Craig A. Nolder

Inequalities for differential forms

Hardy-Littlewood Inequalities.- Norm Comparison Theorems.- Poincare-type inequalities.- Caccioppoli Inequalities.- Imbedding Theorems.- Reverse Holder Inequaltiies.- Inequalities for Operators.- Estimates for Jacobians.- Lipschitz and BMO norms.- References.- Index

A-Harmonic Equations and the Dirac Operator

We show how -harmonic equations arise as components of Dirac systems. We generalize -harmonic equations to -Dirac equations. Removability theorems are proved for solutions to -Dirac equations.

Ls(μ)-averaging domains

Abstract In this paper we first introduce L s ( μ )-averaging domains which are generalizations of L s -averaging domains introduced by S.G. Staples. We characterize L s ( μ )-averaging domains using the quasihyperbolic metric. As applications, we prove norm inequalities for conjugate A -harmonic tensors in L s ( μ )-averaging domains which can be considered as generalizations of the Hardy and Littlewood theorem for conjugate harmonic functions. Finally, we give applications to quasiconformal and quasiregular mappings.

A quasiregular analogue of a theorem of Hardy and Littlewood

Suppose that f is analytic in the unit disk. A theorem of Hardy and Littlewood relates the Holder continuity of f over the unit disk to the growth of the derivative. We prove here a quasiregular analogue of this result in certain domains in n-dimensional space. We replace values of the derivatives with a local integral average. In the process we generalize a result on the continuity of quasiconformal mappings due to Nakki and Palka

Reverse Hölder inequalities

In this chapter, we will present various versions of the reverse Holder inequality which serve as powerful tools in mathematical analysis. The original study of the reverse Holder inequality can be traced back in Muckenhoupt–s work in [145]. During recent years, different versions of the reverse Holder inequality have been established for different classes of functions, such as eigenfunctions of linear second-order elliptic operators [281], functions with discrete-time variable [282], and continuous exponential martingales [119].

A characterization of certain measures using quasiconformal mappings

Suppose that ,u is a finite positive measure on the unit disk. Carleson showed that the L 2(j)-norm is bounded by the H -norm uniformly over the space of analytic functions on the unit disk if and only if ,u is a Carleson measure. Analogues of this result exist for Bergmann spaces of analytic functions in the disk and in the unit ball of Cn . We prove here real variable analogues of certain Bergmann space results using quasiconformal and quasiregular mappings. 1.

Subordination, self-similarity, and option pricing.

We use additive processes to price options on the Standard and Poors 500 index (SPX) for the sake of comparison of pricing performance across both model class and family of time-one distribution. Each of the additive processes in this study is defined using one of the following: subordination, Satos (2002) construction of self-similar additive processes from self-decomposable distributions, or both. We find that during the year 2005: (1) for a given family of time-one distributions, four-parameter self-similar additive models consistently yielded lower pricing errors than those of four-parameter subordinated, and time-inhomogeneous additive models, (2) for a given class of additive models, the time-one marginal given by the normal inverse Gaussian distribution consistently yielded lower pricing errors than those of the variance gamma distribution. Market and model benchmarks for the additive models under consideration are obtained via the bid-ask spreads of the options and Levy stochastic volatility model prices, respectively.

Sensitivities of options via Malliavin calculus: applications to markets of exponential Variance Gamma and Normal Inverse Gaussian processes

This paper presents new sensitivities for options when the underlying follows an exponential Lévy process, specifically Variance Gamma and Normal Inverse Gaussian processes. The calculation of these sensitivities is based on a finite-dimensional Malliavin calculus and finite difference methods via Monte-Carlo simulations. In order to compare the real performance of this method we use the inverse Fourier method to calculate the exact values of the sensitivities of European call and digital options written on the S&P 500 index. Our results show that variations of the localized Malliavin calculus approach outperform the finite difference method in calculations of the Greeks and the new sensitivities that we introduce.

An ^{} definition of interpolating Blaschke products

We give a new characterization of interpolating Blaschke products in terms of LP-norms of their reciprocals. We also obtain a characterization of finite unions of interpolating sequences.

Local distortion of M-conformal mappings

A conformal mapping in a plane domain locally maps circles to circles. More generally, quasiconformal mappings locally map circles to ellipses of bounded distortion. In this work, we study the corresponding situation for solutions to Stein-Weiss systems in the ( n + 1 ) D Euclidean space. This class of solutions is a transformation of a subset of monogenic locally quasiconformal mappings with nonvanishing Jacobian. In the theoretical part of this work, we prove that an M-conformal mapping locally maps the unit hypersphere onto explicitly characterized hyperellipsoids and vice versa. Then we discuss quasiconformal radial mappings and their relations with the Cauchy kernel and p-monogenic mappings. This is followed by the consideration of quadratic M-conformal mappings. In the applications part of this work, we provide the reader with some plot examples that demonstrate the effectiveness of our approach.