Analytic models for commuting operator tuples on bounded symmetric domains

For a domain in and a Hilbert space of analytic functions on which satisfies certain conditions, we characterize the commuting -tuples of operators on a separable Hilbert space  such that is unitarily equivalent to the restriction of to an invariant subspace, where is the operator -tuple on the Hilbert space tensor product  . For the unit disc and the Hardy space , this reduces to a well-known theorem of Sz.-Nagy and Foias; for a reproducing kernel Hilbert space on such that the reciprocal of its reproducing kernel is a polynomial in and  , this is a recent result of Ambrozie, Müller and the second author. In this paper, we extend the latter result by treating spaces for which ceases to be a polynomial, or even has a pole: namely, the standard weighted Bergman spaces (or, rather, their analytic continuation) on a Cartan domain corresponding to the parameter in the continuous Wallach set, and reproducing kernel Hilbert spaces for which is a rational function. Further, we treat also the more general problem when the operator is replaced by ,  being a certain generalization of a unitary operator tuple. For the case of the spaces on Cartan domains, our results are based on an analysis of the homogeneous multiplication operators on , which seems to be of an independent interest.

     

The Dirichlet problem and nondivergence harmonic measure

We consider the Dirichlet problem

for two second-order elliptic operators , , in a bounded Lipschitz domain . The coefficients belong to the space of bounded mean oscillation with a suitable small modulus. We assume that is regular in for some , , that is, for all continuous boundary data . Here is the surface measure on and is the nontangential maximal operator. The aim of this paper is to establish sufficient conditions on the difference of the coefficients that will assure the perturbed operator to be regular in for some , .

     

Hyperbolic mean growth of bounded holomorphic functions in the ball

We consider the hyperbolic Hardy class , . It consists of holomorphic in the unit complex ball for which and

where denotes the hyperbolic distance of the unit disc. The hyperbolic version of the Littlewood-Paley type -function and the area function are defined in terms of the invariant gradient of , and membership of is expressed by the property of the functions. As an application, we can characterize the boundedness and the compactness of the composition operator , defined by , from the Bloch space into the Hardy space .

     

Non-solvability for a class of left-invariant second-order differential operators on the Heisenberg group

We study the question of local solvability for second-order, left-invariant differential operators on the Heisenberg group , of the form

where is a complex matrix. Such operators never satisfy a cone condition in the sense of Sjöstrand and Hörmander. We may assume that cannot be viewed as a differential operator on a lower-dimensional Heisenberg group. Under the mild condition that and their commutator are linearly independent, we show that is not locally solvable, even in the presence of lower-order terms, provided that . In the case we show that there are some operators of the form described above that are locally solvable. This result extends to the Heisenberg group a phenomenon first observed by Karadzhov and Müller in the case of It is interesting to notice that the analysis of the exceptional operators for the case turns out to be more elementary than in the case When the analysis of these operators seems to become quite complex, from a technical point of view, and it remains open at this time.

     

Limits of interpolatory processes

Given distinct real numbers and a positive approximation of the identity , which converges weakly to the Dirac delta measure as goes to zero, we investigate the polynomials which solve the interpolation problem

with prescribed data . More specifically, we are interested in the behavior of when the data is of the form for some prescribed function . One of our results asserts that if is sufficiently nice and has sufficiently well-behaved moments, then converges to a limit which can be completely characterized. As an application we identify the limits of certain fundamental interpolatory splines whose knot set is , where is an arbitrary finite subset of the integer lattice , as their degree goes to infinity.

     

On the capacity of sets of divergence associated with the spherical partial integral operator

In this article, we study the pointwise convergence of the spherical partial integral operator when it is applied to functions with a certain amount of smoothness. In particular, for , , , we prove that -quasieverywhere on , where is such that almost everywhere. A weaker version of this result in the range as well as some related localisation principles are also obtained. For and , we construct a function such that diverges everywhere.

     

On one-dimensional self-similar tilings and -tiles

Let be an integer base, a digit set and the set of radix expansions. It is well known that if has nonvoid interior, then can tile with some translation set ( is called a tile and a tile digit set). There are two fundamental questions studied in the literature: (i) describe the structure of ; (ii) for a given , characterize so that is a tile.

We show that for a given pair , there is a unique self-replicating translation set , and it has period for some . This completes some earlier work of Kenyon. Our main result for (ii) is to characterize the tile digit sets for when are distinct primes. The only other known characterization is for , due to Lagarias and Wang. The proof for the case depends on the techniques of Kenyon and De Bruijn on the cyclotomic polynomials, and also on an extension of the product-form digit set of Odlyzko.

     

Hyperbolic -spheres with conical singularities, accessory parameters and Kähler metrics on

We show that the real-valued function on the moduli space of pointed rational curves, defined as the critical value of the Liouville action functional on a hyperbolic -sphere with conical singularities of arbitrary orders , generates accessory parameters of the associated Fuchsian differential equation as their common antiderivative. We introduce a family of Kähler metrics on parameterized by the set of orders , explicitly relate accessory parameters to these metrics, and prove that the functions are their Kähler potentials.

     

Serre's generalization of Nagao's theorem: An elementary approach

Let be a smooth projective curve over a field . For each closed point of let be the coordinate ring of the affine curve obtained by removing from . Serre has proved that is isomorphic to the fundamental group, , of a graph of groups , where is a tree with at most one non-terminal vertex. Moreover the subgroups of attached to the terminal vertices of are in one-one correspondence with the elements of , the ideal class group of . This extends an earlier result of Nagao for the simplest case .

Serre's proof is based on applying the theory of groups acting on trees to the quotient graph , where is the associated Bruhat-Tits building. To determine he makes extensive use of the theory of vector bundles (of rank 2) over . In this paper we determine using a more elementary approach which involves substantially less algebraic geometry.

The subgroups attached to the edges of are determined (in part) by a set of positive integers , say. In this paper we prove that is bounded, even when Cl is infinite. This leads, for example, to new free product decomposition results for certain principal congruence subgroups of , involving unipotent and elementary matrices.

     

The -module structure of -modules

Let be a regular ring, essentially of finite type over a perfect field . An -module is called a unit -module if it comes equipped with an isomorphism , where denotes the Frobenius map on , and is the associated pullback functor. It is well known that then carries a natural -module structure. In this paper we investigate the relation between the unit -structure and the induced -structure on . In particular, it is shown that if is algebraically closed and is a simple finitely generated unit -module, then it is also simple as a -module. An example showing the necessity of being algebraically closed is also given.

     

An extension theorem for separately holomorphic functions with pluripolar singularities

Let be a pseudoconvex domain and let be a locally pluriregular set, . Put

Let be an open neighborhood of and let be a relatively closed subset of . For let be the set of all for which the fiber is not pluripolar. Assume that are pluripolar. Put

Then there exists a relatively closed pluripolar subset of the ``envelope of holomorphy' of such that:

,

for every function separately holomorphic on there exists exactly one function holomorphic on with on , and

is singular with respect to the family of all functions .

     

Stratified transversality by isotopy

For an abstract stratified set or a -regular stratification, hence for any -, - or -regular stratification, we prove that after stratified isotopy of , a stratified subspace of , or a stratified map , can be made transverse to a fixed stratified map .

     

Generating continuous mappings with Lipschitz mappings

If is a metric space, then and denote the semigroups of continuous and Lipschitz mappings, respectively, from to itself. The relative rank of modulo is the least cardinality of any set where generates . For a large class of separable metric spaces we prove that the relative rank of modulo is uncountable. When is the Baire space , this rank is . A large part of the paper emerged from discussions about the necessity of the assumptions imposed on the class of spaces from the aforementioned results.

     

On Fano manifolds with Nef tangent bundles admitting 1-dimensional varieties of minimal rational tangents

Let be a Fano manifold of Picard number 1 with numerically effective tangent bundle. According to the principal case of a conjecture of Campana-Peternell's, should be biholomorphic to a rational homogeneous manifold , where is a simple Lie group, and is a maximal parabolic subgroup.

In our opinion there is no overriding evidence for the Campana-Peternell Conjecture for the case of Picard number 1 to be valid in its full generality. As part of a general programme that the author has undertaken with Jun-Muk Hwang to study uniruled projective manifolds via their varieties of minimal rational tangents, a new geometric approach is adopted in the current article in a special case, consisting of (a) recovering the generic variety of minimal rational tangents , and (b) recovering the structure of a rational homogeneous manifold from . The author proves that, when and the generic variety of minimal rational tangents is 1-dimensional, is biholomorphic to the projective plane , the 3-dimensional hyperquadric , or the 5-dimensional Fano homogeneous contact manifold of type , to be denoted by .

The principal difficulty is part (a) of the scheme. We prove that is a rational curve of degrees , and show that resp. 2 resp. 3 corresponds precisely to the cases of resp. resp. . Let be the normalization of a choice of a Chow component of minimal rational curves on . Nefness of the tangent bundle implies that is smooth. Furthermore, it implies that at any point , the normalization of the corresponding Chow space of minimal rational curves marked at is smooth. After proving that is a rational curve, our principal object of study is the universal family of , giving a double fibration , which gives -bundles. There is a rank-2 holomorphic vector bundle on whose projectivization is isomorphic to . We prove that is stable, and deduce the inequality from the inequality resulting from stability and the existence theorem on Hermitian-Einstein metrics. The case of is ruled out by studying the structure of the curvature tensor of the Hermitian-Einstein metric on in the special case where .

     

On the inversion of the convolution and Laplace transform

We present a new inversion formula for the classical, finite, and asymptotic Laplace transform of continuous or generalized functions . The inversion is given as a limit of a sequence of finite linear combinations of exponential functions whose construction requires only the values of evaluated on a Müntz set of real numbers. The inversion sequence converges in the strongest possible sense. The limit is uniform if is continuous, it is in if , and converges in an appropriate norm or Fréchet topology for generalized functions . As a corollary we obtain a new constructive inversion procedure for the convolution transform ; i.e., for given and we construct a sequence of continuous functions such that .

     

Strongly self-absorbing -algebras

Say that a separable, unital -algebra is strongly self-absorbing if there exists an isomorphism such that and are approximately unitarily equivalent -homomorphisms. We study this class of algebras, which includes the Cuntz algebras , , the UHF algebras of infinite type, the Jiang-Su algebra and tensor products of with UHF algebras of infinite type. Given a strongly self-absorbing -algebra we characterise when a separable -algebra absorbs tensorially (i.e., is -stable), and prove closure properties for the class of separable -stable -algebras. Finally, we compute the possible -groups and prove a number of classification results which suggest that the examples listed above are the only strongly self-absorbing -algebras.

     

A classification and examples of rank one chain domains

A chain order of a skew field is a subring of so that implies Such a ring has rank one if , the Jacobson radical of is its only nonzero completely prime ideal. We show that a rank one chain order of is either invariant, in which case corresponds to a real-valued valuation of or is nearly simple, in which case and are the only ideals of or is exceptional in which case contains a prime ideal that is not completely prime. We use the group of divisorial of with the subgroup of principal to characterize these cases. The exceptional case subdivides further into infinitely many cases depending on the index of in Using the covering group of and the result that the group ring is embeddable into a skew field for a skew field, examples of rank one chain orders are constructed for each possible exceptional case.

     

On the essential commutant of QC

Let (QC) (resp. ) be the -algebra generated by the Toeplitz operators QC (resp. ) on the Hardy space of the unit circle. A well-known theorem of Davidson asserts that (QC) is the essential commutant of . We show that the essential commutant of (QC) is strictly larger than . Thus the image of in the Calkin algebra does not satisfy the double commutant relation. We also give a criterion for membership in the essential commutant of (QC).

     

Some new results in multiplicative and additive Ramsey theory

There are several notions of largeness that make sense in any semigroup, and others such as the various kinds of density that make sense in sufficiently well-behaved semigroups including and . It was recently shown that sets in which are multiplicatively large must contain arbitrarily large geoarithmetic progressions, that is, sets of the form , as well as sets of the form . Consequently, given a finite partition of , one cell must contain such configurations. In the partition case we show that we can get substantially stronger conclusions. We establish some combined additive and multiplicative Ramsey theoretic consequences of known algebraic results in the semigroups and , derive some new algebraic results, and derive consequences of them involving geoarithmetic progressions. For example, we show that given any finite partition of there must be, for each , sets of the form together with , the arithmetic progression , and the geometric progression in one cell of the partition. More generally, we show that, if is a commutative semigroup and a partition regular family of finite subsets of , then for any finite partition of and any , there exist and such that is contained in a cell of the partition. Also, we show that for certain partition regular families and of subsets of , given any finite partition of some cell contains structures of the form for some .