The Riesz Transform, Rectifiability, and Removability for Lipschitz Harmonic Functions

Quasicompact and Riesz unital endomorphisms of real Lipschitz algebras of complex-valued functions

We first show that a bounded linear operator $ T $ on a real Banach space $ E $ is quasicompact (Riesz, respectively) if and only if $T': E_{mathbb{C}}longrightarrow E_{mathbb{C}}$ is quasicompact  (Riesz, respectively), where the complex Banach space $E_{mathbb{C}}$ is a suitable complexification of $E$ and $T'$ is the complex linear operator on $E_{mathbb{C}}$ associated with $T$. Next, we pr...

THE WEAK-A∞ PROPERTY OF HARMONIC AND p-HARMONIC MEASURES IMPLIES UNIFORM RECTIFIABILITY

Let E ⊂ Rn+1, n ≥ 2, be an Ahlfors-David regular set of dimension n. We show that the weak-A∞ property of harmonic measure, for the open set Ω := Rn+1 \ E, implies uniform rectifiability of E. More generally, we establish a similar result for the Riesz measure, p-harmonic measure, associated to the p-Laplace operator, 1 < p < ∞.

On the Uniform Rectifiability of Ad Regular Measures with Bounded Riesz Transform Operator: the Case of Codimension

We prove that if μ is a d-dimensional Ahlfors-David regular measure in R, then the boundedness of the d-dimensional Riesz transform in L(μ) implies that the non-BAUP David-Semmes cells form a Carleson family. Combined with earlier results of David and Semmes, this yields the uniform rectifiability of μ.

Extending Functions in the Model Subspaces of H^2(R) to C

Uniform Rectifiability, Carleson Measure Estimates, and Approximation of Harmonic Functions

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...