The $${\overline{\partial}}$$-equation on homogeneous varieties with an isolated singularity

Let X be a regular irreducible variety in , Y the associated homogeneous variety in , and N the restriction of the universal bundle of to X. In the present paper, we compute the obstructions to solving the -equation in the L ^p -sense on Y for 1 ≤  p ≤  ∞ in terms of cohomology groups . That allows to identify obstructions explicitly if X is specified more precisely, for example if it is equivalent to or an elliptic curve.      

The Stokes operator in weighted Lq-spaces II: weighted resolvent estimates and maximal Lp-regularity

In this paper we establish a general weighted L ^q -theory of the Stokes operator in the whole space, the half space and a bounded domain for general Muckenhoupt weights . We show weighted L ^q -estimates for the Stokes resolvent system in bounded domains for general Muckenhoupt weights. These weighted resolvent estimates imply not only that the Stokes operator generates a bounded analytic semigroup but even yield the maximal L ^p -regularity of in the respective weighted L ^q -spaces for arbitrary Muckenhoupt weights . This conclusion is archived by combining a recent characterisation of maximal L ^p -regularity by -bounded families due to Weis [Operator-valued Fourier multiplier theorems and maximal L _p -regularity. Preprint (1999)] with the fact that for L ^q -spaces -boundedness is implied by weighted estimates.     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-invariants of Riemannian foliations

For a Riemannian foliation on a closed manifold M, we define L ²-spectral sequence Betti numbers and spectral sequence Novikov–Shubin invariants. The spectral sequence of the lift of to the universal covering of M is used in the definitions. These invariants are natural extensions of the L ²-Betti numbers and the Novikov–Shubin invariants of differentiable manifolds. It is shown that these numbers are invariant by foliated homotopy equivalences, and they are computed for several examples.      

Norm-preserving extensions of functionals and denting points of convex sets

Let W and Z be Banach spaces, and let and be closed subspaces. Let be a subspace of , the Banach space of bounded linear operators from W* to Z**, containing . We describe, for and , all norm-preserving extensions of to the space in terms of convergence of convex combinations. We also characterize denting points of bounded convex subsets of Banach spaces in similar terms. Various applications are presented. Supported by Estonian Science Foundation Grant 5704.     

Compact corigid objects in triangulated categories and co-<Emphasis Type="Italic">t</Emphasis>-structures

In the work of Hoshino, Kato and Miyachi, [11], the authors look at t-structures induced by a compact object, , of a triangulated category, , which is rigid in the sense of Iyama and Yoshino, [12]. Hoshino, Kato and Miyachi show that such an object yields a non-degenerate t-structure on whose heart is equivalent to Mod(End()^op). Rigid objects in a triangulated category can the thought of as behaving like chain differential graded algebras (DGAs). Analogously, looking at objects which behave like cochain DGAs naturally gives the dual notion of a corigid object. Here, we see that a compact corigid object, , of a triangulated category, , induces a structure similar to a t-structure which we shall call a co-t-structure. We also show that the coheart of this non-degenerate co-t-structure is equivalent to Mod(End()^op), and hence an abelian subcategory of .      

Kaplansky classes and derived categories

We put a monoidal model category structure on the category of chain complexes of quasi-coherent sheaves over a quasi-compact and semi-separated scheme X. The approach generalizes and simplifies the method used by the author in (Trans Am Math Soc 356(8) 3369–3390, 2004) and (Trans Am Math Soc 358(7), 2855–2874, 2006) to build monoidal model structures on the category of chain complexes of modules over a ring and chain complexes of sheaves over a ringed space. Indeed, much of the paper is dedicated to showing that in any Grothendieck category , any nice enough class of objects induces a model structure on the category Ch() of chain complexes. The main technical requirement on is the existence of a regular cardinal κ such that every object satisfies the following property: Each κ-generated subobject of F is contained in another κ-generated subobject S for which . Such a class is called a Kaplansky class. Kaplansky classes first appeared in Enochs and López-Ramos (Rend Sem Mat Univ Padova 107, 67–79, 2002) in the context of modules over a ring R. We study in detail the connection between Kaplansky classes and model categories. We also find simple conditions to put on which will guarantee that our model structure is monoidal. We will see that in several categories the class of flat objects form such Kaplansky classes, and hence induce monoidal model structures on the associated chain complex categories. We will also see that in any Grothendieck category , the class of all objects is a Kaplansky class which induces the usual (non-monoidal) injective model structure on Ch().     

Normal generation and Clifford index on algebraic curves

For a smooth curve C it is known that a very ample line bundle on C is normally generated if Cliff() < Cliff(C) and there exist extremal line bundles (:non-normally generated very ample line bundle with Cliff() = Cliff(C)) with . However it has been unknown whether there exists an extremal line bundle with . In this paper, we prove that for any positive integers (g, c) with g = 2c + 5 and (mod 2) there exists a smooth curve of genus g and Clifford index c carrying an extremal line bundle with . In fact, a smooth quadric hypersurface section C of a general projective K3 surface always has an extremal line bundle with . More generally, if C has a line bundle computing the Clifford index c of C with , then C has such an extremal line bundle . For all authors, this work was supported by Korea Research Foundation Grant funded by Korea Government (MOEHRD, Basic Reasearch Promotion Fund)(KRF-2005-070-C00005).     

Positive elimination in valued fields

Let K be an algebraically closed field with a valuation ring or a real closed field with a convex valuation ring . We show that the projection of a basic (see “Introduction”) subset of to K ⁿ is again basic.     

On rectifiable curves with <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-bounds on global curvature: self-avoidance,regularity, and minimizing knots

We discuss the analytic properties of curves γ whose global curvature function ρ _G [γ]⁻¹ is p-integrable. It turns out that the L ^p -norm is an appropriate model for a self-avoidance energy interpolating between “soft” knot energies in form of singular repulsive potentials and “hard” self-obstacles, such as a lower bound on the global radius of curvature introduced by Gonzalez and Maddocks. We show in particular that for all p > 1 finite -energy is necessary and sufficient for W ^2,p -regularity and embeddedness of the curve. Moreover, compactness and lower-semicontinuity theorems lead to the existence of -minimizing curves in given isotopy classes. There are obvious extensions to other variational problems for curves and nonlinearly elastic rods, where one can introduce a bound on to preclude self-intersections.     

The Product Formula in Unitary Deformation K-theory

For finitely generated groups G and H, we prove that there is a weak equivalence G H (G × H) of ku-algebra spectra, where denotes the “unitary deformation K-theory” functor. Additionally, we give spectral sequences for computing the homotopy groups of G and HG in terms of connective K-theory and homology of spaces of G-representations.     

On the Structure of the C *-Algebra Generated by Toeplitz Operators with Piece-wise Continuous Symbols

We study the C ^*-algebra generated by Toeplitz operators with piece-wise continuous symbols acting on the Bergman space on the unit disk in . We describe explicitly each operator from this algebra and characterize Toeplitz operators which belong to the algebra. To the memory of G. S. Litvinchuk     

On the Essential Commutant of Toeplitz Operators in Several Complex Variables

Using the joint local mean oscillation, Jingbo Xia [13] showed that the essential commutant of , where is the subalgebra of L ^∞ generated by all functions which are bounded and have at most one discontinuity, is (QC). Even though Xia’s method cannot be used, we are able to generalize his result to Toeplitz operators in higher dimensions with a different approach. This result is stronger than the well-known result stating that the essential commutant of the full Toeplitz algebra is (QC).      

Pointwise convergence for semigroups in vector-valued <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> spaces

Suppose that {T _t  : t  ≥  0} is a symmetric diffusion semigroup on L ²(X) and denote by its tensor product extension to the Bochner space , where belongs to a certain broad class of UMD spaces. We prove a vector-valued version of the Hopf–Dunford–Schwartz ergodic theorem and show that this extends to a maximal theorem for analytic continuations of on . As an application, we show that such continuations exhibit pointwise convergence.     

A note on simultaneous Diophantine approximation on planar curves

Let denote the set of simultaneously - approximable points in and denote the set of multiplicatively ψ-approximable points in . Let be a manifold in . The aim is to develop a metric theory for the sets and analogous to the classical theory in which is simply . In this note, we mainly restrict our attention to the case that is a planar curve . A complete Hausdorff dimension theory is established for the sets and . A divergent Khintchine type result is obtained for ; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on of is full. Furthermore, in the case that is a rational quadric the convergent Khintchine type result is obtained for both types of approximation. Our results for naturally generalize the dimension and Lebesgue measure statements of Beresnevich et al. (Mem AMS, 179 (846), 1–91 (2006)). Moreover, within the multiplicative framework, our results for constitute the first of their type. The research of Victor V. Beresnevich was supported by an EPSRC Grant R90727/01. Sanju L. Velani is a Royal Society University Research Fellow. For Iona and Ayesha on No. 3.     

Non-free isometric immersions of Riemannian manifolds

Let (V, g) be a Riemannian manifold and let be the isometric immersion operator which, to a map , associates the induced metric on V, where denotes the Euclidean scalar product in . By Nash–Gromov implicit function theorem is infinitesimally invertible over the space of free maps. In this paper we study non-free isometric immersions . We show that the operator (where denotes the space of C ^∞- smooth quadratic forms on ) is infinitesimally invertible over a non-empty open subset of and therefore is an open map in the respective fine topologies.      

On $$ \mathcal{C} $$*-sets and decompositions of continuous and ηζ-continuous functions

The main purpose of this paper is to introduce the concepts of *-sets, *-continuous functions and to obtain new decompositions of continuous and ηζ-continuous functions. Moreover, properties of *-sets and some properties of -sets are discussed.      

Embedding FD(ω) into {\mathcal{P}_s} densely

Let be the lattice of degrees of non-empty subsets of 2 ^ω under Medvedev reducibility. Binns and Simpson proved that FD(ω), the free distributive lattice on countably many generators, is lattice-embeddable below any non-zero element in . Cenzer and Hinman proved that is dense, by adapting the Sacks Preservation and Sacks Coding Strategies used in the proof of the density of the c.e. Turing degrees. With a construction that is a modification of the one by Cenzer and Hinman, we improve on the result of Binns and Simpson by showing that for any , we can lattice embed FD(ω) into strictly between and . We also note that, in contrast to the infinite injury in the proof of the Sacks Density Theorem, in our proof all injury is finite, and that this is also true for the proof of Cenzer and Hinman, if a straightforward simplification is made. Thanks to my adviser Peter Cholak for his guidance in my research. I also wish to thank the anonymous referee for helpful comments and suggestions. My research was partially supported by NSF grants DMS-0245167 and RTG-0353748 and a Schmitt Fellowship at the University of Notre Dame.     

On the Intertwining of \partial{\mathcal{D}}-Isometries

Let be a strictly pseudoconvex bounded domain in with C ² boundary . If a subnormal m-tuple T of Hilbert space operators has the spectral measure of its minimal normal extension N supported on , then T is referred to as a -isometry. Using some non-trivial approximation theorems in the theory of several complex variables, we establish a commutant lifting theorem for those -isometries whose (joint) Taylor spectra are contained in a special superdomain Ω of . Further, we provide a function-theoretic characterization of those subnormal tuples whose Taylor spectra are contained in Ω and that are quasisimilar to a certain (fixed) -isometry T (of which the multiplication tuple on the Hardy space of the unit ball in is a rather special example). Submitted: September 9, 2007. Revised: October 10, 2007. Accepted: October 24, 2007.     

On ideals and quotients of A \mathcal{T} -algebras

Some results on A -algebras are given. We study the problem when ideals, quotients and hereditary subalgebras of A -algebras are A -algebras or A -algebras, and give a necessary and sufficient condition of a hereditary subalgebra of an A -algebra being an A -algebra.     

Algebraic-geometry codes, one-point codes, and evaluation codes

One-point codes are those algebraic-geometry codes for which the associated divisor is a non-negative multiple of a single point. Evaluation codes were defined in order to give an algebraic generalization of both one-point algebraic-geometry codes and Reed–Muller codes. Given an -algebra A, an order function on A and given a surjective -morphism of algebras , the ith evaluation code with respect to is defined as the code . In this work it is shown that under a certain hypothesis on the -algebra A, not only any evaluation code is a one-point code, but any sequence of evaluation codes is a sequence of one-point codes. This hypothesis on A is that its field of fractions is a function field over and that A is integrally closed. Moreover, we see that a sequence of algebraic-geometry codes G _i with associated divisors is the sequence of evaluation codes associated to some -algebra A, some order function and some surjective morphism with if and only if it is a sequence of one-point codes.