Related papers: Analytic Hardy fields
We show that all maximal Hardy fields are elementarily equivalent as differential fields to the differential field $\mathbb T$ of transseries, and give various applications of this result and its proof.
We discuss the conjecture that every maximal Hardy field has the Intermediate Value Property for differential polynomials, and its equivalence to the statement that all maximal Hardy field are elementarily equivalent to the differential…
Differentially algebraic Hardy field extensions of short Hardy fields are short. This is proved in the more general setting of $H$-fields. As an application we extend a theorem of Rosenlicht (1981) by showing that each short asymptotic…
Let $\mathbb{T}$ be the differential field of logarithmic-exponential transseries. We consider the expansion of $\mathbb{T}$ by the binary map that sends a real number $r$ and a positive transseries $f$ to the transseries $f^r$. Building on…
We show that all maximal Hardy fields are elementarily equivalent as differential fields, and give various applications of this result and its proof. We also answer some questions on Hardy fields posed by Boshernitzan.
We show how to fill "countable" gaps in Hardy fields. We use this to prove that any two maximal Hardy fields are back-and-forth equivalent.
We define the universal exponential extension of an algebraically closed differential field and investigate its properties in the presence of a nice valuation and in connection with linear differential equations. Next we prove normalization…
Every maximal Hardy field has a proper elementary differential subfield that is Dedekind complete in the maximal Hardy field. This pair of Hardy fields is a transserial tame pair, shown to have a complete and model complete elementary…
We show that the natural embedding of the differential field of transseries into Conway's field of surreal numbers with the Berarducci-Mantova derivation is an elementary embedding. We also prove that any Hardy field embeds into the field…
We define analytic indices which involve the eta form and the analytic torsion form. We show that these indices are independent of the geometric choices made in their definitions, and hence are topological in nature.
We study relative differential closure in the context of Hardy fields. Using our earlier work on algebraic differential equations over Hardy fields, this leads to a proof of a conjecture of Boshernitzan (1981): the intersection of all…
We develop the theory of variable exponent Hardy spaces. Analogous to the classical theory, we give equivalent definitions in terms of maximal operators. We also show that distributions in these spaces have an atomic decomposition including…
We show that every Hardy field extends to an $\omega$-free Hardy field. This result relates to classical oscillation criteria for second-order homogeneous linear differential equations. It is essential in [10], and here we apply it to…
We prove that the Hausdorff dimension of the set of badly approximable systems of m linear forms in n variables over the field of Laurent series with coefficients from a finite field is maximal. This is a analogue of Schmidt's…
Let $K$ be a field of characteristic zero complete with respect to a non-trivial, non-Archimedean valuation. We relate the sheaf $\widehat{\mathcal{D}}$ of infinite order differential operators on smooth rigid $K$-analytic spaces to the…
Let $X$ be a separable Banach function space on the unit circle $\mathbb{T}$ and $H[X]$ be the abstract Hardy space built upon $X$. We show that the set of analytic polynomials is dense in $H[X]$ if the Hardy-Littlewood maximal operator is…
We construct a Hardy field that contains Ilyashenko's class of germs at infinity of almost regular functions as well as all log-exp-analytic germs. In addition, each germ in this Hardy field is uniquely characterized by an asymptotic…
In quantum logical terms, Hardy-type arguments can be uniformly presented and extended as collections of intertwined contexts and their observables. If interpreted classically those structures serve as graph-theoretic "gadgets" that enforce…
We present a natural family of Hilbert function spaces on the d-dimensional complex unit ball and classify which of them satisfy that subsets of the ball yield isometrically isomorphic subspaces if and only if there is an analytic…
We describe recent nonlinear analytic approximation tools in the classical setting of Hardy spaces in the upper half plane and show how to transfer them to the higher dimensional real setting of harmonic functions in upper half spaces. It…