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

4 ANDREI K. LERNER2. Preliminaries2.1. Banach function spaces. For a general account of Banach functionspaces we refer to [2, Ch. 1]. Here we mention only several factswhich will be used in this paper.By the Lorentz-Luxemburg theorem [2, p. 10], X = X ′′ and ∥f∥ X =∥f∥ X ′′. In particular, this implies that∫(2.1) ∥f∥ X = sup |f(x)g(x)|dx.∥g∥ X ′=1 R nBy Fatou’s lemma [2, p. 5], if f n → f a.e., and if lim infn→∞ ∥f n∥ X < ∞,then f ∈ X, and(2.2) ∥f∥ X ≤ lim infn→∞ ∥f n∥ X .2.2. Adjoint of M. Although M is not a linear operator, it turns outthat it is possible to linearize M with good pointwise control of theadjoint of the linearization. The following theorem is contained in .Theorem 2.1. Given f ∈ L 1 loc (Rn ), there is a linear operator M f suchthatc 1 Mf(x) ≤ M f f(x) ≤ c 2 Mf(x),and for any g ∈ L 1 loc (Rn ),(2.3) (M ⋆ fg) # (x) ≤ c 3 Mg(x),where M ⋆ f is the adjoint of M f, and the constants c i depend only on n.Note also that the construction of M f shows that for any g ∈ L 1 (R n ),(2.4) ∥M ⋆ fg∥ L 1 (R n ) ≤ c∥g∥ L 1 (R n ),where the constant c depends only on n.Since the main properties of M f expressed in (2.3) and (2.4) do notdepend on f, we shall drop the subscript f and use the notions Mand M ⋆ .2.3. Local maximal functions. The non-increasing rearrangement(see, e.g., [2, p. 39]) of a measurable function f on R n is defined by{}f ∗ (t) = inf λ > 0 : |{x ∈ R n : |f(x)| > λ}| ≤ t (0 < t < ∞).Given a measurable function f, the local maximal functions m λ fand M # λf are defined by( ) ∗ ( )m λ f(x) = sup fχQ λ|Q| (0 < λ < 1)Q∋x

FEFFERMAN-STEIN INEQUALITY 5and( )M # ∗ ( )λf(x) = sup inf (f − c)χQ λ|Q| (0 < λ < 1),Q∋x crespectively, where the supremum is taken over all cubes containingthe point x.These functions were introduced by Strömberg ; in particular,the definition of M # λf was motivated by an alternate characterizationof BMO given by John . Roughly speaking, f # is the Hardy-Littlewood maximal operator of M # λf (see [10, 14]):(2.5) c 1 MM # λ f(x) ≤ f # (x) ≤ c 2 MM # λ f(x).The following theorem was proved in .Theorem 2.2. For any measurable function f ∈ S 0 (R n ) and any g ∈L 1 loc (Rn ),∫|f(x)g(x)|dx ≤ cR n ∫M # λ f(x)Mg(x)dx,R nwhere the constants c and λ depend only on n.The following result is contained in [16, Proposition 4.2].Proposition 2.3. For any locally integrable f and for all x ∈ R n ,(2.6) Mf(x) ≤ 3f # (x) + m 1/2 f(x),and(2.7) M # λ (m 1/2f)(x) ≤ 4M # λ/2·9 n f(x).Inequality (2.7) combined with (2.5) yields(2.8) (m 1/2 f) # (x) ≤ c n f # (x).It follows directly from the definitions that(2.9) {x ∈ R n : m λ f(x) > α} ⊂ {x ∈ R n : Mχ {|f|>α} (x) ≥ λ}.Lemma 2.4. For any non-negative functions f and g,∫∫(m λ f)gdx ≤ c f(Mg)dx,R n R nwhere a constant c depends on λ and n.Proof. This is just a combination of (2.9) with the following inequalityof Fefferman and Stein :∫g dx ≤ c ∫|φ|(Mg) dx.{x:Mφ>ξ} ξ R n

6 ANDREI K. LERNERWe have∫R n (m λ f)g dx =≤∫ ∞0∫{m λ f>α}∫ ∞ ∫c λ,nwhich proves the lemma.0{f>α}g dxdα ≤∫ ∞0∫{Mχ {f>α} (x)≥λ}Mg dxdα = c λ,n∫R n f(Mg)dx,3. Proofs of Theorems 1.1 and 1.2g dxdαProof of Theorem 1.1. Let us show first that (i) ⇒ (ii). We can supposethat f, g ≥ 0. Also, it is enough to assume, for instance, thatg is compactly supported, and hence g ∈ L 1 (R n ). The general casewill follow by the standard limiting argument. If g ∈ L 1 (R n ), thenM ⋆ g ∈ L 1 (R n ), and thus M ⋆ g ∈ S 0 (R n ). Therefore, by Theorem 2.1,∥M ⋆ g∥ X ≤ c∥(M ⋆ g) # ∥ X ≤ c∥Mg∥ X .From this, applying Theorem 2.1 again, we get∫∫Mf(x)g(x)dx ≤ c Mf(x)g(x)dxR n R∫n= c f(x)M ⋆ g(x)dxR n≤ c∥f∥ X ′∥M ⋆ g∥ X≤ c∥f∥ X ′∥Mg∥ X .We prove now (ii) ⇒ (i). Using Theorem 2.2 and (2.5), we get∫∫|f(x)φ(x)|dx ≤ c M # λ f(x)Mφ(x)dxR n R n≤From this, by (2.1), we obtain (i).c∥φ∥ X ′∥MM # λ f∥ X≤ c∥φ∥ X ′∥f # ∥ X .Remark 3.1. The proof of Theorem 2.2 in  shows that actually onecan replace Mg by the dyadic maximal function M ∆ g, namely, we have∫∫|f(x)g(x)|dx ≤ c M # λ f(x)M ∆ g(x)dx.R n R nTherefore, taking into account the proof of Theorem 1.1, in order toverify the Fefferman-Stein property of X, it is enough to check that∫(M ∆ f)|g|dx ≤ c∥f∥ X ′∥Mg∥ X .R n□□

FEFFERMAN-STEIN INEQUALITY 7In order to prove Theorem 1.2, we shall need the following lemma,which is interesting in its own right.Lemma 3.2. Inequalities(3.1) ∥f∥ X ≤ c∥f # ∥ X (f ∈ S 0 (R n ))and(3.2) ∥Mf∥ X ≤ c∥f # ∥ X (f ∈ S 0 (R n ))are equivalent.Proof. Since |f| ≤ Mf a.e., we trivially have that (3.2)⇒(3.1).Suppose that (3.1) holds. By (2.6),(3.3) ∥Mf∥ X ≤ 3∥f # ∥ X + ∥m 1/2 f∥ X .Next, from (2.9) and from the weak type (1, 1) property of M,µ m1/2 f(α) ≤ c n µ f (α) (α > 0).Therefore, f ∈ S 0 (R n ) ⇒ m 1/2 f ∈ S 0 (R n ). Hence, combining (3.1)with (2.8), we get∥m 1/2 f∥ X ≤ c∥(m 1/2 f) # ∥ X ≤ c∥f # ∥ X ,which along with (3.3) implies (3.2).Proof of Theorem 1.2. Inequality∫(3.4)M r f(x)|g(x)|dx ≤ c∥f∥ X ′∥Mg∥ X .R ntrivially implies (1.2), so we have to show only that (1.2)⇒(3.4).We can suppose g ≥ 0. It suffices to prove that there exists a constantA > 0 such that for any k ∈ N,∫(3.5)M k f(x)g(x)dx ≤ A k ∥f∥ X ′∥Mg∥ X ,R nwhere M k is the operator M iterated k times. Indeed, if (3.5) is true,then (3.4) follows by the standard way by means of the Rubio de Franciaalgorithm . We set(Rf)(x) = |f(x)| +∞∑k=11(2A) k M k f(x).Then |f| ≤ Rf and M(Rf)(x) ≤ 2A(Rf)(x). We have that (Rf) isan A 1 weight, and hence it satisfies the reverse Hölder inequality (see,e.g., ), which means that there exists r > 1 such thatM r f(x) ≤ M r (Rf)(x) ≤ cM(Rf)(x) ≤ 2Ac(Rf)(x).□

FEFFERMAN-STEIN INEQUALITY 9to ask whether there exists a Banach function space X different fromL ∞ such that ∥M∥ l X = 1.If X = L ∞ , we trivially have ∥M∥ l L = 1. However, the existence∞of X different from L ∞ for which ∥M∥ l X = 1 is not an obvious fact.For example, by Riesz’s sunrise lemma [8, p. 93],|{x ∈ R : Mf(x) > α}| ≥ 1 ∫|f|.αIntegrating this inequality givesand hence for any p > 1,∥Mf∥ L p (R) ≥∥M∥ l L p (R) ≥{x∈R:|f(x)|>α}( p) 1/p∥f∥Lp − 1p (R),( p) 1/p> 1.p − 1Using a non-standard proof of the Fefferman-Stein inequality combinedwith Theorem 1.1, we will prove the following.Theorem 4.1. If M is bounded on X and it is not bounded on X ′ ,then ∥M∥ l X = 1.Remark 4.2. Assume that X is a r.i. space. Let α X and β X be thelower and upper Boyd indices, respectively [2, p. 149]. In general,0 ≤ α X ≤ β X ≤ 1. By the Lorentz-Shimogaki theorem [2, p. 154], Mis bounded on X iff β X < 1. Also, it is well known that β X ′ = 1 − α X .Therefore, by Theorem 4.1, if α X = 0 and β X < 1, then ∥M∥ l X = 1.We start with the following simple observation: if M is bounded onX, then (1.2) is equivalent to the boundedness of M on X ′ . Therefore,we immediately obtain the following corollary of Theorem 1.1.Corollary 4.3. Let M be bounded on X. Then M is bounded on X ′if and only if there exists c > 0 such that for any f ∈ S 0 (R n ),∥f∥ X ≤ c∥f # ∥ X .Next, in order to prove Theorem 4.1, we shall need the following tworesults.Theorem 4.4. For any f ∈ L 1 loc (Rn ) and for a.e. x ∈ R n ,(4.1) MMf(x) ≤ cMf # (x) + Mf(x),where the constant c depends only on n.

10 ANDREI K. LERNERLemma 4.5. If f ∈ S 0 (R n ) ∩ L ∞ , then there is a sequence of boundedand compactly supported measurable functions {f j } such that for a.e.x ∈ R n ,(4.2) limj→∞f j (x) = f(x)and(4.3) (f j ) # (x) ≤ cf # (x),where the constant c depends only on n.Before proving Theorem 4.4 and Lemma 4.5 let us show how theproof of Theorem 4.1 follows.Proof of Theorem 4.1. We shall prove an equivalent statement sayingthat if A ≡ ∥M∥ l X > 1, then M is bounded on X′ . By Corollary4.3, it is enough to prove the Fefferman-Stein inequality on X.Since (|f|) # (x) ≤ 2f # (x), we can assume that f ≥ 0.If A > 1, then by (4.1),A∥Mf∥ X ≤ ∥MMf∥ X ≤ c∥Mf # ∥ X + ∥Mf∥ X .Suppose that ∥f∥ X < ∞. Then ∥Mf∥ X < ∞, and we obtain(4.4) ∥f∥ X ≤ ∥Mf∥ X ≤ cA − 1 ∥Mf# ∥ X ≤ c ′ ∥f # ∥ X .Take now an arbitrary f ∈ S 0 (R n ) ∩ L ∞ . By Lemma 4.5, there isa sequence {f j } satisfying (4.2) and (4.3). Since each f j is boundedand compactly supported, we have that ∥f j ∥ X < ∞ (we have used herethat if |E| < ∞, then ∥χ E ∥ X < ∞ [2, p. 2]). Therefore, by (4.3) and(4.4),∥f j ∥ X ≤ c∥f # ∥ X .From this, applying (4.2) and (2.2), we get the Fefferman-Stein inequalityon X for any f ∈ S 0 (R n ) ∩ L ∞ .Finally, if f is an arbitrary function from S 0 (R n ), consider f N (x) =min(f(x), N). Then clearly f N ∈ S 0 (R n ) ∩ L ∞ . Also (see, e.g., [8,p. 519]), (f N ) # (x) ≤ cf # (x). Therefore,∥f N ∥ X ≤ c∥f # ∥ X .Applying (2.2) again completes the proof.Proof of Theorem 4.4. This theorem was proved in  in the onedimensionalcase. The proof given there can be extended to any n ≥ 1.For the sake of completeness we give here a slightly different proof.Using (2.6), we obtain(4.5) MMf(x) ≤ 3Mf # (x) + Mm 1/2 f(x).□

FEFFERMAN-STEIN INEQUALITY 11Let x, y ∈ Q and let Q ′ be an arbitrary cube containing y. We havethat either Q ⊂ 3Q ′ or Q ′ ⊂ 3Q. If Q ⊂ 3Q ′ , then(fχ Q ′) ∗ (|Q ′ |/2) ≤ ((f − f 3Q ′)χ Q ′) ∗ (|Q ′ |/2) + |f| 3Q ′≤ 2 ∫|f − f|Q ′ 3Q ′| + |f| 3Q ′| Q ′If Q ′ ⊂ 3Q, thenTherefore,≤2 · 3 n f # (x) + Mf(x).(fχ Q ′) ∗ (|Q ′ |/2) ≤ ((f − f 3Q )χ Q ′) ∗ (|Q ′ |/2) + |f| 3Q≤ m 1/2 ((f − f 3Q )χ 3Q )(y) + Mf(x).m 1/2 f(y) ≤ m 1/2 ((f − f 3Q )χ 3Q )(y) + 2 · 3 n f # (x) + Mf(x).From this, using Lemma 2.4 with g ≡ 1, we get∫1m 1/2 f(y)dy ≤ 1|Q||Q| ∥m 1/2((f − f 3Q )χ 3Q )∥ L 1Q+ 2 · 3 n f # (x) + Mf(x)∫c≤ |f − f 3Q | + 2 · 3 n f # (x) + Mf(x)|3Q|≤3Qcf # (x) + Mf(x),and henceMm 1/2 f(x) ≤ cf # (x) + Mf(x).Combining this with (4.5) completes the proof.It remains to prove Lemma 4.5. We shall need the notion of a medianvalue. Given a cube Q and a measurable function f, by a median valueof f over Q we mean a, possibly nonunique, real number m f (Q) suchthat|{x ∈ Q : f(x) > m f (Q)}| ≤ |Q|/2and|{x ∈ Q : f(x) < m f (Q)}| ≤ |Q|/2.It is easy to show (see, e.g., ) that for any constant c,(4.6) |m f (Q) − c| ≤ ( (f − c)χ Q) ∗ (|Q|/2).Fix an open cube Q 0 . Given x ∈ Q 0 , let Q x be the unique cubecentered at x such that l(Q x ) = dist(∂Q 0 , Q x ), where ∂Q and l(Q) arethe boundary and the side length of Q, respectively. SetA Q0 f(x) = ( f(x) − m f (Q x ) ) χ Q0 (x).□

12 ANDREI K. LERNERProposition 4.6. For all x ∈ R n ,(4.7) (A Q0 f) # (x) ≤ cf # (x),where the constant c depends only on n.Proof. Take an arbitrary cube Q containing x, and consider∫1Ω(Q) ≡ inf |A Q0 f(y) − c|dy.c |Q|QIf Q ∩ Q 0 = ∅, we trivially have Ω(Q) = 0. Therefore, assume thatQ ∩ Q 0 ≠ ∅. There are two cases.Case 1. Suppose that there exists y 0 ∈ Q ∩ Q 0 such that l(Q) ≤l(Q y0 )/2. Then Q ⊂ 2Q y0 ⊂ Q 0 . Next, a simple geometrical observationshows that for any y ∈ 2Q y0 we get l(Q y0 )/3 ≤ l(Q y ) ≤ 5l(Q y0 )/2.Hence, Q y ⊂ 5Q y0 and |Q y0 | ≤ 3 n |Q y |. Therefore, by (4.6), for anyy ∈ Q,|m f (Q y ) − c| ≤ ( ) ∗ ((f − c)χ Qy |Qy |/2 )2≤ |f − c| ≤|Q y |∫Q 2 · ∫15n |f − c|.y|5Q y0 | 5Q y0ThusΩ(Q) ≤ infc≤∫1|f(y) − c|dy + inf|Q| Qcf # (x) + infc∫1|m f (Q y ) − c|dy|Q| Q2 · 15 n ∫|f − c| ≤ (2 · 15 n + 1)f # (x).|5Q y0 | 5Q y0Case 2. Assume now that l(Q y ) < 2l(Q) for any y ∈ Q ∩ Q 0 . ThenQ y ⊂ 3Q, and hence, by (4.6),|f 3Q − m f (Q y )| ≤ ( (f − f 3Q )χ Qy) ∗ (|Qy |/2 )≤m 1/2((f − f3Q )χ 3Q)(y).Therefore, applying Lemma 2.4 with g ≡ 1, we getΩ(Q) ≤ 1 ∫|A Q0 f(y)|dy|Q| Q≤ 1 ∫|f(y) − f 3Q |dy + 1 ∫|f 3Q − m f (Q y )|dy|Q| Q|Q| Q≤ 3 n f # (x) + 1|Q| ∥m ( )1/2 (f − f3Q )χ 3Q ∥L 1 ≤ cf # (x).Combining both cases yields∫1|A Q0 f(y) − (A Q0 f) Q |dy ≤ 2Ω(Q) ≤ cf # (x),|Q|Q

proving (4.7).FEFFERMAN-STEIN INEQUALITY 13Proof of Lemma 4.5. Set Q j = (−j, j) n and f j = A Qj f. It is clear thatf j is bounded and compactly supported. Also, by (4.7),(f j ) # (x) ≤ cf # (x).Further, for any x ∈ Q j/2 and for a cube Q x centered at x withl(Q x ) = dist(∂Q j , Q x ) we have |Q x | ≥ (j/3) n . Hence, by (4.6), forx ∈ Q j/2 ,|f(x) − f j (x)| = |m f (Q x )| ≤ f ∗( (j/3) n /2 ) .Since f ∈ S 0 (R n ) is equivalent to f ∗ (+∞) = 0 (see, e.g., [16, Prop. 2.1]),we obtain from this (4.2), and therefore the proof is complete. □5. The case X = L p (w)We consider here the case when X = L p (w), 1 < p < ∞, where wis a weight, that is, a non-negative locally integrable function. In thiscase X ′ = L p′ (σ), where 1/p ′ + 1/p = 1, and σ = w − 1p−1 .Conditions on a weight w for which the weighted Fefferman-Steininequality(5.1) ∥f∥ L p (w) ≤ c∥f # ∥ L p (w) (1 < p < ∞)holds are discussed in the next section. Theorems 1.1 and 1.2 provideseveral reformulations of (5.1). Here we obtain yet another inequalitiesequivalent to (5.1).Theorem 5.1. The following statements are equivalent:(i) there exists c > 0 such that (5.1) holds for any f ∈ S 0 (R n );(ii) there exist c > 0 and r > 1 such that for any f ∈ L 1 loc (Rn ),∫(M r (Mf) p−1 w ) ∫|f| dx ≤ c (Mf) p w dx;R n R n(iii) there exist c > 0 and r > 1 such that for any f ∈ L 1 loc (Rn ),∫∫M p,r (f, w)|f| dx ≤ c (Mf) p w dx,R n R nwhere( ∫ ) p−1 ( ∫ ) 1/r 11M p,r (f, w)(x) = sup |f|w r .Q∋x |Q| Q |Q| QProof. By Theorems 1.1 and 1.2, if (i) holds, then∫R n M r φ|f| dx ≤ c∥φ∥ L p ′ (σ) ∥Mf∥ L p (w).□

16 ANDREI K. LERNERLet T be a Calderón-Zygmund singular integral operator, that is,T = p.v.f ∗ K with kernel K satisfying the standard conditions∥ ̂K∥ L ∞ ≤ c, |K(x)| ≤ c/|x| n ,|K(x) − K(x − y)| ≤ c|y|/|x| n+1 for |y| < |x|/2.Actually, the results described below hold for more general Calderón-Zygmund operators as well.The weighted theory of the Fefferman-Stein inequality has been developedin parallel to the one of Coifman’s inequality relating singularintegrals and the maximal function. Namely, it was proved by Coifman (see also ) that if w ∈ A ∞ , then for any appropriate f,(6.3) ∥T f∥ L p (w) ≤ c∥Mf∥ L p (w) (1 < p < ∞).This result was based on a good-λ inequality related T f and Mf.However, the Fefferman-Stein inequality originally was also proved withthe help of a good-λ inequality related f and f # . Therefore, it hasbeen quickly realized that if w ∈ A ∞ , then (5.1) holds. After that,Sawyer  observed that the weak A ∞ condition is enough for (6.3).The same argument applies to (5.1).In , Muckenhoupt established that in the case when T is theHilbert transform, the C p condition is necessary for (6.3), and he conjecturedthat C p is also sufficient. Note that this question is still open.In , Sawyer proved that if ε > 0, then the C p+ε condition is sufficientfor (6.3). Using almost the same arguments, Yabuta  showedthat C p is necessary for (5.1) and C p+ε is sufficient.Here we give a completely different proof of a slightly improved versionof Yabuta’s result. Given p > 1, let φ p be a non-decreasing,doubling (i.e., φ p (2t) ≤ cφ p (t)) function on (0, 1) satisfying∫ 10φ p (t) dt < ∞.tp+1 We say that a weight w satisfies the ˜C p condition if there are positiveconstants c, δ such that for any cube Q and any subset E ⊂ Q,w(E) ≤ c(|E|/|Q|) δ ∫R n φ p (Mχ Q )w.Theorem 6.1. The C p condition is necessary for∫∫(6.4)M p,r (f, w)|f| dx ≤ c (Mf) p w dx,R n R nand the ˜C p is sufficient.

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 .Remark 6.3. Theorem 6.1 yields a new approach to Sawyer’s result as well. Indeed, it is well known that inequalities (6.3) and (5.1) arevery closely related in view of the following pointwise inequality :(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

18 ANDREI K. LERNERFurther, if x ∈ Q, then( ∫ ) p−1 ∫11|f|φ p (Mχ Q )w|Q| Q |Q| R n( ∫ ) ( p−1 ∫11≤ c |f|w + 1 ∞∑∫ )φ p (2 −kn ) w|Q| Q |Q| Q |Q|k=12 k Q\2 k−1 Q∞∑( ∫ ) p−1 ( ∫ )11≤ c 2 kpn φ p (2 −kn )|f|w|2 k Q|k=12 k Q |2 k Q| 2 k Q( ∫ 1≤ c φ p (t) dt )Mt p+1 p,1 (f, w)(x).0We now observe that it is enough to prove (6.4) for compactly supportedf. Also, one can assume that the right-hand side of (6.4) isfinite, otherwise there is nothing to prove. This means, in particular,thatIt follows from this that1sup0∈Q,|Q|≥1supQ∋x∫R n|Q| p ∫Qw(x)dx < ∞.1 + |x|pn∫w ≤ cR nw(x)dx < ∞,1 + |x|pnwhich ∫ easily implies that M p,1 (f, w)(x) < ∞ a.e. Since (wχ Q ) ∗ (λ|Q|) ≤1w, we obtain also thatλ|Q| Q( ∫ p−1 1|f|)(wχ Q ) ∗ (λ|Q|) < ∞ a.e.|Q|QTherefore, (6.7) shows that M p,r (f, w)(x) < ∞ a.e.Hence, applying (6.7) again and using Hölder’s inequality, we getM p,r (f, w)(x) ≤ cλ 1/r−(1−δ) M p,1 (f, w)(x) + m λ((Mf) p−1 w ) (x)≤cλ 1/r−(1−δ) M p,r (f, w)(x) + m λ((Mf) p−1 w ) (x).From this, taking λ small enough, we obtain(6.8) M p,r (f, w)(x) ≤ cm λ((Mf) p−1 w ) (x).This inequality combined with Lemma 2.4 yields∫∫(M p,r (f, w)|f| dx ≤ c m λ (Mf) p−1 w ) |f| dxR n R∫n≤ c (Mf) p w dx,R nand therefore the theorem is proved.□

FEFFERMAN-STEIN INEQUALITY 19We make several concluding remarks. The question about a necessaryand sufficient condition on w for which (6.4) (or, equivalently, theFefferman-Stein inequality (5.1)) holds remains open. The ˜C p conditionis probably not a necessary condition for (6.4). Indeed, the proofof Theorem 6.1 shows that the ˜C p condition implies (6.8). This alongwith Lemma 2.4 gives that for all suitable f and g,∫∫M p,r (f, w)|g| dx ≤ c (Mf) p−1 (Mg)w dx,R n R nwhich seems to be much stronger than (6.4).Next, the methods used in the proof of Theorem 6.1 show easily thatthe C p condition is equivalent to (6.6). Moreover, the C p condition isequivalent to the following statement: there exist c > 0 and r > 1 suchthat for each cube Q and any g ∈ L 1 loc (Rn ),(6.9)( 1|Q|∫Q) p ( ∫ 1|g||Q|Qw r ) 1/r≤ c 1|Q|∫R n (M(gχQ ) ) pw.Indeed, (6.9) with g ≡ 1 gives (6.6). On the other hand, if x Q is thecenter of Q, then( ∫ ) p ∫( ∫1|g| (Mχ Q ) p w ≤ c|Q| Q R∫R |g|) pQw|x − x n n Q | n + |Q|∫(≤ c M(gχQ ) ) pw,R nwhich along with (6.6) implies (6.9).Denote by M p the class of weights w for which the following Fefferman-Stein-type inequality holds (cf. ): there is c > 0 such that for anysequence of functions {f j } with pairwise disjoint supports,∑∫((Mf j ) p w ≤ c M(j R∫R ∑ pw.f j )) n n jThen the C p ∩ M p condition yields (5.1). To show this, we keep thesame notation as in the proof of Theorem 5.1. Using (6.9) along withthe M p condition, we get∫R n (T l f) p w ≤ c2 −l/r′ ∑ k,j≤c2 −l/r′ ∑ k,j( ∫ 1) p ( ∫ 1) 1/r|Q|Q k j | fEjk |Q k j | w r kj |Q k j(M(fχE kj)∫R ) ∫pw ≤ c2−l/r ′nR n (Mf) p w,

20 ANDREI K. LERNERand now we can follow the proof of Theorem 5.1. The above argumentraises a natural question whether C p ⇒ M p . Observe that the sharpfunction estimate of the vector-valued maximal operator  shows thatC p+ε ⇒ M p for any ε > 0.Acknowledgment. I am grateful to the referee for useful remarks andcorrections.References J. Alvarez and C. Pérez, Estimates with A ∞ weights for various singular integraloperators, Boll. Un. Mat. Ital. (7), 8-A (1994), no. 1, 123–133. C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, NewYork, 1988. R.R. Coifman, Distribution function inequalities for singular integrals, Proc.Nat. Acad. Sci. USA 69 (1972), 2838–2839. R.R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functionsand singular integrals, Studia. Math. 15 (1974), 241–250. G.P. Curbera, J. Garc´a-Cuerva, J.M. Martell and C. Pérez, Extrapolation withweights, rearrangement-invariant function spaces, modular inequalities and applicationsto singular integrals, Adv. Math. 203 (2006), no. 1, 256–318. C. Fefferman and E.M. Stein, Some maximal inequalities, Amer. J. Math., 93(1971), 107–115. C. Fefferman and E.M. Stein, H p spaces of several variables, Acta Math. 129(1972), 137–193. L. Grafakos, Classical and modern Fourier analysis, Prentice Hall, 2004. P. Janakiraman, Limiting weak-type behavior for singular integral and maximaloperators, Trans. Amer. Math. Soc. 358 (2006), no. 5, 1937–1952. B. Jawerth and A. Torchinsky, Local sharp maximal functions, J. Approx.Theory, 43 (1985), 231–270. F. John, Quasi-isometric mappings, Seminari 1962-1963 di Analisi, Algebra,Geometria e Topologia, Rome, 1965. L. Kahanpää and L. Mejlbro, Some new results on the Muckenhoupt conjectureconcerning weighted norm inequalities connecting the Hilbert transform withthe maximal function, Proceedings of the second Finnish-Polish summer schoolin complex analysis (Jyväskylä, 1983), 53–72, Bericht, 28, Univ. Jyväskylä,Jyväskylä, 1984. S. Korry, Fixed points of the Hardy-Littlewood maximal operator, Collect.Math. 52 (2001), no. 3, 289–294. A.K. Lerner, On the John-Strömberg characterization of BMO for nondoublingmeasures, Real. Anal. Exchange, 28 (2003), no. 2, 649–660. A.K. Lerner, Weighted norm inequalities for the local sharp maximal function,J. Fourier Anal. Appl. 10 (2004), no. 5, 465–474. A.K. Lerner, Weighted rearrangement inequalities for local sharp maximalfunctions, Trans. Amer. Math. Soc., 357 (2005), no. 6, 2445–2465. A.K. Lerner, BMO-boundedness of the maximal operator for arbitrary measures,Israel J. Math., 159 (2007), no. 1, 243–252. A.K. Lerner and S. Ombrosi, A boundedness criterion for general maximaloperators, Publ. Mat., 54 (2010) no. 1, 53–71.

FEFFERMAN-STEIN INEQUALITY 21 A.K. Lerner, S. Ombrosi, C. Pérez, R.H. Torres and R. Trujillo-Gonzalez, Newmaximal functions and multiple weights for the multilinear Calderón-Zygmundtheory, Adv. Math., 220 (2009), no. 4, 1222–1264. A.K. Lerner and C. Pérez, A new characterization of the Muckenhoupt A pweights through an extension of the Lorentz-Shimogaki theorem, Indiana Univ.Math. J., 56 (2007), no. 6, 2697–2722. J. Martín and J. Soria, Characterization of rearrangement invariant spaceswith fixed points for the Hardy-Littlewood maximal operator, Ann. Acad. Sci.Fenn. Math., 31 (2006), no. 1, 39–46. B. Muckenhoupt, Norm inequalities relating the Hilbert transform to the Hardy-Littlewood maximal function, Functional analysis and approximation (Oberwolfach,1980), 219–231, Internat. Ser. Numer. Math., 60, Birkhäuser, Basel-Boston, Mass., 1981. J.L. Rubio de Francia, Factorization theory and A p weights, Amer. J. Math.106 (1984), 533–547. E.T. Sawyer, Two weight norm inequalities for certain maximal and integraloperators, Harmonic analysis (Minneapolis, Minn., 1981), 102–127, LectureNotes in Math., 908, Springer, Berlin-New York, 1982. E.T. Sawyer, Norm inequalities relating singular integrals and the maximalfunction, Studia Math., 75 (1983), 253–263. J.-O. Strömberg, Bounded mean oscillation with Orlicz norms and duality ofHardy spaces, Indiana Univ. Math. J., 28 (1979), 511–544. A. de la Torre, On the adjoint of the maximal function, Function spaces, differentialoperators and nonlinear analysis (Paseky nad Jizerou, 1995), 189–194,Prometheus, Prague, 1996. K. Yabuta, Sharp maximal function and C p condition, Arch. Math. 55 (1990),no. 2, 151–155.Department of Mathematics, Bar-Ilan University, 52900 Ramat Gan,IsraelE-mail address: aklerner@netvision.net.il

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