Let E ⊂ C be a Borel set with finite length, that is, 0 < H 1 (E) < ∞. By a theorem of David and Léger, the L 2 (H 1 ⌊E)-boundedness of the singular integral associated to the Cauchy kernel (or even to one of its coordinate parts x/|z| 2 , y/|z| 2 , z = (x, y) ∈ C) implies that E is rectifiable. We extend this result to any kernel of the form x 2n−1 /|z| 2n , z = (x, y) ∈ C, n ∈ N. We thus provide the first non-trivial examples of operators not directly related with the Cauchy transform whose L 2 -boundedness implies rectifiability.2010 Mathematics Subject Classification. Primary 42B20, 42B25. Key words and phrases. Calderón-Zygmund singular integrals, rectifiability. Most of this work had been carried out in the first semester of 2011 while V.C was visiting the Centre de Recerca Matemàtica in Barcelona and he feels grateful for the hospitality.. David showed in [D1] and [D2] that all such singular integrals are bounded in L 2 (µ) when µ is d-uniformly rectifiable. In the other direction David and Semmes proved that the L 2 (µ)-boundedness of all singular integrals in the class described above forces the measure µ to be d-uniformly rectifiable. The fundamental question they posed reads as follows:

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In this paper we study fractal solutions of linear and nonlinear dispersive PDE on the torus. In the first part we answer some open questions on the fractal solutions of linear Schrödinger equation and equations with higher order dispersion. We also discuss applications to their nonlinear counterparts like the cubic Schrödinger equation (NLS) and the Korteweg-de Vries equation (KdV).In the second part, we study fractal solutions of the vortex filament equation and the associated Schrödinger map equation (SM). In particular, we construct global strong solutions of the SM in H s for s > 3 2 for which the evolution of the curvature is given by a periodic nonlinear Schrödinger evolution. We also construct unique weak solutions in the energy level. Our analysis follows the frame construction of Chang et al. [9] and Nahmod et al. [26].

Let G be any Carnot group. We prove that, if a subset of G is contained in a rectifiable curve, then it satisfies Peter Jones' geometric lemma with some natural modifications. We thus prove one direction of the Traveling Salesman Theorem in G. Our proof depends on new Alexandrov-type curvature inequalities for the Hebisch-Sikora metrics. We also apply the geometric lemma to prove that, in every Carnot group, there exist −1-homogeneous Calderón-Zygmund kernels such that, if a set E ⊂ G is contained in a 1-regular curve, then the corresponding singular integral operators are bounded in L 2 (E). In contrast to the Euclidean setting, these kernels are nonnegative and symmetric.

We prove that if µ is a Radon measure on the Heisenberg group H n such that the density Θ s (µ, ·), computed with respect to the Korányi metric dH , exists and is positive and finite on a set of positive µ measure, then s is an integer. The proof relies on an analysis of uniformly distributed measures on (H n , dH ). We provide a number of examples of such measures, illustrating both the similarities and the striking differences of this sub-Riemannian setting from its Euclidean counterpart.