Integral representations on equipotential and harmonic sets

Abstract

The sets we are going to consider here are of the form ${z\in\mathbb C \mid (z) 1}$ (equipotential) and ${z\in\mathbb C \mid IM A(z)=0}$ (harmonic) with $A$ being a polynomial with complex coefficients. There are two themes which we want to focus on and which come out from invariance property of inner products on $\mathbb C[Z]$ related to the aforesaid sets. First, we formalize the construction of integral representation of the inner products in question with respect to matrix measure. Then we show that these inner products when represented in a Sobolev way are precisely those with discrete measures in the higher order terms of the representation. In this way we fill up the case already considered in [3] by extending it from the real line to harmonic sets on the complex plane as well as we describe completely what happens in this matter on equipotential sets. As a kind of smooth introduction to the above we are giving an account of standard integral representations on the complex plane in general and of those supported by these two kinds of real algebraic sets.