Subharmonic Functions

We have seen that harmonic functions on an open set Ω can be characterized as those finite-valued, continuous functions h on Ω which satisfy the mean value property: h (x) = M (h;x,r) whenever % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qadaqdaaWdaeaapeGaamOqamaabmaapaqaa8qacaWG4bGaaiilaiaa % dkhaaiaawIcacaGLPaaaaaGaeyOGIWSaeuyQdCfaaa!3EE1!$$ \overline {B\left( {x,r} \right)} \subset \Omega $$. Subharmonic functions correspond to one half of this definition — they are upper-finite, upper semicontinuous functionss which satisfy the mean value inequality s (x) ≤ M (s;x,r) whenever % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qadaqdaaWdaeaapeGaamOqamaabmaapaqaa8qacaWG4bGaaiilaiaa % dkhaaiaawIcacaGLPaaaaaGaeyOGIWSaeuyQdCfaaa!3EE1!$$ \overline {B\left( {x,r} \right)} \subset \Omega $$. They are allowed to take the value −∞ 00 so that we can include such fundamental examples as % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qaciGGSbGaai4BaiaacEgadaqbdaWdaeaapeGaamiEaaGaayzcSlaa % wQa7amaabmaapaqaa8qacaWGobGaeyypa0JaaGOmaaGaayjkaiaawM % caaaaa!4164!$$ \log \left\| x \right\|\left( {N = 2} \right)$$ and % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacqGHsisldaqbdaWdaeaapeGaamiEaaGaayzcSlaawQa7a8aadaah % aaWcbeqaa8qacaaIYaGaeyOeI0IaamOtaaaakmaabmaapaqaaiaad6 % eacqGHLjYScaaIZaaapeGaayjkaiaawMcaaaaa!4314!$$ - {\left\| x \right\|^{2 - N}}\left( {N \geqslant 3} \right)$$. Also, semicontinuity (rather th an continuity) is the appropriate condition for certain key results (for example, Theorems 3.1.4 and 3.3.1) to hold. The reason for the name “subharmonic” will become apparent in Section 3.2.