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 [27].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 [26]; in particular,the definition of M # λf was motivated by an alternate characterizationof BMO given by John [11]. 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 [15].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 [6]:∫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 [15] 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 [23]. 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., [4]), 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 [17] 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., [16]) 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[3] (see also [4]) 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 [24] observed that the weak A ∞ condition is enough for (6.3).The same argument applies to (5.1).In [22], 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 [25], Sawyer proved that if ε > 0, then the C p+ε condition is sufficientfor (6.3). Using almost the same arguments, Yabuta [28] 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 [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

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. [6]): 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 [5] shows thatC p+ε ⇒ M p for any ε > 0.Acknowledgment. I am grateful to the referee for useful remarks andcorrections.References[1] 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.[2] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, NewYork, 1988.[3] R.R. Coifman, Distribution function inequalities for singular integrals, Proc.Nat. Acad. Sci. USA 69 (1972), 2838–2839.[4] R.R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functionsand singular integrals, Studia. Math. 15 (1974), 241–250.[5] 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.[6] C. Fefferman and E.M. Stein, Some maximal inequalities, Amer. J. Math., 93(1971), 107–115.[7] C. Fefferman and E.M. Stein, H p spaces of several variables, Acta Math. 129(1972), 137–193.[8] L. Grafakos, Classical and modern Fourier analysis, Prentice Hall, 2004.[9] P. Janakiraman, Limiting weak-type behavior for singular integral and maximaloperators, Trans. Amer. Math. Soc. 358 (2006), no. 5, 1937–1952.[10] B. Jawerth and A. Torchinsky, Local sharp maximal functions, J. Approx.Theory, 43 (1985), 231–270.[11] F. John, Quasi-isometric mappings, Seminari 1962-1963 di Analisi, Algebra,Geometria e Topologia, Rome, 1965.[12] 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.[13] S. Korry, Fixed points of the Hardy-Littlewood maximal operator, Collect.Math. 52 (2001), no. 3, 289–294.[14] A.K. Lerner, On the John-Strömberg characterization of BMO for nondoublingmeasures, Real. Anal. Exchange, 28 (2003), no. 2, 649–660.[15] A.K. Lerner, Weighted norm inequalities for the local sharp maximal function,J. Fourier Anal. Appl. 10 (2004), no. 5, 465–474.[16] A.K. Lerner, Weighted rearrangement inequalities for local sharp maximalfunctions, Trans. Amer. Math. Soc., 357 (2005), no. 6, 2445–2465.[17] A.K. Lerner, BMO-boundedness of the maximal operator for arbitrary measures,Israel J. Math., 159 (2007), no. 1, 243–252.[18] A.K. Lerner and S. Ombrosi, A boundedness criterion for general maximaloperators, Publ. Mat., 54 (2010) no. 1, 53–71.

**FEFFERMAN**-**STEIN** **INEQUALITY** 21[19] 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.[20] 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.[21] 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.[22] 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.[23] J.L. Rubio de Francia, Factorization theory and A p weights, Amer. J. Math.106 (1984), 533–547.[24] 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.[25] E.T. Sawyer, Norm inequalities relating singular integrals and the maximalfunction, Studia Math., 75 (1983), 253–263.[26] J.-O. Strömberg, Bounded mean oscillation with Orlicz norms and duality ofHardy spaces, Indiana Univ. Math. J., 28 (1979), 511–544.[27] 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.[28] 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