Variation formulas for H-rectifiable sets

Information Theory on Rectifiable Sets

Entropy and—maybe even more so—mutual information are invaluable tools for analyzing properties of probability distributions, especially in coding theory. While there are general definitions for both concepts available for arbitrary probability distributions, these tend to be hard to work with and the literature (e.g., [CT06]) focuses on either discrete, or continuous random variables. In this ...

Lp-ESTIMATES FOR THE VARIATION FOR SINGULAR INTEGRALS ON UNIFORMLY RECTIFIABLE SETS

The L (1 < p < ∞) and weak-L estimates for the variation for CalderónZygmund operators with smooth odd kernel on uniformly rectifiable measures are proven. The L boundedness and the corona decomposition method are two key ingredients of the proof.

Integration formulas for the conditional transform involving the first variation

In this paper, we show that the conditional transform with respect to the Gaussian process involving the first variation can be expressed in terms of the conditional transform without the first variation. We then use this result to obtain various integration formulas involving the conditional $diamond$-product and the first variation.

Non-rectifiable Limit Sets of Dimension One

Asymptotics of Best-packing on Rectifiable Sets

We investigate the asymptotic behavior, as N grows, of the largest minimal pairwise distance of N points restricted to an arbitrary compact rectifiable set embedded in Euclidean space, and we find the limit distribution of such optimal configurations. For this purpose, we compare best-packing configurations with minimal Riesz s-energy configurations and determine the s-th root asymptotic behavi...