Let X = (Xₜ,ℱₜ) be a continuous BMO-martingale, that is, ${\left|\right|X\left|\right|}_{BMO}\equiv su{p}_T\left|\right|E\left[\right|X_{\infty }-X_T\left|\right|{ℱ}_T{\left]\right||}_{\infty }<\infty$, where the supremum is taken over all stopping times T. Define the critical exponent b(X) by $b\left(X\right)=b>0:su{p}_T\left|\right|E\left[exp\left(b²(⟨X{⟩}_{\infty }-⟨X{⟩}_T)\right)\right|{ℱ}_T]{\left|\right|}_{\infty }<\infty$, where the supremum is taken over all stopping times T. Consider the continuous martingale q(X) defined by $q\left(X\right)ₜ=E\left[⟨X{⟩}_{\infty }\right|ℱₜ]-E\left[⟨X{⟩}_{\infty }\right|ℱ₀]$. We use q(X) to characterize the distance between ⟨X⟩ and the class $L^{\infty }$ of all bounded martingales in the space of continuous BMO-martingales, and we show that the inequalities $1/4d₁(q\left(X\right),L^{\infty })\le b\left(X\right)\le 4/d₁(q\left(X\right),L^{\infty })$ hold for every continuous BMO-martingale X.

When $1\&lt;p\&lt;\infty$ and $p\ne 2$ the $p$-harmonic measure on the boundary of the half plane ${ℝ}_{+}^2$ is not additive on null sets. In fact, there are finitely many sets $E_1$, $E_2$,...,$E_{\kappa }$ in $ℝ$, of $p$-harmonic measure zero, such that $E_1\cup E_2\cup ...\cup E_{\kappa }=ℝ$.

We describe the geometrical structure on a complex quasi-Banach space $X$ that is necessay and sufficient for the existence of boundary limits for bounded, $X$-valued analytic functions on the open unit disc of the complex plane. It is shown that in such spaces, closed bounded subsets have many plurisubharmonic barriers and that bounded upper semi-continuous functions on these sets have arbitrarily small plurisubharmonic perturbations that attain their maximum. This yields a certain representation of...

Some martingale analogues of Sawyer's two-weight norm inequality for the Hardy-Littlewood maximal function Mf are shown for the Doob maximal function of martingales.