Bernstein widths of Hardy-type operators in a non-homogeneous case

Abstract

Let I = [ a , b ] ⊂ R , let 1 < p ⩽ q < ∞ , let u and v be positive functions with u ∈ L p ′ ( I ) , v ∈ L q ( I ) and let T : L p ( I ) → L q ( I ) be the Hardy-type operator given by ( T f ) ( x ) = v ( x ) ∫ a x f ( t ) u ( t ) d t , x ∈ I . We show that the Bernstein numbers b n of T satisfy lim n → ∞ n b n = c p q ( ∫ I ( u v ) r d t ) 1 / r , 1 / r = 1 / p ′ + 1 / q , where c p q is an explicit constant depending only on p and q.