0000000000083845
AUTHOR
Thomas Zürcher
Space-filling vs. Luzin's condition (N)
Let us assume that we are given two metric spaces, where the Hausdorff dimension of the first space is strictly smaller than the one of the second space. Suppose further that the first space has sigma-finite measure with respect to the Hausdorff measure of the corresponding dimension. We show for quite general metric spaces that for any measurable surjection from the first onto the second space, there is a set of measure zero that is mapped to a set of positive measure (both measures are the Hausdorff measures corresponding to the Hausdorff dimension of the first space). We also study more general situations where the measures on the two metric spaces are not necessarily the same and not ne…
Generalized Hausdorff dimension distortion in Euclidean spaces under Sobolev mappings
Abstract We investigate how the integrability of the derivatives of Orlicz–Sobolev mappings defined on open subsets of R n affect the sizes of the images of sets of Hausdorff dimension less than n. We measure the sizes of the image sets in terms of generalized Hausdorff measures.
A Stieltjes Approach to Static Hedges
Static hedging of complicated payoff structures by standard instruments becomes increasingly popular in finance. The classical approach is developed for quite regular functions, while for less regular cases, generalized functions and approximation arguments are used. In this note, we discuss the regularity conditions in the classical decomposition formula due to P. Carr and D. Madan (in Jarrow ed, Volatility, pp. 417–427, Risk Publ., London, 1998) if the integrals in this formula are interpreted as Lebesgue integrals with respect to the Lebesgue measure. Furthermore, we show that if we replace these integrals by Lebesgue–Stieltjes integrals, the family of representable functions can be exte…
Generalized Dimension Distortion under Mappings of Sub-Exponentially Integrable Distortion
We prove a dimension distortion estimate for mappings of sub-exponentially integrable distortion in Euclidean spaces, which is essentially sharp in the plane.
Generalized dimension distortion under planar Sobolev homeomorphisms
We prove essentially sharp dimension distortion estimates for planar Sobolev-Orlicz homeomorphisms.
Higher Order Sobolev-Type Spaces on the Real Line
This paper gives a characterization of Sobolev functions on the real line by means of pointwise inequalities involving finite differences. This is also shown to apply to more general Orlicz-Sobolev, Lorentz-Sobolev, and Lorentz-Karamata-Sobolev spaces.