An extremal problem related to Kolmogoroff’s inequality for bounded functions

Abstract

LetA andB be positive numbers andm andn positive integers,m<n. Then there is for complex valued functions φ onR with sufficient differentiability and boundedness properties a representation

wherev ₁ andv ₂ are bounded Borel measures withv ₁ absolutely continuous, such that there exists a function φ with ∣φ⁽ⁿ⁾∣ ⩽A and ∣φ∣ ⩽A onR and satisfying

$$\varphi ^{(m)} (0) = A\int_R {\left| {d\nu _1 } \right|} + B\int_R {\left| {d\nu _2 } \right|} .$$

This result is formulated and proved in a general setting also applicable to derivatives of fractional order. Necessary and sufficient conditions are given in order that the measures and the optimal functions have the same essential properties as those which occur in the particular case stated above.