On a Conjecture of Ditzian and Runovskii

Abstract

Let M θ be the mean operator on the unit sphere in R n , n⩾3, which is an analogue of the Steklov operator for functions of single variable. Denote by D the Laplace–Beltrami operator on the sphere which is an analogue of second derivative for functions of single variable. Ditzian and Runovskii have a conjecture on the norm of the operator θ 2 D( M θ ) m , m⩾2 from X= L p (1⩽ p⩽∞) to itself which can be expressed as lim m→∞ sup{∥θ 2D(M θ) m∥ (X,X): θ∈(0,π)}=0. . We give a proof of this conjecture.