SOME REMARKS ON THE FEFFERMAN-STEIN INEQUALITY 1 ...

FEFFERMAN-STEIN INEQUALITY 17Remark 6.2. It is easy to see that φ p (t) ≤ ct p , and hence ˜C p ⊂ C p . Onthe other hand, taking φ p (t) such that t p+ε ≤ cφ p (t) for any ε > 0 (forexample, φ p (t) = t p log −2 (1 + 1/t)), we get ∪ ε>0 C p+ε ⊂ ˜C p . Hence, byTheorem 5.1, we have an improvement of [28].Remark 6.3. Theorem 6.1 yields a new approach to Sawyer’s result [25]as well. Indeed, it is well known that inequalities (6.3) and (5.1) arevery closely related in view of the following pointwise inequality [1]:(6.5) (|T f| α ) # (x) ≤ c(Mf) α (x) (0 < α < 1).The C p+ε condition implies (5.1) with p + ε ′ , ε ′ < ε, instead of p.Combining this with (6.5), where α = p/(p + ε ′ ), we get (6.3).Proof of Theorem 6.1. Setting in (6.4) f = χ Q , we obtain(6.6)( ∫ ) 1/r 1w r ≤ c 1 ∫(Mχ Q ) p w.|Q| Q |Q| R nFrom this, by Hölder’s inequality we get the C p condition with δ = 1/r ′ .Suppose now that w ∈ ˜C p . Then for 0 < t < |Q| (cf. [2, p. 53]),∫ t0From this,(wχ Q ) ∗ (τ)dτ =(wχ Q ) ∗ (t) ≤ 1 t∫ t0sup w(E) ≤ c(t/|Q|) δ φ p (Mχ Q )w.E⊂Q,|E|=t∫R n(wχ Q ) ∗ (τ)dτ ≤c ∫1t 1−δ |Q| δHence, fixing some 1 < r < 1 , for 0 < λ < 1 we get1−δ∫ ∫ |Q|w r = (wχ Q ) ∗ (t) r dtQ=≤0∫ λ|Q|0(wχ Q ) ∗ (t) r dt +cλ 1−r(1−δ) |Q|∫ |Q|λ|Q|(wχ Q ) ∗ (t) r dtR n φ p (Mχ Q )w.( ∫) r 1φ p (Mχ Q )w + |Q|(wχ Q ) ∗ (λ|Q|) r .|Q| R nTherefore,( ∫ ) 1/r ∫1(6.7) w r ≤ c λ1/r−(1−δ)φ p (Mχ Q )w + (wχ Q ) ∗ (λ|Q|).|Q| Q|Q| R n