We introduce tangent measures in the sense of David Preiss. Unable to display preview. Download preview PDF. Skip to main content. Advertisement Hide. Tangent Measures, Densities, and Singular Integrals. Conference paper. This is a preview of subscription content, log in to check access. Bedford and A.
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Pertti Mattila. Now in paperback, the main theme of this book is the study of geometric properties of general sets and measures in euclidean spaces. Applications of this theory include fractal-type objects such as strange attractors for dynamical systems and those fractals used as models in the sciences.
The author provides a firm and unified foundation and develops all the necessary main tools, such as covering theorems, Hausdorff measures and their relations to Riesz capacities and Fourier transforms.
The last third of the book is devoted to the Beisovich-Federer theory of rectifiable sets, which form in a sense the largest class of subsets of euclidean space posessing many of the properties of smooth surfaces. These sets have wide application including the higher-dimensional calculus of variations. Their relations to complex analysis and singular integrals are also studied.
Essentially self-contained, this book is suitable for graduate students and researchers in mathematics. General measure theory. Covering and differentiation. Hausdorff measures of level sets. The lower density of Lipschitz images. Remarks on Lipschitz maps.
Tangent Measures, Densities, and Singular Integrals
Capacities and Hausdorff measures. Frostmans lemma in Rn. Dimensions of product sets. Weighted Hausdorff measures.
Frostmans lemma in compact metric spaces. Existence of subsets with finite Hausdorff measure. Orthogonal projections capacities and Hausdorff dimension. Selfsimilar sets with overlap.
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Plane sections capacities and Hausdorff measures. Conical densities. Hausdorff dimension and capacities of intersections. Examples and remarks.
Rectifiable Sets, Densities and Tangent Measures (Paperback)
Preliminary results on tangent measures. Densities and tangent measures. Marstrands theorem. A metric on measures.
Tangent measures to tangent measures are tangent measures. Proof of Theorem Linear approximation properties. Rectifiability and measures in cones. Approximate tangent planes. Remarks on rectifiability.
- Density bounds and tangent measures.
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Uniform rectifiability. Weak linear approximation densities and projections. Rectifiability and tangent measures. Rectifiability and density one. Rectifiability and packing measures. Remarks on projections.