In our previous paper we studied some questions related to the (sic)(1) and Lipschitz harmonic capacities. A serious error was found in the arguments. In this note we explain how this error which we have not been able to fix, affects the results claimed in that paper.

In our previous paper we studied some questions related to the C1 and Lipschitz harmonic capacities. A serious error was found in the arguments. In this note we explain how this error, which we have not been able to fix, affects the results claimed in that paper.

The Lipschitz and C^1 harmonic capacities K and K_c in R^n can be considered as high-dimensional versions of the so-called analytic and continuous analytic capacities G and A (respectively). In this paper we provide a dual characterization of K_c in the spirit of the classical one for the capacity A by means of the Garabedian function.

If mu is a finite complex measure in the complex plane C we denote by C-mu its Cauchy integral defined in the sense of principal value. The measure it is called reflectionless if it is continuous (has no atoms) and C-mu = 0 at mu-almost every point. We show that if mu is reflectionless and its Cauchy maximal function C-*(mu) is summable with respect to |mu| then mu is trivial. An example of a reflectionless measure whose maximal function belongs to the "weak" L-1 is also constructed, proving that the above result is sharp in its scale. We also give a partial geometric description of the set of reflectionless measures on the line and discuss connections of our results with the notion of sets of finite perimeter in the sense of De Giorgi.

We characterize the systems of translates of the Poisson kernel spanning L-P(R), 1 <= p < infinity. An equivalent formulation in terms of discrete uniqueness sets for harmonic functions is given, together with a Blaschke-type condition for the zero variety of bounded harmonic functions in the unit disk.

We obtain the complete characterization of those domains G ⊂ ℂ which admit the so-called estimate of the Cauchy integral, that is to say, $$\left| {\smallint _{\partial G} {\text{f}}\left( z \right)dz} \right| \leqslant C\left( G \right)\left\| {\text{f}} \right\|\infty \gamma \left( E \right)$$ for all E ⊂ G and f ∊ H∞ (G E), where γ\(E) is the analytic capacity of E. The corresponding result for continuous functions f and the continuous analytic capacity α(E) is also proved.

For Jordan domains D in R-2 of Dini-Lyapunov type, we show that any function subharmonic in D and of class C-1((D) over bar) can be extended to a function subharmonic and of class C-1 on the whole of R-2 with a uniform estimate of its gradient. We construct a large class of Jordan domains (including domains with C-1-smooth boundaries) for which this extension property fails. We also prove a localization theorem on C-1-subharmonic extension from any closed Jordan domain.

For a Jordan Dini-Lyapunov type domain D in 2 we prove the possibility to extend any function subharmonic in D and of the class C1(D) to a function subharmonic and of the class C1 on the whole of 2 with the uniform estimate of its gradient. We also obtain a localization theorem for C1-subharmonic extension from closed Jor- dan domains, and give examples of C1-smooth Jordan domains which don't have this extension property.

Several explanations concerning notation, terminology, and background are in order. First notation: by 7Hi we have denoted the one-dimensional Hausdorff measure (i.e. length), and A(z,r) stands for the closed disc with center z and radius r. A curve F is called AD-regular, that is, Ahlfors-David-regular, if it satisfies (1.2) (with E = F). Since the lower bound is automatic for curves, this means that 'Hl (r n A(z, r)) 0. General sets satisfying (1.2) are called AD-regular. It is simplest to define the L2-boundedness of the Cauchy singular integral operator via the truncated integrals: we say that CE is bounded in L2(E) (without really defining the operator CE itself) if there is M < ox such that

Ifμ\muis a finite complex Borel measure andΓ\Gammaa Lipschitz graph in the complex plane, then forλ>0\lambda > 0\[|{z∈Γ:supε>0|∫|ζ−z|⩾ε(ζ−z)−1dμζ|>λ}|⩽c(Γ)λ−1||μ||1.\left | {\left \{ {z \in \Gamma :\sup \limits _{\varepsilon > 0} \left | {\int _{|\zeta - z| \geqslant \varepsilon } {{{(\zeta - z)}^{ - 1}}} d\mu \zeta } \right | > \lambda } \right \}} \right | \leqslant c(\Gamma ){\lambda ^{ - 1}}||\mu |{|_1}.\]It follows that for any finite Borel measureμ\muand any rectifiable curveΓ\Gammathe finite principal value\[limε↓0∫|ζ−z|⩾ε(ζ−z)−1dμζ\lim \limits _{\varepsilon \downarrow 0} \int _{|\zeta - z| \geqslant \varepsilon } {{{(\zeta - z)}^{ - 1}}d\mu \zeta }\]exists for almost all (with respect to length)z∈Γz \in \Gamma.

Mateu and Orobitg proved (in Lipschitz approximation by harmonic functions and some applications to spectral synthesis, Indiana Univ. Math. J. 39 (1990)) that given λ > 1 \lambda > 1 and d − 1 > α ⩽ d d - 1 > \alpha \leqslant d there exist constants C C and N N (depending on λ \lambda and α \alpha ) with the following property: For any compact set K K in R d {\mathbb {R}^d} one can find a (finite) family of balls { B ( x i , r i ) } \{ B({x_i},{r_i})\} such that (i) K ⊂ ⋃ B ( x i , r i ) K \subset \bigcup {B({x_i},{r_i})} , (ii) ∑ r i α ⩽ C M α ( K ) \sum {r_i^\alpha \leqslant C{M^\alpha }(K)} , M α {M^\alpha } denoting the α \alpha -dimensional Hausdorff content, and (iii) the dilated balls { B ( x i , λ r i ) } \{ B({x_i},\lambda {r_i})\} are an almost disjoint family with constant N N . In this paper we prove that such a result is false for α ⩽ d − 1 \alpha \leqslant d - 1 .