Content deleted Content added
Adding local short description: "Class of mathematical functions", overriding Wikidata description "function f such that, if f≤h on the boundary of a ball for any harmonic function h, then f≤h also inside the ball" (Shortdesc helper) |
m use template instead of HTML entity |
||
Line 66:
<math display="block"> \|M \varphi\|_{L^2(\mathbf{T})}^2 \le C^2 \, \int_0^{2\pi} \varphi(e^{i\theta})^2 \, d\theta.</math>
If ''f'' is a function holomorphic in Ω and 0 < ''p'' < ∞, then the preceding inequality applies to ''φ'' = |''f''
<math display="block"> \int_0^{2\pi} \left( \sup_{0 \le r < 1} \left|F(r e^{i\theta})\right| \right)^p \, d\theta \le C^2 \, \sup_{0 \leq r < 1} \int_0^{2\pi} \left|F(re^{i\theta})\right|^p \, d\theta.</math>
With more work, it can be shown that ''F'' has radial limits ''F''(''e''{{i sup|''iθ''}}) almost everywhere on the unit circle, and (by the [[dominated convergence theorem]]) that ''F<sub>r</sub>'', defined by ''F<sub>r</sub>''(''e''{{i sup|''iθ''}}) = ''F''(''r''
== Subharmonic functions on Riemannian manifolds ==
| |||