Pertti Mattila

Spherical averages of Fourier transforms of measures with finite energy; dimensions of intersections and distance sets

Let μ, be a positive Radon measure with compact support in the euclidean n -space ℝ n . Introducing the Fourier transform and the averages over the spheres we can write the α-energy, 0 α n , of μ as where the positive constants c 1 and c 2 depend only on n and α. The second equality is based on the Plancherel formula and the fact that where .

Removable sets for Lipschitz harmonic functions in the plane

The main motivation for this work comes from the century-old Painleve problem: try to characterize geometrically removable sets for bounded analytic functions in C.

Lipschitz equivalence of subsets of self-conformal sets

We give sufficient conditions to guarantee that if two self-conformal sets E and F have Lipschitz equivalent subsets of positive measure, then there is a bilipschitz map of E into, or onto, F.

Strong Marstrand theorems and dimensions of sets formed by subsets of hyperplanes

We present strong versions of Marstrands projection theorems and other related theorems. For example, if E is a plane set of positive and finite s-dimensional Hausdorff measure, there is a set X of directions of Lebesgue measure 0, such that the projection onto any line with direction outside X, of any subset F of E of positive s-dimensional measure, has Hausdorff dimension min(1,s), i.e. the set of exceptional directions is independent of F. Using duality this leads to results on the dimension of sets that intersect families of lines or hyperplanes in positive Lebesgue measure.

Boundedness and convergence for singular integrals of measures separated by Lipschitz graphs

We shall consider the truncated singular integral operators T_{\mu, K}^{\epsilon}f(x)=\int_{\mathbb{R}^{n}\setminus B(x,\epsilon)}K(x-y)f(y)d\mu y and related maximal operators

Recent Progress on Dimensions of Projections

T_{\mu,K}^{\ast}f(x)=\underset{\epsilon >0}{\sup}| T_{\mu,K}^{\epsilon}f(x)|

Singular Integrals on Self-similar Subsets of Metric Groups

. We shall prove for a large class of kernels

AHLFORS-DAVID REGULAR SETS AND BILIPSCHITZ MAPS

K

Hausdorff dimension and capacities of intersections of sets in n-space

and measures

Projection and slicing theorems in Heisenberg groups

\mu