Solution to a conjecture on the norm of the Hardy operator minus the identity

Abstract

We prove that for a decreasing weight w, the following inequality is sharp:where B p is the Ariño and Muckenhoupt class of weights, and p 2. The case w ≡ 1 gives a positive answer to a conjecture formulated in Kruglyak and Setterqvist (2008) [8], where this estimate is proved only when p 2 is an integer. Simple examples show that, for 1 < p < 2, or if w is not decreasing, the result is false. Finally, using a different argument, we also prove that in the case p = 1, and for arbitrary weights w ∈ B 1 , w B 1 is the best constant in the corresponding inequality.

Characterization of the restricted type spaces R(X)

“…Let X be a rearrangement invariant space (r.i.) on (0, ∞), satisfying that the function 1/(1 + s) ∈ X. Associated with X, we can consider the space R(X), introduced in [17] (which appears naturally in the study of the norm of the Hardy operator minus the identity in the cone of radially decreasing functions [3]). This is defined as the minimal Lorentz function space Λ W X , with (1) W X (t) = 1 1 + · Another important space associated to ϕ is the weak-type Lorentz space…”

“…It was proved in [14] that every W X as in (1) satisfies W X (t) ≥ Ct log(1 + 1/t), for some constant C > 0. Our main interest is to consider a converse result, namely, whether every Lorentz function space Λ ϕ whose fundamental function ϕ satisfies the inequality (3) ϕ(t) ≥ Ct log(1 + 1/t),…”

Optimal rearrangement invariant range for Hardy-type operators

“…Lorentz spaces. Note that Proposition 4.1, yields that S : Λ ϕ → L 1 + L ∞ is bounded if and only if the function ϕ satisfies that: (4) ϕ(t) ≥ Ct log(1 + 1/t).…”

“…Proposition 4.4. Let ϕ be a quasi-concave function satisfying(4). The r.i. optimal range space is given byR[S, Λ ϕ ] = Λ ϕ , Λ ϕ ] ⊂ Λϕ holds, if and only if the Hardy operator S : Λ ϕ → Λ ϕ is bounded.…”

Isometries on and monotone functions

“…In , , , several estimates concerning the norm of the Hardy averaging operator minus the identity I , have been established on , and sharp bounds are known, on the cone of decreasing functions, for the inequality: …”