We denote by H(R) the linear
space of harmonic functions on an open Riemann surface R with the topology of
uniform convergence on every compact subset. A continuous linear functional on
H(R) is referred to as a harmonic functional on R; the totality of such functionals is
the dual space H(R)∗ of the locally convex space H(R). A point evaluation
u → u(z), with z a fixed point of R; and a period u →∫γ∗du, with γ a
fixed cycle on R, are the most common examples of harmonic functionals
frequently occurring in the theory of functions. We denote by ⟨u,h∗⟩ the
value of a harmonic functional h∗ on R at u in H(R). The main purpose of
the present study is to establish the following representation of harmonic
functionals:
Representation Theorem. Every harmonic functional h∗on an open Riemann surfaceR can be represented by means of a function h harmonic at the point at infinity of Ras
(1)
for every u in H(R), where W is any relatively compact subregion of R such that therelative boundary ∂W is smooth and h is harmonic on R − W. If h1and h2arefunctions representing h∗in the above sense, then h1− h2can be continuedharmonically to all of R.
