ASYMPTOTIC BEHAVIOR OF THE APPROXIMATION NUMBERS OF THE HARDY - TYPE OPERATOR FROM L p INTO L

We consider the Hardy-type operator (Tf) (x) := v(x) ∫ x a u(t)f(t)dt, x > a, and establish properties of T as a map from L(a, b) into L(a, b) for 1 < p ≤ q ≤ 2, 2 ≤ p ≤ q < ∞ and 1 < p ≤ 2 ≤ q < ∞. The main result is that, with appropriate assumptions on u and v, the approximation numbers an(T ) of T satisfy the inequality c1 ∫ b a |uv|dt ≤ lim inf n→∞ nan(T ) ≤ lim sup n→∞ nan(T ) ≤ c2 ∫ b a |uv|dt when 1 < p ≤ q ≤ 2 or 2 ≤ p ≤ q < ∞, and in the case 1 < p ≤ 2 ≤ q < ∞ we have lim sup n→∞ nan(T ) ≤ c3 ∫ d 0 |u(t)v(t)|dt and c4 ∫ d 0 |u(t)v(t)|dt ≤ lim inf n→∞ nan(T ), where r = p ′q p′+q and constants c1, c2, c3, c4. Upper and lower estimates for the l s and l norms of {an(T )} are also given.