Boundary Limits

We saw in Chapter 4 that a potential on the unit ball B has radial limit 0 at σ-almost every boundary point (Littlewood’s theorem), and that a positive harmonic function on B has finite non-tangential limits at σ-almost every boundary point (Fatou’s theorem). The notions of radial and non-tangential limits are clearly unsuitable for the study of boundary behaviour in general domains. To overcome this difficulty, we will develop the ideas of the preceding two chapters by defining the “minimal fine topology” on the Martin compactification, % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacuqHPoWvgaqcaaaa!37B2!$$ \hat \Omega $$, of a Greenian domain Ω. (When restricted to Ω, this topology reduces to the fine topology we have already studied.) Thus we will be able to discuss minimal fine limits of functions at (minimal) Martin boundary points and the concept of a set being “minimally thin” at such points.