Extension Of Theorem Research Articles

Ultradifferentiable extension theorems: A survey

We survey ultradifferentiable extension theorems, i.e., quantitative versions of Whitney’s classical extension theorem, with special emphasis on the existence of continuous linear extension operators. The focus is on Denjoy–Carleman classes for which we develop the theory from scratch and discuss important related concepts such as (non-)quasianalyticity. It allows us to give an efficient and, to a fair extent, elementary introduction to Braun–Meise–Taylor classes based on their representation as intersections and unions of Denjoy–Carleman classes.

Optimal $$L^2$$-Extensions on Tube Domains and a Simple Proof of Prékopa’s Theorem

We prove the optimal $$L^2$$ -extension theorem of Ohsawa–Takegoshi type on a tube domain. As an application, we give a simple proof of Prékopa’s theorem.

Von Neumann algebra conditional expectations with applications to generalized representing measures for noncommutative function algebras

We establish several deep existence criteria for conditional expectations on von Neumann algebras, and then apply this theory to develop a noncommutative theory of representing measures of characters of a function algebra. Our main cycle of results describes what may be understood as a ‘noncommutative Hoffman-Rossi theorem’ giving the existence of weak* continuous ‘noncommutative representing measures’ for so-called D-characters. These results may also be viewed as ‘module’ Hahn-Banach extension theorems for weak* continuous ‘characters’ into possibly noninjective von Neumann algebras. In closing we introduce the notion of ‘noncommutative Jensen measures’, and show that as in the classical case representing measures of logmodular algebras are Jensen measures. The proofs of the two main cycles of results rely on the delicate interplay of Tomita-Takesaki theory, noncommutative Radon-Nikodym derivatives, Connes cocycles, Haagerup noncommutative Lp-spaces, Haagerup's reduction theorem, etc.

Elliptic Theory for Sets with Higher Co-dimensional Boundaries

Many geometric and analytic properties of sets hinge on the properties of elliptic measure, notoriously missing for sets of higher co-dimension. The aim of this manuscript is to develop a version of elliptic theory, associated to a linear PDE, which ultimately yields a notion analogous to that of the harmonic measure, for sets of codimension higher than 1. To this end, we turn to degenerate elliptic equations. Let Γ ⊂ R n \Gamma \subset \mathbb {R}^n be an Ahlfors regular set of dimension d > n − 1 d>n-1 (not necessarily integer) and Ω = R n ∖ Γ \Omega = \mathbb {R}^n \setminus \Gamma . Let L = − div ⁡ A ∇ L = - \operatorname {div} A\nabla be a degenerate elliptic operator with measurable coefficients such that the ellipticity constants of the matrix A A are bounded from above and below by a multiple of dist ⁡ ( ⋅ , Γ ) d + 1 − n \operatorname {dist}(\cdot , \Gamma )^{d+1-n} . We define weak solutions; prove trace and extension theorems in suitable weighted Sobolev spaces; establish the maximum principle, De Giorgi-Nash-Moser estimates, the Harnack inequality, the Hölder continuity of solutions (inside and at the boundary). We define the Green function and provide the basic set of pointwise and/or L p L^p estimates for the Green function and for its gradient. With this at hand, we define harmonic measure associated to L L , establish its doubling property, non-degeneracy, change-of-the-pole formulas, and, finally, the comparison principle for local solutions. In another article to appear, we will prove that when Γ \Gamma is the graph of a Lipschitz function with small Lipschitz constant, we can find an elliptic operator L L for which the harmonic measure given here is absolutely continuous with respect to the d d -Hausdorff measure on Γ \Gamma and vice versa. It thus extends Dahlberg’s theorem to some sets of codimension higher than 1.

On Markov Moment Problem, Polynomial Approximation on Unbounded Subsets, and Mazur–Orlicz Theorem

We review earlier and recent results on the Markov moment problem and related polynomial approximation on unbounded subsets. Such results allow proving the existence and uniqueness of the solutions for some Markov moment problems. This is the first aim of the paper. Our solutions have a codomain space a commutative algebra of (linear) symmetric operators acting from the entire real or complex Hilbert space H to H; this algebra of operators is also an order complete Banach lattice. In particular, Hahn–Banach type theorems for the extension of linear operators having a codomain such a space can be applied. The truncated moment problem is briefly discussed by means of reference citations. This is the second purpose of the paper. In the end, a general extension theorem for linear operators with two constraints is recalled and applied to concrete spaces. Here polynomial approximation plays no role. This is the third aim of this work.

Extending Nirenberg–Spencer’s question on holomorphic embeddings to families of holomorphic embeddings

Nirenberg and Spencer posed the question whether the germ of a compact complex submanifold in a complex manifold is determined by its infinitesimal neighborhood of finite order when the normal bundle is sufficiently positive. To study the problem for a larger class of submanifolds, including free rational curves, we reformulate the question in the setting of families of submanifolds and their infinitesimal neighborhoods. When the submanifolds have no nonzero vector fields, we prove that it is sufficient to consider only first-order neighborhoods to have an affirmative answer to the reformulated question. When the submanifolds do have nonzero vector fields, we obtain an affirmative answer to the question under the additional assumption that submanifolds have certain nice deformation properties, which is applicable to free rational curves. As an application, we obtain a stronger version of the Cartan–Fubini-type extension theorem for Fano manifolds of Picard number 1. We also propose a potential application on hyperplane sections of projective K3 surfaces.

Representation of uniform boundedness principle and Hahn–Banach theorem in linear n-normed space

The concept of b-linear functional and its different types of continuity in linear n-normed space are presented and some of their properties are being established. We derive the Uniform Boundedness Principle and Hahn-Banach extension Theorem with the help of bounded b-linear functionals in the case of linear n-normed spaces and discuss some examples and applications. Finally, we present the concept of weak*convergence for the sequence of bounded b-linear functionals in linear n-normed space.

A Lipschitz Version of the $$\lambda $$-Lemma and a Characterization of Homoclinic and Heteroclinic Orbits

In this paper we consider finite dimensional dynamical systems generated by a Lipschitz function. We prove a version of the Whitney’s Extension Theorem on compact manifolds to obtain a version of the well-known \(\lambda \)-lemma for Lipschitz functions. The notions of Lipschitz transversality and hyperbolicity are investigated in the finite dimensional framework with a norm between \(C^1\)-norm and \(C^0\)-norm. As an application, we study homoclinic and heteroclinic orbits obtaining, as a consequence, a stability result for Lipschitz Morse–Smale functions.

Extensions and dilations of module maps

We study completely positive module maps on C*-algebras which are C*-module over another C*-algebra with compatible actions. We extend several well known dilation and extension results to this setup, including the Stinespring dilation theorem and Wittstock, Arveson, and Voiculescu extension theorems.

Extension of cohomology classes and holomorphic sections defined on subvarieties

In this paper, we obtain two extension theorems for cohomology classes and holomorphic sections defined on analytic subvarieties, which are defined as the supports of the quotient sheaves of multiplier ideal sheaves of quasi-plurisubharmonic functions with arbitrary singularities. The first result gives a positive answer to a question posed by Cao-Demailly-Matsumura and unifies a few well-known injectivity theorems. The second result generalizes and optimizes a general L 2 L^2 extension theorem obtained by Demailly.

A remark on characterizations of Griffiths positivity through asymptotic conditions

In this paper, we study characterizations of Griffiths semi-positivity through [Formula: see text]-estimates of the [Formula: see text]-equation and [Formula: see text]-extension theorems for symmetric powers of a holomorphic vector bundle. We also investigate several versions of the converse of the Demailly–Skoda theorem.

Symmetric gradient Sobolev spaces endowed with rearrangement-invariant norms

A unified approach to embedding theorems for Sobolev type spaces of vector-valued functions, defined via their symmetric gradient, is proposed. The Sobolev spaces in question are built upon general rearrangement-invariant norms. Optimal target spaces in the relevant embeddings are determined within the class of all rearrangement-invariant spaces. In particular, all symmetric gradient Sobolev embeddings into rearrangement-invariant target spaces are shown to be equivalent to the corresponding embeddings for the full gradient built upon the same spaces. A sharp condition for embeddings into spaces of uniformly continuous functions, and their optimal targets, are also exhibited. By contrast, these embeddings may be weaker than the corresponding ones for the full gradient. Related results, of independent interest in the theory of symmetric gradient Sobolev spaces, are established. They include global approximation and extension theorems under minimal assumptions on the domain. A formula for the K-functional, which is pivotal for our method based on reduction to one-dimensional inequalities, is provided as well. The case of symmetric gradient Orlicz-Sobolev spaces, of use in mathematical models in continuum mechanics driven by nonlinearities of non-power type, is especially focused.

Volterra equations driven by rough signals

This article is devoted to the extension of the theory of rough paths in the context of Volterra equations with possibly singular kernels. We begin to describe a class of two parameter functions defined on the simplex called Volterra paths. These paths are used to construct a so-called Volterra-signature, analogously to the signature used in Lyon’s theory of rough paths. We provide a detailed algebraic and analytic description of this object. Interestingly, the Volterra signature does not have a multiplicative property similar to the classical signature, and we introduce an integral product behaving like a convolution extending the classical tensor product. We show that this convolution product is well defined for a large class of Volterra paths, and we provide an analogue of the extension theorem from the theory of rough paths (which guarantees in particular the existence of a Volterra signature). Moreover the concept of convolution product is essential in the construction of Volterra controlled paths, which is the natural class of processes to be integrated with respect to the driving noise in our situation. This leads to a rough integral given as a functional of the Volterra signature and the Volterra controlled paths, combined through the convolution product. The rough integral is then used in the construction of unique solutions to Volterra equations driven by Hölder noises with singular kernels. An example concerning Brownian noises and a singular kernel is treated.

Nevanlinna and algebraic hyperbolicity

Motivated by the notion of the algebraic hyperbolicity, we introduce the notion of Nevanlinna hyperbolicity for a pair [Formula: see text], where [Formula: see text] is a projective variety and [Formula: see text] is an effective Cartier divisor on [Formula: see text]. This notion links and unifies the Nevanlinna theory, the complex hyperbolicity (Brody and Kobayashi hyperbolicity), the big Picard-type extension theorem (more generally the Borel hyperbolicity). It also implies the algebraic hyperbolicity. The key is to use the Nevanlinna theory on parabolic Riemann surfaces recently developed by P[Formula: see text]un and Sibony [Value distribution theory for parabolic Riemann surfaces, preprint (2014), arXiv:1403.6596].

Differential forms on log canonical spaces in positive characteristic

Given a logarithmic $1$-form on the snc locus of a log canonical surface pair $(X, D)$ over a perfect field of characteristic $p \ge 7$, we show that it extends with at worst logarithmic poles to any resolution of singularities. We also prove the analogous statement for regular differential forms, under an additional tameness hypothesis. In addition, residue and restriction sequences for tamely dlt pairs are established. We give a number of examples showing that our results are sharp in the surface case, and that they fail in higher dimensions. On the other hand, our techniques yield a new proof of the characteristic zero Logarithmic Extension Theorem in any dimension.

Extension with log-canonical measures and an improvement to the plt extension of Demailly–Hacon–Păun

With a view to proving the conjecture of "dlt extension" related to the abundance conjecture, a sequence of potential candidates for replacing the Ohsawa measure in the Ohsawa-Takegoshi $L^2$ extension theorem, called the "lc-measures", which hopefully could provide the $L^2$ estimate of a holomorphic extension of any suitable holomorphic section on a subvariety with singular locus, are introduced in the first half of the paper. Based on the version of $L^2$ extension theorem proved by Demailly, a proof is provided to show that the lc-measure can replace the Ohsawa measure in the case where the classical Ohsawa-Takegoshi $L^2$ extension works, with some improvements on the assumptions on the metrics involved. The second half of the paper provides a simplified proof of the result of Demailly-Hacon-Paun on the "plt extension" with the superfluous assumption "$\operatorname{supp} D \subset \operatorname{supp}(S+B)$" in their result removed. Most arguments in the proof are readily adopted to the "dlt extension" once the $L^2$ estimates with respect to the lc-measures of holomorphic extensions of sections on subvarieties with singular locus are ready.

Fréchet-valued meromorphic extension along a pencil of complex lines

The aim of paper is to find the condition under which a Fréchet-valued function [Formula: see text] admitting meromorphic extension along some pencil of complex lines can be meromorphically extended to a neighborhood of [Formula: see text] Some auxiliary results concerning the domains of existence for Fréchet-valued meromorphic functions, Rothstein’s theorem, Levi extension theorem for meromorphic functions with values in a locally complete space, convergence of formal power series of Fréchet-valued homogeneous polynomials are also proved in this work.

Springer’s Odd Degree Extension Theorem for quadratic forms over semilocal rings

A fundamental result of Springer says that a quadratic form over a field of characteristic ≠2 is isotropic if it is so after an odd degree field extension. In this paper we generalize Springer’s theorem as follows. Let R be an arbitrary semilocal ring, let S be a finite R-algebra of constant odd degree, which is étale or generated by one element, and let q be a nonsingular R-quadratic form whose base ring extension qS is isotropic. We show that then already q is isotropic.

New common fixed point results for single-valued mappings in cone b-metric spaces over Banach algebra with generalized c-distance

In this work, we obtain common fixed point and fixed point theorems in cone b-metric spaces over Banach algebra with generalized c-distance. The presented theorems extend and generalize several well-known comparable results in literature to generalized c-distance in such space. We present an example to support our extension and generalization theorems.

On ultrafilter extensions of first-order models and ultrafilter interpretations

There exist two known types of ultrafilter extensions of first-order models, both in a certain sense canonical. One of them (Goranko in Filter and ultrafilter extensions of structures: universal-algebraic aspects, preprint, 2007) comes from modal logic and universal algebra, and in fact goes back to Jonsson and Tarski (Am J Math 73(4):891–939, 1951; 74(1):127–162, 1952). Another one (Saveliev in Lect Notes Comput Sci 6521:162–177, 2011; Saveliev in: Friedman, Koerwien, Muller (eds) The infinity project proceeding, Barcelona, 2012) comes from model theory and algebra of ultrafilters, with ultrafilter extensions of semigroups (Hindman and Strauss in Algebra in the Stone–Cech Compactification, W. de Gruyter, Berlin, 2012) as its main precursor. By a classical fact of general topology, the space of ultrafilters over a discrete space is its largest compactification. The main result of Saveliev (Lect Notes Comput Sci 6521:162–177, 2011; in: Friedman, Koerwien, Muller (eds) The infinity project proceeding, Barcelona, 2012), which confirms a canonicity of this extension, generalizes this fact to discrete spaces endowed with an arbitrary first-order structure. An analogous result for the former type of ultrafilter extensions was obtained in Saveliev (in On two types of ultrafilter extensions of binary relations. arXiv:2001.02456 ). Results of such kind are referred to as extension theorems. After a brief introduction, we offer a uniform approach to both types of extensions based on the idea to extend the extension procedure itself. We propose a generalization of the standard concept of first-order interpretations in which functional and relational symbols are interpreted rather by ultrafilters over sets of functions and relations than by functions and relations themselves, and define ultrafilter models with an appropriate semantics for them. We provide two specific operations which turn ultrafilter models into ordinary models, establish necessary and sufficient conditions under which the latter are the two canonical ultrafilter extensions of some ordinary models, and obtain a topological characterization of ultrafilter models. We generalize a restricted version of the extension theorem to ultrafilter models. To formulate the full version, we propose a wider concept of ultrafilter models with their semantics based on limits of ultrafilters, and show that the former concept can be identified, in a certain way, with a particular case of the latter; moreover, the new concept absorbs the ordinary concept of models. We provide two more specific operations which turn ultrafilter models in the narrow sense into ones in the wide sense, and establish necessary and sufficient conditions under which ultrafilter models in the wide sense are the images of ones in the narrow sense under these operations, and also are two canonical ultrafilter extensions of some ordinary models. Finally, we establish three full versions of the extension theorem for ultrafilter models in the wide sense. The results of the first three sections of this paper were partially announced in Poliakov and Saveliev (in: Kennedy, de Queiroz (eds) On two concepts of ultrafilter extensions of first-order models and their generalizations, Springer, Berlin, 2017).