For a set E of positive and finite length, we prove that if the Huovinen transform (the convolution operator with kernel zk/|z|k+1 for an odd number k) associated to exists in principal value, then is rectifiable.

We introduce a notion of sparse rectifiability, stronger than that of uniform rectifiability. As applications we derive, firstly, results ensuring the convergence of the total variation measures |μ| subject to the weak* convergence of the sparsely rectifiable Radon measures μ. Secondly, we apply sparse rectifiability to derive compactness results for special functions of bounded variation (SBV)...

Introduction. The object of this note is to present a short and simple proof of the following two theorems. Theorem 1. Suppose that C is a closed rectifiable Jordan curve in the complex l-plane and that C has a continuously turning tangent in a neighborhood of one of its points, XoSuppose, furthermore, that the tangent angle t(s) as function of the arc length s has a modulus of continuity (¡)(t...

Local volume-constrained minimizers in anisotropic capillarity problems develop free boundaries on the walls of their containers. We prove the regularity of the free boundary outside a closed negligible set, showing in particular the validity of Young’s law at almost every point of the free boundary. Our regularity results are not specific to capillarity problems, and actually apply to sets of ...

Geographical curves are so involved in their detail that their lengths are often infinite or, rather, undefinable. However, many are statistically "selfsimilar," meaning that each portion can be considered a reduced-scale image of the whole. In that case, the degree of complication can be described by a quantity D that has many properties of a "dimension," though it is fractional; that is, it e...

Let E ⊂ Rn+1, n ≥ 2, be a uniformly rectifiable set of dimension n. Then bounded harmonic functions in Ω := Rn+1 \ E satisfy Carleson measure estimates, and are “ε-approximable”. Our results may be viewed as generalized versions of the classical F. and M. Riesz theorem, since the estimates that we prove are equivalent, in more topologically friendly settings, to quantitative mutual absolute con...

We show that any simply connected rectifiable domain Ω can be decomposed into Lipschitz crescents using only crosscuts of the domain and using total length bounded by a multiple of the length of ∂Ω. In particular, this gives a new proof of a theorem of Peter Jones that such a domain can be decomposed into Lipschitz disks. 1991 Mathematics Subject Classification. Primary: 68U05 Secondary: 26B15,...