HAUSDORFF AND PACKING DIMENSIONS, INTERSECTION MEASURES, AND SIMILARITIES
Maarit Järvenpää, Universityof Jy · 1999
Let µ and ν be Radon measures on R n with compact supports. We study the Hausdorff, dimH, and packing dimension, dimp, properties of the intersection measures µ ∩ f♯ν when f runs through the similarities of R n and f♯ν is the image of ν under f. These measures can be regarded as natural measures on spt µ ∩ f(spt ν), where spt is the support of a measure. Using the relations between Hausdorff dimensions of sets and measures, we show that if dimH(µ × ν)= dimH µ +dimH ν>n and if the t-energy of ν is finite for all 0 0} =dimH µ +dimH ν − n. Here θn is the unique orthogonally invariant Radon probability measure on the orthogonal group of R n, denoted by On, L 1 is the Lebesgue measure on the open interval (0, ∞),and τz◦g◦δr: R n → R n is the similarity τz ◦g◦δr(a) =rga+z. By relating packing dimensions of intersection measures to certain integral kernels, we prove that if the s-energy of µ is finite and the t-energy of ν is finite for some 0 n, then for θn ×L 1 almost all (g, r) ∈ On ×(0, ∞) we have ess inf{dimp µ ∩ (τz ◦ g ◦ δr)♯ν: z ∈ R n with µ ∩ (τz ◦ g ◦ δr)♯ν(R n)> 0} = dµ,ν, where dµ,ν is a constant depending onlyon the measures µ and ν. We also deduce corresponding equalities for the upper Hausdorff and upper packing dimensions