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Fine boundary limits of harmonic and caloric functions. (English) Zbl 0594.31017
In two fundamental papers [Bull. Sci. Math., II. Sér. 68, 12-36 (1944; Zbl 0028.36201); J. Anal. Math. 4, 209-221 (1955/56; Zbl 0071.100)] M. Brelot studied the existence and representation of fine limits of bounded harmonic functions at an irregular boundary point and introduced maximal sequences for their investigation. In the present paper, the author shows that Brelot’s classical results remain valid under weaker assumptions and that they can be proved on every strong harmonic space X in the author’s sense (X in particular possesses a countable base, and Doob’s convergence axiom holds [cf. the author, ”Harmonische Räume und ihre Potentialtheorie.” (Lect. Notes Math. 22) (1966; Zbl 0142.384)]). This is undertaken by utilizing ideas of the first above cited paper of Brelot, but avoiding Green functions and the Martin representation (cf. the second paper of Brelot). As demonstration for the larger applicability of this approach caloric functions are considered.
Reviewer: M.Kracht

31C05 Harmonic, subharmonic, superharmonic functions on other spaces
35K05 Heat equation
31D05 Axiomatic potential theory
35J67 Boundary values of solutions to elliptic equations and elliptic systems