Let X be a Hilbert-space valued martingale and Y a real-valued supermartingale which are orthogonal and with Y diierentially subordinate to X. Then where the constant (2) is Catalan's constant whose approximate value is 0:915965594. The constant K is B. Davis' constant in the Kolmogorov's weak-type inequality for conjugate harmonic functions in the unit disk. The inequality is sharp.

is called Fuchsian if the N ×N coefficient matrix A(λ) is a rational function of λ whose singularities are simple poles. Each Fuchsian system generates, via analytic continuation of its fundamental solution Ψ (λ) along closed curves, a representation of the fundamental group of the punctured Riemann sphere (punctured at the poles of A(λ)) in the group of N ×N invertible matrices. This represent...

and Applied Analysis 3 if and only if L(γ K ) is σ-additive, where γ K is the canonical cylindrical finitely additive set-valued function (also called a Gaussian distribution) on K. The following is our standing assumption: Assumption 1. Space E ⊂ H ∩ L is a Hilbert space such that for some δ ∈ (0, 1/2), A −δ : E 󳨀→ H ∩ L 4 is γ-radonifying. (12) Remark 2. Under the above assumption, we have th...