On a Conjecture of Ditzian and Runovskii Original Research Article

Let Mθ be the mean operator on the unit sphere in Rn, 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 θ2D(Mθ)m, m⩾2 from X=Lp (1⩽p⩽∞) to itself which can be expressed aslimm→∞ sup{∥θ2D(Mθ)m∥(X,X): θ∈(0,π)}=0. . We give a proof of this conjecture.