SOME REMARKS ON THE FEFFERMAN-STEINu.math.biu.ac.il/~lernera/Fefferman-Stein.pdf · 2010. 3....

SOME REMARKS ON THE FEFFERMAN-STEININEQUALITY

ANDREI K. LERNER

Abstract. We investigate the Fefferman-Stein inequality relateda function f and the sharp maximal function f# on a Banachfunction space X. It is proved that this inequality is equivalent toa certain boundedness property of the Hardy-Littlewood maximaloperatorM . The latter property is shown to be self-improving. Weapply our results in several directions. First, we show the existenceof nontrivial spaces X for which the lower operator norm of M isequal to 1. Second, in the case when X is the weighted Lebesguespace Lp(w), we obtain a new approach to the results of Sawyerand Yabuta concerning the Cp condition.

1. Introduction

Given a locally integrable function f on Rn, the Fefferman-Steinmaximal functions f# is defined by

f#(x) = supQ∋x

1

|Q|

∫Q

|f(y)− fQ|dy,

where fQ = 1|Q|

∫Qf , and the supremum is taken over all cubes Q with

sides parallel to the axes containing the point x. The sharp functionf# is closely related to the space BMO, namely, we have

BMO(Rn) = f ∈ L1loc(Rn) : f# ∈ L∞(Rn),

and∥f∥BMO = ∥f#∥L∞ .

A fundamental inequality of Fefferman and Stein [7] says that undersome growth restrictions on f ,

(1.1) ∥f∥Lp(Rn) ≤ c∥f#∥Lp(Rn) (1 < p < ∞).

Originally, this result was applied to describing the intermediate spacesbetween BMO and Lp. It was realized later that (1.1) is also a veryconvenient tool for obtaining Lp-norm inequalities involving variousoperators in harmonic analysis. Typically, one can obtain a pointwise

2000 Mathematics Subject Classification. 42B20,42B25,46E30.Key words and phrases. Maximal operators, Banach function spaces, weights.

1

2 ANDREI K. LERNER

estimate (T1f)#(x) ≤ cT2f(x), where T1 is a certain singular-type op-

erator, and T2 is a maximal-type operator. Combining this with (1.1)yields a norm estimate of T1 by T2.

In this paper we investigate the Fefferman-Stein inequality on a Ba-nach function space X instead of Lp. We obtain a characterization ofthis inequality in terms of the Hardy-Littlewood maximal operator Mdefined by

Mf(x) = supQ∋x

1

|Q|

∫Q

|f(y)|dy.

Let X be a Banach function space over Rn equipped with Lebesguemeasure. Denote by X ′ the associate space with norm

∥f∥X′ = sup∥g∥X=1

∫Rn

|f(x)g(x)|dx.

Let S0(Rn) be the space of measurable functions f on Rn such that forany α > 0,

µf (α) = |x ∈ Rn : |f(x)| > α| < ∞.

Our first result is the following.

Theorem 1.1. The following statements are equivalent:

(i) there exists c > 0 such that for any f ∈ S0(Rn),

∥f∥X ≤ c∥f#∥X ;

(ii) there exists c > 0 such that for any f ∈ X ′ and g ∈ L1loc(Rn),

(1.2)

∫Rn

Mf(x)|g(x)|dx ≤ c∥f∥X′∥Mg∥X .

Let Mrf = (M |f |r)1/r. By Holder’s inequality, Mf ≤ Mrf if r > 1.In a recent paper [20] (see also [18] for a different proof), the authorsestablished that the following Hardy-Littlewood-type inequality

∥Mf∥X ≤ c∥f∥Xis self-improving in the sense that it implies the boundedness of Mr

on X for some r > 1. Our next result shows that the Fefferman-Stein-type inequality is also self-improving, although this phenomenonis expressed in a somewhat indirect way.

Theorem 1.2. Inequality (1.2) holds if and only if there exist c > 0and r > 1 such that for any f ∈ X ′ and g ∈ L1

loc(Rn),∫Rn

Mrf(x)|g(x)|dx ≤ c∥f∥X′∥Mg∥X .

FEFFERMAN-STEIN INEQUALITY 3

The proof of Theorem 1.1 is based on two ingredients. The first oneis the adjoint of linearizations of M , and its sharp function estimateestablished by de la Torre [27]. The second one is a certain dualityrelation involving the local sharp maximal function proved by the au-thor in [15]. Theorem 1.2 is based on Theorem 1.1 and on the Rubiode Francia algorithm.

Theorem 1.1 can be applied to some questions related to the behaviorof M on general Banach function spaces. For instance, in Section 4 weshow the existence of nontrivial spaces X (that is, different from L∞) inany dimension for which the lower operator norm of M is equal to 1. Itis interesting to compare this with the fact that M has no fixed pointsin any rearrangement-invariant space, regardless of the dimension (aswas observed in [21]).

In Sections 5 and 6, we consider the case when X is the weightedLebesgue space Lp(w). Observe that the question about a necessaryand sufficient condition on a weight w for which the Fefferman-Steininequality holds on Lp(w) is still open. The currently best knownresult in this direction is due to Yabuta [28] who showed that theCp condition (see (6.2) below) is necessary and the Cp+ε is sufficient.Note that the method in [28] follows the work of Sawyer [25] wherethe same conditions were established regarding Coifman’s inequalityrelated singular integrals and the maximal operator.

Using Theorems 1.1 and 1.2, we first obtain several statements equiv-alent to the Fefferman-Stein inequality. For example, one of them is thefollowing: there exist c > 0 and r > 1 such that for any f ∈ L1

loc(Rn),

(1.3)

∫Rn

Mp,r(f, w)|f | dx ≤ c

∫Rn

(Mf)pw dx,

where

Mp,r(f, w)(x) = supQ∋x

(1

|Q|

∫Q

|f |)p−1(

1

|Q|

∫Q

wr

)1/r

(note that the maximal functions like Mp,r have been recently shownto be very useful in the theory of multilinear singular integrals [19]).

Next, we introduce a Cp condition which is between Cp and Cp+ε for anyε > 0, and which is sufficient for (1.3). This gives a slight improvementof Yabuta’s work [28] as well as a new approach to Sawyer’s result [25].However, a question about the full characterization of the Fefferman-Stein Lp(w)-inequality in terms of w remains open.

4 ANDREI K. LERNER

2. Preliminaries

2.1. Banach function spaces. For a general account of Banach func-tion spaces 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∥g∥X′=1

∫Rn

|f(x)g(x)|dx.

By Fatou’s lemma [2, p. 5], if fn → f a.e., and if lim infn→∞

∥fn∥X < ∞,

then f ∈ X, and

(2.2) ∥f∥X ≤ lim infn→∞

∥fn∥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 ∈ L1loc(Rn), there is a linear operator Mf such

that

c1Mf(x) ≤ Mff(x) ≤ c2Mf(x),

and for any g ∈ L1loc(Rn),

(2.3) (M⋆fg)

#(x) ≤ c3Mg(x),

where M⋆f is the adjoint of Mf , and the constants ci depend only on n.

Note also that the construction ofMf shows that for any g ∈ L1(Rn),

(2.4) ∥M⋆fg∥L1(Rn) ≤ c∥g∥L1(Rn),

where the constant c depends only on n.Since the main properties of Mf expressed in (2.3) and (2.4) do not

depend 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 Rn is defined by

f ∗(t) = infλ > 0 : |x ∈ Rn : |f(x)| > λ| ≤ t

(0 < t < ∞).

Given a measurable function f , the local maximal functions mλfand M#

λ f are defined by

mλf(x) = supQ∋x

(fχQ

)∗(λ|Q|

)(0 < λ < 1)


and

M#λ f(x) = sup

Q∋xinfc

((f − c)χQ

)∗(λ|Q|

)(0 < λ < 1),

respectively, where the supremum is taken over all cubes containingthe point x.

These functions were introduced by Stromberg [26]; in particular,

the definition of M#λ f was motivated by an alternate characterization

of BMO given by John [11]. Roughly speaking, f# is the Hardy-

Littlewood maximal operator of M#λ f (see [10, 14]):

(2.5) c1MM#λ f(x) ≤ f#(x) ≤ c2MM#

λ f(x).

The following theorem was proved in [15].

Theorem 2.2. For any measurable function f ∈ S0(Rn) and any g ∈L1

loc(Rn), ∫Rn

|f(x)g(x)|dx ≤ c

∫Rn

M#λ f(x)Mg(x)dx,

where 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 ∈ Rn,

(2.6) Mf(x) ≤ 3f#(x) +m1/2f(x),

and

(2.7) M#λ (m1/2f)(x) ≤ 4M#

λ/2·9nf(x).

Inequality (2.7) combined with (2.5) yields

(2.8) (m1/2f)#(x) ≤ cnf

#(x).

It follows directly from the definitions that

(2.9) x ∈ Rn : mλf(x) > α ⊂ x ∈ Rn : Mχ|f |>α(x) ≥ λ.

Lemma 2.4. For any non-negative functions f and g,∫Rn

(mλf)gdx ≤ c

∫Rn

f(Mg)dx,

where 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]:∫

x:Mφ>ξg dx ≤ c

ξ

∫Rn

|φ|(Mg) dx.

6 ANDREI K. LERNER

We have∫Rn

(mλf)g dx =

∫ ∞

0

∫mλf>α

g dxdα ≤∫ ∞

0

∫Mχf>α(x)≥λ

g dxdα

≤ cλ,n

∫ ∞

0

∫f>α

Mg dxdα = cλ,n

∫Rn

f(Mg)dx,

which proves the lemma.

3. Proofs of Theorems 1.1 and 1.2

Proof of Theorem 1.1. Let us show first that (i) ⇒ (ii). We can sup-pose that f, g ≥ 0. Also, it is enough to assume, for instance, thatg is compactly supported, and hence g ∈ L1(Rn). The general casewill follow by the standard limiting argument. If g ∈ L1(Rn), thenM⋆g ∈ L1(Rn), and thus M⋆g ∈ S0(Rn). Therefore, by Theorem 2.1,

∥M⋆g∥X ≤ c∥(M⋆g)#∥X ≤ c∥Mg∥X .From this, applying Theorem 2.1 again, we get∫

Rn

Mf(x)g(x)dx ≤ c

∫Rn

Mf(x)g(x)dx

= c

∫Rn

f(x)M⋆g(x)dx

≤ 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∫Rn

|f(x)φ(x)|dx ≤ c

∫Rn

M#λ f(x)Mφ(x)dx

≤ c∥φ∥X′∥MM#λ f∥X

≤ c∥φ∥X′∥f#∥X .From this, by (2.1), we obtain (i). 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∫

Rn

|f(x)g(x)|dx ≤ c

∫Rn

M#λ f(x)M∆g(x)dx.

Therefore, taking into account the proof of Theorem 1.1, in order toverify the Fefferman-Stein property of X, it is enough to check that∫

Rn

(M∆f)|g|dx ≤ c∥f∥X′∥Mg∥X .


In 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 ∈ S0(Rn))

and

(3.2) ∥Mf∥X ≤ c∥f#∥X (f ∈ S0(Rn))

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 + ∥m1/2f∥X .Next, from (2.9) and from the weak type (1, 1) property of M ,

µm1/2f (α) ≤ cnµf (α) (α > 0).

Therefore, f ∈ S0(Rn) ⇒ m1/2f ∈ S0(Rn). Hence, combining (3.1)with (2.8), we get

∥m1/2f∥X ≤ c∥(m1/2f)#∥X ≤ c∥f#∥X ,

which along with (3.3) implies (3.2). Proof of Theorem 1.2. Inequality

(3.4)

∫Rn

Mrf(x)|g(x)|dx ≤ c∥f∥X′∥Mg∥X .

trivially 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 constant

A > 0 such that for any k ∈ N,

(3.5)

∫Rn

Mkf(x)g(x)dx ≤ Ak∥f∥X′∥Mg∥X ,

where Mk 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 Fran-cia algorithm [23]. We set

(Rf)(x) = |f(x)|+∞∑k=1

1

(2A)kMkf(x).

Then |f | ≤ Rf and M(Rf)(x) ≤ 2A(Rf)(x). We have that (Rf) isan A1 weight, and hence it satisfies the reverse Holder inequality (see,e.g., [4]), which means that there exists r > 1 such that

Mrf(x) ≤ Mr(Rf)(x) ≤ cM(Rf)(x) ≤ 2Ac(Rf)(x).

8 ANDREI K. LERNER

Combining this inequality with (3.5), we get∫Rn

Mrf(x)g(x)dx ≤ c

∫Rn

(Rf)(x)g(x)dx

≤ c

∫Rn

|f(x)|g(x)dx+ c∞∑k=1

(1/2)k∥f∥X′∥Mg∥X

≤ c∥f∥X′∥Mg∥X .

We prove (3.5) by induction with respect to k. For k = 1 (3.5) isjust (1.2). Suppose that (3.5) holds for k = l, and let us show that itholds for k = l+1. By the induction assumption and by Theorem 2.1,∫

Rn

M l+1f(x)g(x)dx =

∫Rn

M(M lf)(x)g(x)dx

≤ c

∫Rn

M(M lf)(x)g(x)dx = c

∫Rn

M lf(x)M⋆g(x)dx

≤ cAl∥f∥X′∥M(M⋆g)∥X .

As in the previous proof, one can assume that g ∈ L1(Rn), and henceM⋆g ∈ S0(Rn). By Theorem 1.1 and Lemma 3.2,

∥M(M⋆g)∥X ≤ c∥(M⋆g)#∥X ≤ c∥Mg∥X .

Therefore, ∫Rn

M l+1f(x)g(x)dx ≤ c′Al∥f∥X′∥Mg∥X .

This proves (3.5) with A = c′, and hence the theorem is proved.

4. On the lower operator norm of M

It was shown by Korry [13] (see also [9]) that the centered maximaloperator M c (where the supremum is taken over all balls centered at x)has a fixed point f ∈ Lp(Rn) (that is, M cf = f) if and only if n ≥ 3and p > n/(n − 2). Then Martın and Soria [21] extended this resultto more general rearrangement-invariant (r.i.) spaces. Also, it wasremarked in [21] that the usual uncentered maximal operator M hasno fixed points in any r.i. space, regardless of the dimension.

We consider a related question about the lower operator norm of Mdefined by

∥M∥lX = inf∥f∥X=1

∥Mf∥X .

It is clear that ∥M∥lX ≥ 1 for any X and any dimension. Taking intoaccount the above mentioned results about fixed points, it is natural


to ask whether there exists a Banach function space X different fromL∞ such that ∥M∥lX = 1.

If X = L∞, we trivially have ∥M∥lL∞ = 1. However, the existenceof X different from L∞ for which ∥M∥lX = 1 is not an obvious fact.For example, by Riesz’s sunrise lemma [8, p. 93],

|x ∈ R : Mf(x) > α| ≥ 1

α

∫x∈R:|f(x)|>α

|f |.

Integrating this inequality gives

∥Mf∥Lp(R) ≥( p

p− 1

)1/p∥f∥Lp(R),

and hence for any p > 1,

∥M∥lLp(R) ≥( p

p− 1

)1/p> 1.

Using a non-standard proof of the Fefferman-Stein inequality com-bined with 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∥lX = 1.

Remark 4.2. Assume that X is a r.i. space. Let αX and βX be thelower and upper Boyd indices, respectively [2, p. 149]. In general,0 ≤ αX ≤ βX ≤ 1. By the Lorentz-Shimogaki theorem [2, p. 154], Mis bounded on X iff βX < 1. Also, it is well known that βX′ = 1− αX .Therefore, by Theorem 4.1, if αX = 0 and βX < 1, then ∥M∥lX = 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 ∈ S0(Rn),

∥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 ∈ L1loc(Rn) and for a.e. x ∈ Rn,

(4.1) MMf(x) ≤ cMf#(x) +Mf(x),

where the constant c depends only on n.

10 ANDREI K. LERNER

Lemma 4.5. If f ∈ S0(Rn) ∩ L∞, then there is a sequence of boundedand compactly supported measurable functions fj such that for a.e.x ∈ Rn,

(4.2) limj→∞

fj(x) = f(x)

and

(4.3) (fj)#(x) ≤ cf#(x),


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∥lX > 1, then M is bounded on X ′. By Corol-lary 4.3, 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 ≤ c

A− 1∥Mf#∥X ≤ c′∥f#∥X .

Take now an arbitrary f ∈ S0(Rn) ∩ L∞. By Lemma 4.5, there isa sequence fj satisfying (4.2) and (4.3). Since each fj is boundedand compactly supported, we have that ∥fj∥X < ∞ (we have used herethat if |E| < ∞, then ∥χE∥X < ∞ [2, p. 2]). Therefore, by (4.3) and(4.4),

∥fj∥X ≤ c∥f#∥X .From this, applying (4.2) and (2.2), we get the Fefferman-Stein inequal-ity on X for any f ∈ S0(Rn) ∩ L∞.

Finally, if f is an arbitrary function from S0(Rn), consider fN(x) =min(f(x), N). Then clearly fN ∈ S0(Rn) ∩ L∞. Also (see, e.g., [8,p. 519]), (fN)

#(x) ≤ cf#(x). Therefore,

∥fN∥X ≤ c∥f#∥X .Applying (2.2) again completes the proof. Proof of Theorem 4.4. This theorem was proved in [17] in the one-dimensional case. 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) +Mm1/2f(x).


Let x, y ∈ Q and let Q′ be an arbitrary cube containing y. We havethat either Q ⊂ 3Q′ or Q′ ⊂ 3Q. If Q ⊂ 3Q′, then

(fχQ′)∗(|Q′|/2) ≤ ((f − f3Q′)χQ′)∗(|Q′|/2) + |f |3Q′

≤ 2

|Q′|

∫Q′|f − f3Q′|+ |f |3Q′

≤ 2 · 3nf#(x) +Mf(x).

If Q′ ⊂ 3Q, then

(fχQ′)∗(|Q′|/2) ≤ ((f − f3Q)χQ′)∗(|Q′|/2) + |f |3Q≤ m1/2((f − f3Q)χ3Q)(y) +Mf(x).

Therefore,

m1/2f(y) ≤ m1/2((f − f3Q)χ3Q)(y) + 2 · 3nf#(x) +Mf(x).

From this, using Lemma 2.4 with g ≡ 1, we get

1

|Q|

∫Q

m1/2f(y)dy ≤ 1

|Q|∥m1/2((f − f3Q)χ3Q)∥L1

+ 2 · 3nf#(x) +Mf(x)

≤ c

|3Q|

∫3Q

|f − f3Q|+ 2 · 3nf#(x) +Mf(x)

≤ cf#(x) +Mf(x),

and henceMm1/2f(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 median

value. Given a cube Q and a measurable function f , by a median valueof f over Q we mean a, possibly nonunique, real number mf (Q) suchthat

|x ∈ Q : f(x) > mf (Q)| ≤ |Q|/2and

|x ∈ Q : f(x) < mf (Q)| ≤ |Q|/2.It is easy to show (see, e.g., [16]) that for any constant c,

(4.6) |mf (Q)− c| ≤((f − c)χQ

)∗(|Q|/2).

Fix an open cube Q0. Given x ∈ Q0, let Qx be the unique cubecentered at x such that ℓ(Qx) = dist(∂Q0, Qx), where ∂Q and ℓ(Q) arethe boundary and the side length of Q, respectively. Set

AQ0f(x) =(f(x)−mf (Qx)

)χQ0(x).

12 ANDREI K. LERNER

Proposition 4.6. For all x ∈ Rn,

(4.7) (AQ0f)#(x) ≤ cf#(x),


Proof. Take an arbitrary cube Q containing x, and consider

Ω(Q) ≡ infc

1

|Q|

∫Q

|AQ0f(y)− c|dy.

If Q ∩ Q0 = ∅, we trivially have Ω(Q) = 0. Therefore, assume thatQ ∩Q0 = ∅. There are two cases.

Case 1. Suppose that there exists y0 ∈ Q ∩ Q0 such that ℓ(Q) ≤ℓ(Qy0)/2. Then Q ⊂ 2Qy0 ⊂ Q0. Next, a simple geometrical observa-tion shows that for any y ∈ 2Qy0 we get ℓ(Qy0)/3 ≤ ℓ(Qy) ≤ 5ℓ(Qy0)/2.Hence, Qy ⊂ 5Qy0 and |Qy0 | ≤ 3n|Qy|. Therefore, by (4.6), for anyy ∈ Q,

|mf (Qy)− c| ≤((f − c)χQy

)∗(|Qy|/2)

≤ 2

|Qy|

∫Qy

|f − c| ≤ 2 · 15n

|5Qy0 |

∫5Qy0

|f − c|.

Thus

Ω(Q) ≤ infc

1

|Q|

∫Q

|f(y)− c|dy + infc

1

|Q|

∫Q

|mf (Qy)− c|dy

≤ f#(x) + infc

2 · 15n

|5Qy0 |

∫5Qy0

|f − c| ≤ (2 · 15n + 1)f#(x).

Case 2. Assume now that ℓ(Qy) < 2ℓ(Q) for any y ∈ Q ∩Q0. ThenQy ⊂ 3Q, and hence, by (4.6),

|f3Q −mf (Qy)| ≤((f − f3Q)χQy

)∗(|Qy|/2)

≤ m1/2

((f − f3Q)χ3Q

)(y).

Therefore, applying Lemma 2.4 with g ≡ 1, we get

Ω(Q) ≤ 1

|Q|

∫Q

|AQ0f(y)|dy

≤ 1

|Q|

∫Q

|f(y)− f3Q|dy +1

|Q|

∫Q

|f3Q −mf (Qy)|dy

≤ 3nf#(x) +1

|Q|∥m1/2

((f − f3Q)χ3Q

)∥L1 ≤ cf#(x).

Combining both cases yields

1

|Q|

∫Q

|AQ0f(y)− (AQ0f)Q|dy ≤ 2Ω(Q) ≤ cf#(x),


proving (4.7). Proof of Lemma 4.5. Set Qj = (−j, j)n and fj = AQj

f . It is clear thatfj is bounded and compactly supported. Also, by (4.7),

(fj)#(x) ≤ cf#(x).

Further, for any x ∈ Qj/2 and for a cube Qx centered at x withℓ(Qx) = dist(∂Qj, Qx) we have |Qx| ≥ (j/3)n. Hence, by (4.6), forx ∈ Qj/2,

|f(x)− fj(x)| = |mf (Qx)| ≤ f ∗((j/3)n/2).Since f ∈ S0(Rn) 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 = Lp(w)

We consider here the case when X = Lp(w), 1 < p < ∞, where wis a weight, that is, a non-negative locally integrable function. In this

case X ′ = Lp′(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∥Lp(w) ≤ c∥f#∥Lp(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 ∈ S0(Rn);(ii) there exist c > 0 and r > 1 such that for any f ∈ L1

loc(Rn),∫Rn

Mr

((Mf)p−1w

)|f | dx ≤ c

∫Rn

(Mf)pw dx;

(iii) there exist c > 0 and r > 1 such that for any f ∈ L1loc(Rn),∫

Rn


∫Rn

(Mf)pw dx,

where

Mp,r(f, w)(x) = supQ∋x

(1

|Q|

∫Q

|f |)p−1(

1

|Q|

∫Q

wr

)1/r

.

Proof. By Theorems 1.1 and 1.2, if (i) holds, then∫Rn

Mrφ|f | dx ≤ c∥φ∥Lp′ (σ)∥Mf∥Lp(w).

14 ANDREI K. LERNER

Setting here φ = (Mf)p−1w, we get (ii).Next, (ii) implies (iii) trivially since

Mp,r(f, w)(x) ≤ Mr

((Mf)p−1w

)(x).

It remains to prove that (iii) ⇒ (i). By Remark 3.1, it suffices toshow that

(5.2)

∫Rn

(M∆φ)|f | dx ≤ c∥φ∥Lp′ (σ)∥Mf∥Lp(w).

We can suppose f, φ ≥ 0. By the Calderon-Zygmund decomposition,

Ωk = x ∈ Rn : M∆φ(x) > 2k = ∪jQkj ,

where Qkj are pairwise disjoint dyadic cubes such that

2k <1

|Qkj |

∫Qk

j

φ ≤ 2n+k.

From this, setting Ek = Ωk \ Ωk+1 and Eki = Ek ∩Qk

i , we have∫Rn

(M∆φ)f =∑k

∫Ek

(M∆φ)f ≤ 2∑k,j

φQkj

∫Ek

j

f(5.3)

= 2

∫Rn

φ∑k,j

( 1

|Qkj |

∫Ek

j

f)χQk

j

≤ 2∥φ∥Lp′ (σ)∥∑k,j

( 1

|Qkj |

∫Ek

j

f)χQk

j∥Lp(w).

Next we note that

|Qkj ∩ Ωk+l| =

∑Qk+l

i ⊂Qkj

|Qk+li | < 1

2(k+l)

∑Qk+l

i ⊂Qkj

∫Qk+l

i

φ

≤ 1

2(k+l)

∫Qk

j

φ ≤ 2n−l|Qkj |.

Therefore, setting

Tlf(x) =∑k,j

( 1

|Qkj |

∫Ek

j

f)χQk

j∩Ek+l


and using (iii) and Holder’s inequality, we obtain∫Rn

(Tlf)pw =

∑k,j

( 1

|Qkj |

∫Ek

j

f)pw(Qk

j ∩ Ek+l)

≤ c2−l/r′∑k,j

( 1

|Qkj |

∫Ek

j

f)p−1( 1

|Qkj |

∫Qk

j

wr)1/r ∫

Ekj

f

≤ c2−l/r′∑k,j

∫Ek

j

Mp,r(f, w)f

≤ c2−l/r′∫Rn

Mp,r(f, w)f ≤ c2−l/r′∫Rn

(Mf)pw.

Hence,

∥∑k,j

( 1

|Qkj |

∫Ek

j

f)χQk

j∥Lp(w) ≤

∞∑l=0

∥Tlf∥Lp(w)

≤ c∞∑l=0

2−l/pr′∥Mf∥Lp(w) ≤ c∥Mf∥Lp(w),

which combined with (5.3) yields (5.2), and therefore the theorem isproved.

6. On the Cp condition

Given a measurable set E ⊂ Rn, let w(E) =∫Ew(x)dx. We say that

w satisfies the A∞ condition if there are positive constants c, δ suchthat for any cube Q and any subset E ⊂ Q,

(6.1) w(E) ≤ c(|E|/|Q|)δw(Q).

Given r > 0, denote by rQ the cube concentric with Q having di-ameter r times that of Q. If w(Q) on the right-hand side of (6.1) isreplaced by w(2Q), then the corresponding condition is called the weakA∞ condition.

We say that w satisfies the Cp condition if there are positive constantsc, δ such that for any cube Q and any subset E ⊂ Q,

(6.2) w(E) ≤ c(|E|/|Q|)δ∫Rn

(MχQ)pw.

It is easy to see that

A∞ ⊂ weakA∞ ⊂ Cp.

Also, Cp+ε ⊂ Cp if ε > 0. An interesting analysis of the Cp conditioncan be found in [12]. In particular, it was shown there that Cp =∪ε>0Cp+ε.

16 ANDREI K. LERNER

Let T be a Calderon-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 Calderon-Zygmund operators as well.

The weighted theory of the Fefferman-Stein inequality has been de-veloped in parallel to the one of Coifman’s inequality relating singularintegrals and the maximal function. Namely, it was proved by Coif-man [3] (see also [4]) that if w ∈ A∞, then for any appropriate f ,

(6.3) ∥Tf∥Lp(w) ≤ c∥Mf∥Lp(w) (1 < p < ∞).

This result was based on a good-λ inequality related Tf 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 Cp condition is necessary for (6.3), and he con-jectured that Cp is also sufficient. Note that this question is still open.In [25], Sawyer proved that if ε > 0, then the Cp+ε condition is suffi-cient for (6.3). Using almost the same arguments, Yabuta [28] showedthat Cp is necessary for (5.1) and Cp+ε is sufficient.

Here we give a completely different proof of a slightly improved ver-sion of Yabuta’s result. Given p > 1, let φp be a non-decreasing,doubling (i.e., φp(2t) ≤ cφp(t)) function on (0, 1) satisfying∫ 1

0

φp(t)dt

tp+1< ∞.

We say that a weight w satisfies the Cp condition if there are positiveconstants c, δ such that for any cube Q and any subset E ⊂ Q,

w(E) ≤ c(|E|/|Q|)δ∫Rn

φp(MχQ)w.

Theorem 6.1. The Cp condition is necessary for

(6.4)

∫Rn


∫Rn

(Mf)pw dx,

and the Cp is sufficient.


Remark 6.2. It is easy to see that φp(t) ≤ ctp, and hence Cp ⊂ Cp. Onthe other hand, taking φp(t) such that tp+ε ≤ cφp(t) for any ε > 0 (for

example, φp(t) = tp log−2(1 + 1/t)), we get ∪ε>0Cp+ε ⊂ Cp. 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) (|Tf |α)#(x) ≤ c(Mf)α(x) (0 < α < 1).

The Cp+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 = χQ, we obtain

(6.6)

(1

|Q|

∫Q

wr

)1/r

≤ c1

|Q|

∫Rn

(MχQ)pw.

From this, by Holder’s inequality we get the Cp condition with δ = 1/r′.

Suppose now that w ∈ Cp. Then for 0 < t < |Q| (cf. [2, p. 53]),∫ t

0

(wχQ)∗(τ)dτ = sup

E⊂Q,|E|=t

w(E) ≤ c(t/|Q|)δ∫Rn

φp(MχQ)w.

From this,

(wχQ)∗(t) ≤ 1

t

∫ t

0

(wχQ)∗(τ)dτ ≤ c

t1−δ

1

|Q|δ

∫Rn

φp(MχQ)w.

Hence, fixing some 1 < r < 11−δ

, for 0 < λ < 1 we get∫Q

wr =

∫ |Q|

0

(wχQ)∗(t)rdt

=

∫ λ|Q|

0

(wχQ)∗(t)rdt+

∫ |Q|

λ|Q|(wχQ)

∗(t)rdt

≤ cλ1−r(1−δ)|Q|(

1

|Q|

∫Rn

φp(MχQ)w

)r

+ |Q|(wχQ)∗(λ|Q|)r.

Therefore,

(6.7)

(1

|Q|

∫Q

wr

)1/r

≤ cλ1/r−(1−δ)

|Q|

∫Rn

φp(MχQ)w + (wχQ)∗(λ|Q|).

18 ANDREI K. LERNER

Further, if x ∈ Q, then(1

|Q|

∫Q

|f |)p−1

1

|Q|

∫Rn

φp(MχQ)w

≤ c

(1

|Q|

∫Q

|f |)p−1

(1

|Q|

∫Q

w +1

|Q|

∞∑k=1

φp(2−kn)

∫2kQ\2k−1Q

w

)

≤ c∞∑k=1

2kpnφp(2−kn)

(1

|2kQ|

∫2kQ

|f |)p−1(

1

|2kQ|

∫2kQ

w

)≤ c(∫ 1

0

φp(t)dt

tp+1

)Mp,1(f, w)(x).

We now observe that it is enough to prove (6.4) for compactly sup-ported f . Also, one can assume that the right-hand side of (6.4) isfinite, otherwise there is nothing to prove. This means, in particular,that ∫

Rn

w(x)

1 + |x|pndx < ∞.

It follows from this that

sup0∈Q,|Q|≥1

1

|Q|p

∫Q

w ≤ c

∫Rn

w(x)

1 + |x|pndx < ∞,

which easily implies thatMp,1(f, w)(x) < ∞ a.e. Since (wχQ)∗(λ|Q|) ≤

1λ|Q|

∫Qw, we obtain also that

supQ∋x

(1

|Q|

∫Q

|f |)p−1

(wχQ)∗(λ|Q|) < ∞ a.e.

Therefore, (6.7) shows that Mp,r(f, w)(x) < ∞ a.e.Hence, applying (6.7) again and using Holder’s inequality, we get

Mp,r(f, w)(x) ≤ cλ1/r−(1−δ)Mp,1(f, w)(x) +mλ

((Mf)p−1w

)(x)

≤ cλ1/r−(1−δ)Mp,r(f, w)(x) +mλ

((Mf)p−1w

)(x).

From this, taking λ small enough, we obtain

(6.8) Mp,r(f, w)(x) ≤ cmλ

((Mf)p−1w

)(x).

This inequality combined with Lemma 2.4 yields∫Rn


∫Rn

mλ

((Mf)p−1w

)|f | dx

≤ c

∫Rn

(Mf)pw dx,

and therefore the theorem is proved.


We make several concluding remarks. The question about a neces-sary and sufficient condition on w for which (6.4) (or, equivalently, the

Fefferman-Stein inequality (5.1)) holds remains open. The Cp condi-tion is probably not a necessary condition for (6.4). Indeed, the proof

of Theorem 6.1 shows that the Cp condition implies (6.8). This alongwith Lemma 2.4 gives that for all suitable f and g,∫

Rn

Mp,r(f, w)|g| dx ≤ c

∫Rn

(Mf)p−1(Mg)w dx,

which seems to be much stronger than (6.4).Next, the methods used in the proof of Theorem 6.1 show easily that

the Cp condition is equivalent to (6.6). Moreover, the Cp condition isequivalent to the following statement: there exist c > 0 and r > 1 suchthat for each cube Q and any g ∈ L1

loc(Rn),

(6.9)

(1

|Q|

∫Q

|g|)p(

1

|Q|

∫Q

wr

)1/r

≤ c1

|Q|

∫Rn

(M(gχQ)

)pw.

Indeed, (6.9) with g ≡ 1 gives (6.6). On the other hand, if xQ is thecenter of Q, then(

1

|Q|

∫Q

|g|)p ∫

Rn

(MχQ)pw ≤ c

∫Rn

( ∫Q|g|

|x− xQ|n + |Q|

)p

w

≤ c

∫Rn

(M(gχQ)

)pw,

which along with (6.6) implies (6.9).Denote byMp 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 fj with pairwise disjoint supports,∑

j

∫Rn

(Mfj)pw ≤ c

∫Rn

(M(∑j

fj))pw.

Then the Cp ∩ Mp condition yields (5.1). To show this, we keep thesame notation as in the proof of Theorem 5.1. Using (6.9) along withthe Mp condition, we get∫

Rn

(Tlf)pw ≤ c2−l/r′

∑k,j

( 1

|Qkj |

∫Ek

j

f)p( 1

|Qkj |

∫Qk

j

wr)1/r

|Qkj |

≤ c2−l/r′∑k,j

∫Rn

(M(fχEk

j))pw ≤ c2−l/r′

∫Rn

(Mf)pw,

20 ANDREI K. LERNER

and now we can follow the proof of Theorem 5.1. The above argumentraises a natural question whether Cp ⇒ Mp. Observe that the sharpfunction estimate of the vector-valued maximal operator [5] shows thatCp+ε ⇒ Mp for any ε > 0.

Acknowledgment. I am grateful to the referee for useful remarks andcorrections.

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Department of Mathematics, Bar-Ilan University, 52900 Ramat Gan,Israel

E-mail address: [email protected]

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