The Hardy Operator and Boyd Indices

We give necessary and sufficient conditions for the Hardy operator to be bounded on a rearrangement invariant quasi-Banach space in terms of its Boyd indices. MAIN RESULTS A rearrangement invariant space X on R is a set of measurable functions (modulo functions equal almost everywhere) with a complete quasi-norm ‖ · ‖X such that the following holds: i) if g ≤ f ∗ and f ∈ X, then g ∈ X with ‖g‖X ≤ ‖f‖X ; ii) if f is simple with finite support then f ∈ X; iii) either fn ց 0 implies ‖fn‖ ց 0 iii) or 0 ≤ fn ր f and supn ‖fn‖X < ∞ imply f ∈ X with ‖f‖X = supn ‖fn‖X . Here f ∗ denotes the decreasing rearrangement of |f |, that is, f(s) = sup{ t : measure{|f | > t} > s }. The properties of rearrangement invariant spaces that we will use will be the Boyd indices defined as follows. Given a number 0 < a < ∞, we define the operator Daf(t) = f(at). Then the lower Boyd index of X is defined by pX = sup{ p : ∃c ∀a < 1 ‖Daf‖X ≤ c a −1/p‖f‖X } ∗Research supported in part by N.S.F. Grant DMS 9201357. and the upper Boyd index of X is defined by qX = inf{ q : ∃c ∀a > 1 ‖Daf‖X ≤ c a −1/q‖f‖X }. Thus we see that 1 ≤ pX ≤ qX ≤ ∞. Also, if X is the Lorentz space Lp,q, then pX = qX = p. We also define the Hardy operators as follows. Hf(t) = 1 t1/p ( ∫ t 0 (f (s)) ds )1/r , H(q,r)f(t) = 1 t1/q ( ∫ ∞ t (f (s)) ds )1/r , Hf(t) = sup 0 p. ii) For 0 < q ≤ ∞ and 0 < r < ∞ the operator H(q,r) is bounded from X to X if and only if qX < q. iii) for 0 < p < ∞ the operator H is bounded from X to X if pX > p. iv) For 0 < q < ∞ the operator H(q,∞) is bounded from X to X if qX < q. Note that the reverse implications are not true in parts (iii) and (iv). For example, the operators H and H(p,∞) are both bounded on the space Lp,∞. From this we can immediately generalize a result of Boyd (1967, 1969) to the following. THEOREM 3 If 0 < p < q ≤ ∞ and 0 < r1, r2, s1, s2 ≤ ∞, and if T : Lp,r1 ∩ Lq,s1 → Lp,r2 ∩Lq,s2 is a quasi-linear operator such that ‖Tf‖p,r1 ≤ c ‖f‖p,r2 and ‖Tf‖q,s1 ≤ c ‖f‖q,s2 for all f ∈ Lp,r1 ∩ Lq,s1, and if X is a quasi-Banach r.i. space with Boyd indices strictly between p and q, then ‖Tf‖X ≤ c ‖f‖X for all f ∈ Lp,r1 ∩ Lq,s1 . Proof: From Theorem 1, we see that H1(Tf)(t) + H(q,s1)(Tf)(t) ≤ c(H 2f(t) + H(q,s2)f(t)). Now the result follows easily from Theorem 2 which implies that for i = 1, 2 ‖Hi + H(q,si)‖X ≈ ‖f‖X . Thus, as applications, we may obtain the following generalization of a result of Fehér, Gaspar and Johnen (1973). THEOREM 4 The Hilbert transform is bounded on a quasi-Banach r.i. space X if and only if pX > 1 and qX < ∞. Proof: The implication from right to left follows immediately from Theorem 3. As for the other way, this follows from the easy estimate: P.V. ∫ ∞ −∞ f (y − x) y dy ≥ 1 2 (Hf(x) + Hf(x)) x > 0. We also obtain a result in the spirit of Ariño and Muckenhoupt (1990). THEOREM 5 The Hardy–Littlewood maximal function is bounded from X(R) to X(R) if and only if pX > 1. Proof: Combine the argument given in Ariño and Muckenhoupt (1990) with Theorem 2 above. Before proceeding with the proof of Theorem 2, we will require the following lemma. LEMMA 6 Suppose that X is a quasi-Banach r.i. space. Then given any p > 0, there is a number 0 < u ≤ p such that for any f1, f2, . . . , fn ∈ X we have