arXiv subjects
Publications and source records attributed to Mark Melnikov.
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 $μ$ is a finite complex measure in the complex plane $\C$ we denote by $C^μ$ its Cauchy integral defined in the sense of principal value. The measure $μ$ is called reflectionless if it is continuous (has no atoms) and $C^μ=0$ at $μ$-almost every point. We show that if $μ$ is reflectionless and its Cauchy maximal function $C^μ_*$ is summable with respect to $|μ|$ then $μ$ 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.