–5, or monomeric Meis and are shown to interact with members of the Pbx and Meis families. Ectopic expression of Brn2 plus HoxA1 but not either factor alone, is sufficient to induce efficient expression from the endogenous Pax3 promoter in P19 EC stem cells under conditions where they would not otherwise express Pax3. Finally, in transgenic mice, mutation of either of the Pou-domain protein binding sites results in reduced expression throughout the neural tube while mutation of the Pbx/Hox binding site results in loss of expression in the anterior domain in which Hox family members from paralogous groups 1–5 are expressed. These observations demonstrate that binding elements for both neural and anterior–posterior position specific transcription factors mediate domains of Pax3 expression.

Affine and Holomorphic Equivalence of Tube Domains in \Bbb C2

It is shown that, except for several explicitly described cases, two hyperbolic tube domains in \(\mathbb{C}^2 \) are biholomorphically equivalent if and only if they are affinely equivalent.

Hox/Pbx and Brn binding sites mediate Pax3 expression in vitro and in vivo

Pax3 is a paired-homeodomain class transcription factor that serves a role in dorsal–ventral and medial–lateral patterning during vertebrate embryogenesis. Its expression is localized to dorsal domains within the developing neural tube and lateral domains within the developing somite. Additionally, modulation of its expression occurs along the rostral-caudal axis. Previous studies [Development 124 (1997) 617] have localized sequence elements required for expression of Pax3 in the neural tube and neural crest to a 1.6 kbp promoter fragment. In the present study, four discrete DNA elements within the 1.6 kbp promoter fragment are shown by electrophoretic mobility shift assays (EMSA) to exhibit sequence specific interactions with proteins present in nuclear extracts from P19 EC cells induced to express Pax3 by treatment with retinoic acid (RA). Proteins interacting at each of these elements are identified based on biochemical purification using DNA affinity chromatography or a candidate approach. These identifications were confirmed by the ability of specific antibodies to super-shift DNA-protein complexes in EMSA. Two of the four DNA sequence elements are shown to interact with the neural specific Pou-domain class III transcription factors Brn1 and Brn2. The remaining sites contain either consensus binding elements for heterodimers of Pbx and an anterior set of Hox family members, from paralogous groups 1–5, or monomeric Meis and are shown to interact with members of the Pbx and Meis families. Ectopic expression of Brn2 plus HoxA1 but not either factor alone, is sufficient to induce efficient expression from the endogenous Pax3 promoter in P19 EC stem cells under conditions where they would not otherwise express Pax3. Finally, in transgenic mice, mutation of either of the Pou-domain protein binding sites results in reduced expression throughout the neural tube while mutation of the Pbx/Hox binding site results in loss of expression in the anterior domain in which Hox family members from paralogous groups 1–5 are expressed. These observations demonstrate that binding elements for both neural and anterior–posterior position specific transcription factors mediate domains of Pax3 expression.

Maximum boundary regularity of bounded Hua-harmonic functions on tube domains

In this paper we prove that bounded Hua-harmonic functions on tube domains that satisfy some boundary regularity condition are necessarily pluriharmonic. In doing so, we show that a similar theorem is true on one-dimensional extensions of the Heisenberg group or equivalently on the Siegel upper half-plane.

Multidimensional Tauberian theorems for Banach-space valued generalized functions

Multidimensional Tauberian theorems for the standard averages of tempered Banach-space valued distributions are stated and proved. These results enable one to determine from the asymptotic behaviour of the averages the asymptotic behaviour of the generalized function itself. The role of the asymptotic scale in these results is performed by the class of regularly varying functions. Special attention is paid to averaging kernels such that several of their moments or linear combinations of moments vanish. Important in these results is the structure of the zero set of the Fourier transformations of the kernels in question. The results so established are applied to the study of the asymptotic properties of solutions of the Cauchy problem for the heat equation in the class of tempered distributions, to the problem of the diffusion of a many-component gas, and to the problem of the absence of the phenomenon of compensation of singularities for holomorphic functions in tube domains over acute cones.

Differential recursion relations for Laguerre functions on Hermitian matrices

In our previous papers [1, 2] we studied Laguerre functions and polynomials on symmetric cones Ω = H/L. The Laguerre functions ℓ n ν, n ∈ Λ, form an orthogonal basis in L 2(Ω, dμν) L and are related via the Laplace transform to an orthogonal set in the representation space of a highest weight representations (πν, ℋν) of the autormorphism group G corresponding to a tube domain T(Ω). In this article, we consider the case where Ω is the space of positive definite Hermitian matrices and G = SU(n, n). We describe the Lie algebraic realization of πν acting in L 2 (Ω, dμν) and use that to determine explicit differential equations and recurrence relations for the Laguerre functions.

Auto/Cross-Regulation of Hoxb3 Expression in Posterior Hindbrain and Spinal Cord

The complex and dynamic pattern of Hoxb3 expression in the developing hindbrain and the associated neural crest of mouse embryos is controlled by three separate cis-regulatory elements: element I (region A), element IIIa, and the r5 enhancer (element IVa). We have examined the cis-regulatory element IIIa by transgenic and mutational analysis to determine the upstream trans-acting factors and mechanisms that are involved in controlling the expression of the mouse Hoxb3 gene in the anterior spinal cord and hindbrain up to the r5/r6 boundary, as well as the associated neural crest which migrate to the third and posterior branchial arches and to the gut. By deletion analysis, we have identified the sequence requirements within a 482-bp element III482. Two Hox binding sites are identified in element III482 and we have shown that in vitro both Hoxb3 and Hoxb4 proteins can interact with these Hox binding sites, suggesting that auto/cross-regulation is required for establishing the expression of Hoxb3 in the neural tube domain. Interestingly, we have identified a novel GCCAGGC sequence motif within element III482, which is also required to direct gene expression to a subset of the expression domains except for rhombomere 6 and the associated neural crest migrating to the third and posterior branchial arches. Element III482 can direct a higher level of reporter gene expression in r6, which led us to investigate whether kreisler is involved in regulating Hoxb3 expression in r6 through this element. However, our transgenic and mutational analysis has demonstrated that, although kreisler binding sites are present, they are not required for the establishment or maintenance of reporter gene expression in r6. Our results have provided evidence that the expression of Hoxb3 in the neural tube up to the r5/r6 boundary is auto/cross-regulated by Hox genes and expression of Hoxb3 in r6 does not require kreisler.

Representations in L2-Spaces on Infinite-Dimensional Symmetric Cones

In this paper we study representations of the automorphism groups of classical infinite-dimensional tube domains. In particular we construct the L2-realization of all unitary highest weight representations, including the vector-valued case. We also find a projective representation of the full identity component of the affine automorphism group of the Hilbert–Schmidt version of the tube domain with trivial cocycle on the subgroup corresponding to the trace class version, but non-trivial on the large group. Finally we show that the operator-valued measures corresponding to the vector valued highest weight representations have densities of a rather weak type with respect to Wishart distributions which makes it possible to determine their “supports.”

HYPOELLIPTICITY OF THE KOHN LAPLACIAN FOR THREE-DIMENSIONAL TUBULAR CAUCHY–RIEMANN STRUCTURES

A necessary and sufficient geometric condition is established for the Kohn Laplacian to be hypoelliptic, modulo its nullspace, for boundaries of arbitrary infinitely differentiable pseudoconvex tube domains in two complex variables. Hypoellipticity is also proved to be equivalent to validity of a superlogarithmic estimate, for this special class of structures.AMS 2000 Mathematics subject classification: Primary 32W05; 32T27. Secondary 35J70.

HYPOELLIPTICITY OF THE KOHN LAPLACIAN FOR THREE-DIMENSIONAL TUBULAR CAUCHY–RIEMANN STRUCTURES

A necessary and sufficient geometric condition is established for the Kohn Laplacian to be hypoelliptic, modulo its nullspace, for boundaries of arbitrary infinitely differentiable pseudoconvex tube domains in two complex variables. Hypoellipticity is also proved to be equivalent to validity of a superlogarithmic estimate, for this special class of structures.AMS 2000 Mathematics subject classification: Primary 32W05; 32T27. Secondary 35J70.

Pumiceous rhyolitic peperite in ancient submarine volcanic successions

Pumiceous peperite is associated with a rhyolitic sill that intruded wet, unconsolidated, submarine stratified pumice breccia in the Cambrian Mount Read Volcanics, Australia. Other examples of pumiceous peperite described in this paper occur at the pumiceous base of a rhyolitic lava and the margins of a rhyolitic cryptodome complex in the Miocene Green Tuff Belt, Japan. Intervals of pumiceous peperite are thin (<15 m), laterally discontinuous, massive, poorly sorted and the clast-to-matrix ratio varies significantly over short distances. They are composed of feldspar-phyric tube pumice clasts and domains of clastic sediment that include stratified pumice breccia, pumiceous siltstone and bioturbated mudstone. The sediment domains may be massive or may display relic depositional structures. Sediment adjacent to pumice clasts is silicified, possibly reflecting induration on contact with hot pumice. Pumiceous peperite has gradational contacts with in situ and clast-rotated pumiceous hyaloclastite and coherent pumiceous rhyolite. Gradational relationships between pumiceous peperite and coherent pumiceous rhyolite, pumice clasts with chilled margins and curviplanar surfaces, and abundant jigsaw-fit texture imply that quench fragmentation and autobrecciation were the dominant mechanisms of clast formation. The formation of pumiceous peperite is favoured by a combination of low confining pressure and delayed quenching allowing vesiculation. Pumiceous peperite may easily be misinterpreted or overlooked as it resembles other pumice-rich facies that are common in submarine volcanic successions. The correct identification of pumiceous peperite can provide evidence to constrain the timing of intrusive episodes in volcanic successions.

Capelli identity and relative discrete series of line bundles over tube domains

We use the Capelli identity to give an explicit realization of some relative discrete series of the $L^2$-space of sections of a line bundle over a tube domain. This amounts to a geometric construction of some Opdam shift operators.

Holomorphic Almost Periodic Functions in Tube Domains and Their Amoebas

We extend the notion of amoeba to holomorphic almost periodic functions in tube domains. In this setting, the order of a function in a connected component of the complement of its amoeba is just the mean motion of this function. We also find a correlation between the orders in different components.

Zeros of holomorphic almost periodic mappings with independent components

For a holomorphic almost periodic mapping f=(f 1,…,f q from a tube domain into Cq ,1≤q≤n, the distribution of its zero set is studied by means of the current being the mean value of the function over . We show that for generic mappings on a tube subdomain TG′ of GG if and only if the components of f can be separately shifted along R n in a way to produce a mapping with zeros in TG′ .