On the structure of nonarchimedean exponential fields I

Given an ordered fieldK, we compute the natural valuation and skeleton of the ordered multiplicative group (K ^>0, ·, 1, <) in terms of those of the ordered additive group (K,+,0,<). We use this computation to provide necessary and sufficient conditions on the value groupv(K) and residue field , for theL -equivalence of the above mentioned groups. We then apply the results to exponential fields, and describev(K) in that case. Finally, ifK is countable or a power series field, we derive necessary and sufficient conditions onv(K) and forK to be exponential. In the countable case, we get a structure theorem forv(K).This paper represents some results of the author's doctoral thesisThis paper was written while the author was supported by a research grant from the University of Heidelberg     

Uniform distribution and Hardy fields

Under certain regularity conditions on the growth at infinity of a real functionf (belonging to some Hardy field), asymptotic distribution modulo 1 of the values off is studied. In particular, necessary and sufficient conditions (in terms of the extent to whichf can be approximated by rational polynomals) are provided for the sequence {f(n)} _n≥1 to be uniformly distributed or dense modulo 1. Supported in part by NSF-DMS-9003450.     

Operator theory in the Hardy space over the bidisk (I)

In this paper, we identify the vector valued Hardy space with the Hardy space over the bidisk and construct a universal model for thecontractive analytic functions. We will also study some elementary properties of the submodules and show, in some cases, how the operator theoretical properties are related to the module theoretical properties. The last part focus on the study of double commutativity of compression operators.     

Independence relations for exponential fields

We give four different independence relations on any exponential field. Each is a canonical independence relation on a suitable Abstract Elementary Class of exponential fields, showing that two of these are NSOP₁-like and non-simple, a third is stable, and the fourth is the quasiminimal pregeometry of Zilber's exponential fields, previously known to be stable (and uncountably categorical). We also characterise the fourth independence relation in terms of the third, strong independence.     

Locally solvable vector fields and Hardy spaces

We characterize the boundary value of homegeneous solutions of planar one-sided locally solvable vector fields with analytic coefficients with the property that the Lp norm of their traces is locally uniformly bounded, 0<p?1. For p≠1/n, , the boundary value must locally belong to the local Hardy space hp(R) of Goldberg while for p=1/n, , the answer calls for a new class of atomic Hardy spaces if the vector field is of infinite type at some boundary point.     

Function theory in real Hardy spaces

We show that many classical results in Hardy space theory have exact analogues when the Fourier coefficients are restricted to be real. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Mixed exponential sums over finite fields

In this paper we calculate the conductor of a character that consists of the product of an additive and a multiplicative character. This computation improves the bound for exponential sums given by G. I. Perelmuter. This calculation gives an easy method to compute the conductor associated to a character of the Galois group of the composite of an Artin-Schreier extension and a Kummer extension.

     

Hardy type inequalities with exponential weights for a class of convolution operators

In this paper we consider operators of the form H=λ(-i∇), with λ analytic in a strip and with some specific growth conditions at infinity, and prove Hardy type estimates in L ²(ℝ ⁿ ) with exponential weights. In fact we extend our previous results [19] from bounded analytic functions on a strip to analytic functions with polynomial growth in that strip.     

Multipliers on Herz type Hardy spaces over local fields

In this paper, we establish two multiplier theorems for Herz type Hardy spaces, and as an application, we discuss the boundedness of pseudo-differential operators in these spaces.     

Multipliers on Herz type Hardy spaces over local fields

In this paper, we establish two multiplier theorems for Herz type Hardy spaces, and as an application, we discuss the boundedness of pseudo-differential operators in these spaces.     

Valuation theory: A constructive view

The theory of valuations on fields is developed in the constructive spirit of Errett Bishop. As a consequence of the general theory we are able to construct all nonarchimedean valuations on algebraic number fields and compute their ramification indices and residue class degrees. The notion of a field with a valuation for which the infimum of the values of any polynomial function can be computed plays an important role. Numerous limiting counterexamples are provided.     

Integration of locally exponential Lie algebras of vector fields

A locally convex Lie algebra is said to be locally exponential if it belongs to some local Lie group in canonical coordinates. In this note we give criteria for locally exponential Lie algebras of vector fields on an infinite-dimensional manifold to integrate to global Lie group actions. Moreover, we show that all necessary conditions are satisfied if the manifold is finite-dimensional connected and σ-compact, which leads to a generalization of Palais’ Integrability Theorem.      

Works of M. I. Yadrenko in the theory of random fields

A survey of the works of M. I. Yadrenko in the theory of random fields on finite-dimensional and infinite-dimensional spaces is presented.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 11, pp. 1460–1465, November. 1992.