Rational Approximation and Weak Analyticity. I.
Rational Approximation and Weak Analyticity. II.
Rectifiability, analytic capacity, and singular integrals.
Rectifiability and parameterization of intrinsic regular surfaces in the Heisenberg group
We construct an intrinsic regular surface in the first Heisenberg group equipped wiht its Carnot-Carathéodory metric which has euclidean Hausdorff dimension . Moreover we prove that each intrinsic regular surface in this setting is a -dimensional topological manifold admitting a -Hölder continuous parameterization.
Rectifiability of solutions of the one-dimensional -Laplacian.
Regularity of sets with constant intrinsic normal in a class of Carnot groups
In this Note, we define a class of stratified Lie groups of arbitrary step (that are called “groups of type ” throughout the paper), and we prove that, in these groups, sets with constant intrinsic normal are vertical halfspaces. As a consequence, the reduced boundary of a set of finite intrinsic perimeter in a group of type is rectifiable in the intrinsic sense (De Giorgi’s rectifiability theorem). This result extends the previous one proved by Franchi, Serapioni & Serra Cassano in step...
Remarks on the n-dimensional geometric measure of compacta
Remarks on the nonexistence of doubling measures.
Removable sets for Lipschitz harmonic functions in the plane.
The main motivation for this work comes from the century-old Painlevé problem: try to characterize geometrically removable sets for bounded analytic functions in C.
Research Article. Multiscale Analysis of 1-rectifiable Measures II: Characterizations
A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures μ in n-dimensional Euclidean space for all n ≥ 2 in terms of positivity of the lower density and finiteness of a geometric square function, which loosely speaking, records in an L2 gauge the extent to which μ admits approximate tangent lines, or has rapidly growing density ratios, along its support. In contrast with the classical theorems...
Restricted Radon transforms and unions of hyperplanes.
We study Lp(Rn) → Ldμ(σ)α,∞(Ldt∞) estimates for the Radon transform in certain cases where the dimension of the measure μ on Σ(n-1) is less than n-1.