Rectifiability and analytic capacity in the complex plane

Analytic capacity and removable sets In this chapter we shall discuss a classical problem in complex analysis and its relations to the rectifiability of sets in the complex plane C . The problem is the following: which compact sets E ⊃ C are removable for bounded analytic functions in the following sense? (19.1) If U is an open set in C containing E and f : U\E → C is a bounded analytic function, then f has an analytic extension to U . This problem has been studied for almost a century, but a geometric characterization of such removable sets is still lacking. We shall prove some partial results and discuss some other results and conjectures. For many different function classes a complete so…

Hausdorff measures and dimension

Covering and differentiation

Dimension of a measure

Intersections with planes

Suuret teknologiat : miten teknologiaa kuvattiin Helsingin Sanomien uutisoinnissa ja kirjoittelussa vuosina 1889-1930

Tutkimuksessa tarkastellaan teknologian historiaa 1900-luvun vaihteen Suo-messa. Päälähteenä toimii vuodesta 1889 lähtien julkaistu sanomalehti, Helsingin Sanomat. Tutkimuksessa selvitetään aineistolähtöisen grounded theory -menetelmän avulla niitä piirteitä ja tapoja, joilla uusia yhteiskuntaan merkittävästi vaikuttaneita teknologioita kuvattiin. Suuria teknologioita, joilla tarkoitetaan merkittäviä yhteiskuntaan juurtuneita teknologioita, alettiin kuvata ajankohtaisemmin ja niitä yhteiskuntaan liittäen 1930 – lukua lähestyttäessä. Tämä oli seurausta teknologian yleistymisestä ja aikaisempaa merkittävämmästä näkyvyydestä yhteiskunnassa. Teknologian kuvaaminen muuttui selkeimmin auton kohda…

The packing dimension of projections and sections of measures

AbstractWe show that for a probability measure μ on ℝnfor almost all m–dimensional subspaces V, provided dimH μ≤m. Here projv denotes orthogonal projection onto V, and dimH and dimp denote the Hausdorff and packing dimension of a measure. In the case dimH μ > m we show that at μ-almost all points x the slices of μ by almost all (n − m)-planes Vx through x satisfyWe give examples to show that these inequalities are sharp.

Other measures and dimensions

Rectifiability, weak linear approximation and tangent measures

Principal Values of Cauchy Integrals, Rectifiable Measures and Sets

The extensive studies started by A. P. Calderon in the sixties and continued by many authors up today have revealed that the Cauchy integrals $$ {C_{\Gamma }}f(z) = \int_{\Gamma } {\frac{{f\left( \zeta \right)d\zeta }}{{\zeta - z}}} $$ behave very well on sufficiently regular, not necessarily smooth, curves F, see [CCFJR], [D] and [MT].

Rectifiability and singular integrals

Local structure of s-dimensional sets and measures

Rectifiable sets and approximate tangent planes

Rectifiable Measures in R n and Existence of Principal Values for Singular Integrals

Linear Approximation Property, Minkowski Dimension, and Quasiconformal Spheres

Intersections of general sets

Tangent measures and densities

Energies, capacities and subsets of finite measure

Density theorems for Hausdorff and packing measures

Singular integrals, analytic capacity and rectifiability

In this survey we study some interplay between classical complex analysis (removable sets for bounded analytic functions), harmonic analysis (singular integrals), and geometric measure theory (rectifiability).

Tangent Measures, Densities, and Singular Integrals

We introduce tangent measures in the sense of David Preiss. We discuss their applications to the density and rectifiability properties of general Borel measures in ℝ n as well as to the behaviour of certain singular integrals with respect to such measures.

Rectifiability and densities

Singular integrals and rectifiability

We shall discuss singular integrals on lower dimensional subsets of Rn. A survey of this topic was given in [M4]. The first part of this paper gives a quick review of some results discussed in [M4] and a survey of some newer results and open problems. In the second part we prove some results on the Riesz kernels in Rn. As far as I know, they have not been explicitly stated and proved, but they are very closely related to some earlier results and methods. [Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial (Madrid), 2002].

Hausdorff measures, Hölder continuous maps and self-similar fractals

Let f: A → ℝn be Hölder continuous with exponent α, 0 < α ≼ 1, where A ⊂ ℝm has finite m-dimensional Lebesgue measure. Then, as is easy to see and well-known, the s-dimensional Hausdorif measure HS(fA) is finite for s = m/α. Many fractal-type sets fA also have positive Hs measure. This is so for example if m = 1 and f is a natural parametrization of the Koch snow flake curve in ℝ2. Then s = log 4/log 3 and α = log 3/log 4. In this paper we study the question of what s-dimensional sets in can intersect some image fA in a set of positive Hs measure where A ⊂ ℝm and f: A → ℝn is (m/s)-Hölder continuous. In Theorem 3·3 we give a general density result for such Holder surfacesfA which implies…

Menger curvature and $C^{1}$ regularity of fractals

General measure theory

Measure and dimension functions: measurability and densities

During the past several years, new types of geometric measure and dimension have been introduced; the packing measure and dimension, see [Su], [Tr] and [TT1]. These notions are playing an increasingly prevalent role in various aspects of dynamics and measure theory. Packing measure is a sort of dual of Hausdorff measure in that it is defined in terms of packings rather than coverings. However, in contrast to Hausdorff measure, the usual definition of packing measure requires two limiting procedures, first the construction of a premeasure and then a second standard limiting process to obtain the measure. This makes packing measure somewhat delicate to deal with. The question arises as to whe…