Vol. 93, No. 1, 1981

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Harmonic functionals on open Riemann surfaces

Mitsuru Nakai and Leo Sario

Vol. 93 (1981), No. 1, 147–161

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,hthe 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 surface R can be represented by means of a function h harmonic at the point at infinity of R as

    ∗   ∫
⟨u,h ⟩ =  ∂Wu ∗dh − h∗ du

for every u in H(R), where W is any relatively compact subregion of R such that the relative boundary ∂W is smooth and h is harmonic on R W. If h1 and h2 are functions representing h in the above sense, then h1 h2 can be continued harmonically to all of R.

Mathematical Subject Classification 2000
Primary: 30F15
Secondary: 31C05
Received: 27 August 1979
Published: 1 March 1981
Mitsuru Nakai
Leo Sario