SOME REMARKS ON THE FEFFERMAN-STEIN
INEQUALITY
ANDREI K. LERNER
Abstract. We investigate the Fefferman-Stein inequality related
a function f and the sharp maximal function f # on a Banach
function space X. It is proved that this inequality is equivalent to
a certain boundedness property of the Hardy-Littlewood maximal
operator M . The latter property is shown to be self-improving. We
apply our results in several directions. First, we show the existence
of nontrivial spaces X for which the lower operator norm of M is
equal to 1. Second, in the case when X is the weighted Lebesgue
space Lp (w), we obtain a new approach to the results of Sawyer
and Yabuta concerning the Cp condition.
1. Introduction
Given a locally integrable function f on Rn , the Fefferman-Stein
maximal functions f # is defined by
∫
1
#
|f (y) − fQ |dy,
f (x) = sup
Q∋x |Q| Q
∫
1
where fQ = |Q|
f , and the supremum is taken over all cubes Q with
Q
sides parallel to the axes containing the point x. The sharp function
f # is closely related to the space BM O, namely, we have
BM O(Rn ) = {f ∈ L1loc (Rn ) : f # ∈ L∞ (Rn )},
and
∥f ∥BM O = ∥f # ∥L∞ .
A fundamental inequality of Fefferman and Stein [7] says that under
some growth restrictions on f ,
(1.1)
∥f ∥Lp (Rn ) ≤ c∥f # ∥Lp (Rn )
(1 < p < ∞).
Originally, this result was applied to describing the intermediate spaces
between BM O and Lp . It was realized later that (1.1) is also a very
convenient tool for obtaining Lp -norm inequalities involving various
operators 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 (T1 f )# (x) ≤ cT2 f (x), where T1 is a certain singular-type operator, 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 Banach function space X instead of Lp . We obtain a characterization of
this inequality in terms of the Hardy-Littlewood maximal operator M
defined by
∫
1
M f (x) = sup
|f (y)|dy.
Q∋x |Q| Q
Let X be a Banach function space over Rn equipped with Lebesgue
measure. Denote by X ′ the associate space with norm
∫
∥f ∥X ′ = sup
|f (x)g(x)|dx.
∥g∥X =1
Rn
Let S0 (Rn ) be the space of measurable functions f on Rn such that for
any 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)
M f (x)|g(x)|dx ≤ c∥f ∥X ′ ∥M g∥X .
Rn
Let Mr f = (M |f |r )1/r . By Hölder’s inequality, M f ≤ Mr f if r > 1.
In a recent paper [20] (see also [18] for a different proof), the authors
established that the following Hardy-Littlewood-type inequality
∥M f ∥X ≤ c∥f ∥X
is self-improving in the sense that it implies the boundedness of Mr
on X for some r > 1. Our next result shows that the FeffermanStein-type inequality is also self-improving, although this phenomenon
is expressed in a somewhat indirect way.
Theorem 1.2. Inequality (1.2) holds if and only if there exist c > 0
and r > 1 such that for any f ∈ X ′ and g ∈ L1loc (Rn ),
∫
Mr f (x)|g(x)|dx ≤ c∥f ∥X ′ ∥M g∥X .
Rn
FEFFERMAN-STEIN INEQUALITY
3
The proof of Theorem 1.1 is based on two ingredients. The first one
is the adjoint of linearizations of M , and its sharp function estimate
established by de la Torre [27]. The second one is a certain duality
relation involving the local sharp maximal function proved by the author in [15]. Theorem 1.2 is based on Theorem 1.1 and on the Rubio
de Francia algorithm.
Theorem 1.1 can be applied to some questions related to the behavior
of M on general Banach function spaces. For instance, in Section 4 we
show the existence of nontrivial spaces X (that is, different from L∞ ) in
any dimension for which the lower operator norm of M is equal to 1. It
is interesting to compare this with the fact that M has no fixed points
in any rearrangement-invariant space, regardless of the dimension (as
was observed in [21]).
In Sections 5 and 6, we consider the case when X is the weighted
Lebesgue space Lp (w). Observe that the question about a necessary
and sufficient condition on a weight w for which the Fefferman-Stein
inequality holds on Lp (w) is still open. The currently best known
result in this direction is due to Yabuta [28] who showed that the
Cp 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] where
the same conditions were established regarding Coifman’s inequality
related singular integrals and the maximal operator.
Using Theorems 1.1 and 1.2, we first obtain several statements equivalent to the Fefferman-Stein inequality. For example, one of them is the
following: there exist c > 0 and r > 1 such that for any f ∈ L1loc (Rn ),
∫
(1.3)
Rn
∫
Mp,r (f, w)|f | dx ≤ c
(M f )p w dx,
Rn
where
(
Mp,r (f, w)(x) = sup
Q∋x
1
|Q|
)p−1 (
)1/r
∫
1
r
|f |
w
|Q| Q
Q
∫
(note that the maximal functions like Mp,r have been recently shown
to be very useful in the theory of multilinear singular integrals [19]).
ep condition which is between Cp and Cp+ε for any
Next, we introduce a C
ε > 0, and which is sufficient for (1.3). This gives a slight improvement
of Yabuta’s work [28] as well as a new approach to Sawyer’s result [25].
However, a question about the full characterization of the FeffermanStein 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 function spaces we refer to [2, Ch. 1]. Here we mention only several facts
which 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
Rn
By Fatou’s lemma [2, p. 5], if fn → f a.e., and if lim inf ∥fn ∥X < ∞,
n→∞
then f ∈ X, and
(2.2)
∥f ∥X ≤ lim inf ∥fn ∥X .
n→∞
2.2. Adjoint of M . Although M is not a linear operator, it turns out
that it is possible to linearize M with good pointwise control of the
adjoint 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
c1 M f (x) ≤ Mf f (x) ≤ c2 M f (x),
and for any g ∈ L1loc (Rn ),
(2.3)
(M⋆f g)# (x) ≤ c3 M g(x),
where M⋆f is the adjoint of Mf , and the constants ci depend only on n.
Note also that the construction of Mf shows that for any g ∈ L1 (Rn ),
(2.4)
∥M⋆f g∥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 M
and 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) = inf λ > 0 : |{x ∈ Rn : |f (x)| > λ}| ≤ t
(0 < t < ∞).
Given a measurable function f , the local maximal functions mλ f
and Mλ# f are defined by
)∗ (
)
(
mλ f (x) = sup f χQ λ|Q| (0 < λ < 1)
Q∋x
FEFFERMAN-STEIN INEQUALITY
and
5
(
)∗ (
)
Mλ# f (x) = sup inf (f − c)χQ λ|Q| (0 < λ < 1),
Q∋x c
respectively, where the supremum is taken over all cubes containing
the point x.
These functions were introduced by Strömberg [26]; in particular,
the definition of Mλ# f was motivated by an alternate characterization
of BM O given by John [11]. Roughly speaking, f # is the HardyLittlewood maximal operator of Mλ# f (see [10, 14]):
(2.5)
c1 M Mλ# f (x) ≤ f # (x) ≤ c2 M Mλ# f (x).
The following theorem was proved in [15].
Theorem 2.2. For any measurable function f ∈ S0 (Rn ) and any g ∈
L1loc (Rn ),
∫
∫
|f (x)g(x)|dx ≤ c
Mλ# f (x)M g(x)dx,
Rn
Rn
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)
M f (x) ≤ 3f # (x) + m1/2 f (x),
and
(2.7)
#
Mλ# (m1/2 f )(x) ≤ 4Mλ/2·9
n f (x).
Inequality (2.7) combined with (2.5) yields
(m1/2 f )# (x) ≤ cn f # (x).
(2.8)
It follows directly from the definitions that
(2.9)
{x ∈ Rn : mλ f (x) > α} ⊂ {x ∈ Rn : M χ{|f |>α} (x) ≥ λ}.
Lemma 2.4. For any non-negative functions f and g,
∫
∫
(mλ f )gdx ≤ c
f (M g)dx,
Rn
Rn
where a constant c depends on λ and n.
Proof. This is just a combination of (2.9) with the following inequality
of Fefferman and Stein [6]:
∫
∫
c
g dx ≤
|φ|(M g) dx.
ξ Rn
{x:M φ>ξ}
6
ANDREI K. LERNER
We have
∫
∫
(mλ f )g dx =
Rn
∞
0
∫
∫
≤ cλ,n
∫
{mλ f >α}
∞∫
0
g dxdα ≤
{f >α}
∞
∫
g dxdα
0
{M χ{f >α} (x)≥λ}
∫
M g dxdα = cλ,n
f (M g)dx,
Rn
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 suppose that f, g ≥ 0. Also, it is enough to assume, for instance, that
g is compactly supported, and hence g ∈ L1 (Rn ). The general case
will follow by the standard limiting argument. If g ∈ L1 (Rn ), then
M⋆ g ∈ L1 (Rn ), and thus M⋆ g ∈ S0 (Rn ). Therefore, by Theorem 2.1,
∥M⋆ g∥X ≤ c∥(M⋆ g)# ∥X ≤ c∥M g∥X .
From this, applying Theorem 2.1 again, we get
∫
∫
M f (x)g(x)dx ≤ c
Mf (x)g(x)dx
Rn
Rn
∫
= c
f (x)M⋆ g(x)dx
Rn
≤ c∥f ∥X ′ ∥M⋆ g∥X
≤ c∥f ∥X ′ ∥M g∥X .
We prove now (ii) ⇒ (i). Using Theorem 2.2 and (2.5), we get
∫
∫
|f (x)φ(x)|dx ≤ c
Mλ# f (x)M φ(x)dx
Rn
Rn
≤ c∥φ∥X ′ ∥M Mλ# 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 one
can replace M g by the dyadic maximal function M ∆ g, namely, we have
∫
∫
|f (x)g(x)|dx ≤ c
Mλ# f (x)M ∆ g(x)dx.
Rn
Rn
Therefore, taking into account the proof of Theorem 1.1, in order to
verify the Fefferman-Stein property of X, it is enough to check that
∫
(M ∆ f )|g|dx ≤ c∥f ∥X ′ ∥M g∥X .
Rn
FEFFERMAN-STEIN INEQUALITY
7
In order to prove Theorem 1.2, we shall need the following lemma,
which is interesting in its own right.
Lemma 3.2. Inequalities
∥f ∥X ≤ c∥f # ∥X
(3.1)
(f ∈ S0 (Rn ))
and
∥M f ∥X ≤ c∥f # ∥X
(3.2)
(f ∈ S0 (Rn ))
are equivalent.
Proof. Since |f | ≤ M f a.e., we trivially have that (3.2)⇒(3.1).
Suppose that (3.1) holds. By (2.6),
(3.3)
∥M f ∥X ≤ 3∥f # ∥X + ∥m1/2 f ∥X .
Next, from (2.9) and from the weak type (1, 1) property of M ,
µm1/2 f (α) ≤ cn µf (α) (α > 0).
Therefore, f ∈ S0 (Rn ) ⇒ m1/2 f ∈ S0 (Rn ). Hence, combining (3.1)
with (2.8), we get
∥m1/2 f ∥X ≤ c∥(m1/2 f )# ∥X ≤ c∥f # ∥X ,
which along with (3.3) implies (3.2).
Proof of Theorem 1.2. Inequality
∫
(3.4)
Mr f (x)|g(x)|dx ≤ c∥f ∥X ′ ∥M g∥X .
Rn
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)
M k f (x)g(x)dx ≤ Ak ∥f ∥X ′ ∥M g∥X ,
Rn
where 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 Francia algorithm [23]. We set
∞
∑
1
(Rf )(x) = |f (x)| +
M k f (x).
k
(2A)
k=1
Then |f | ≤ Rf and M (Rf )(x) ≤ 2A(Rf )(x). We have that (Rf ) is
an A1 weight, and hence it satisfies the reverse Hölder inequality (see,
e.g., [4]), which means that there exists r > 1 such that
Mr f (x) ≤ Mr (Rf )(x) ≤ cM (Rf )(x) ≤ 2Ac(Rf )(x).
8
ANDREI K. LERNER
Combining this inequality with (3.5), we get
∫
∫
Mr f (x)g(x)dx ≤ c
(Rf )(x)g(x)dx
Rn
Rn
∞
∑
∫
≤c
Rn
|f (x)|g(x)dx + c
(1/2)k ∥f ∥X ′ ∥M g∥X
k=1
≤ c∥f ∥X ′ ∥M g∥X .
We prove (3.5) by induction with respect to k. For k = 1 (3.5) is
just (1.2). Suppose that (3.5) holds for k = l, and let us show that it
holds for k = l + 1. By the induction assumption and by Theorem 2.1,
∫
∫
l+1
M f (x)g(x)dx =
M (M l f )(x)g(x)dx
Rn
Rn
∫
∫
l
≤c
M(M f )(x)g(x)dx = c
M l f (x)M⋆ g(x)dx
Rn
l
Rn
≤ cA ∥f ∥X ′ ∥M (M⋆ g)∥X .
As in the previous proof, one can assume that g ∈ L1 (Rn ), and hence
M⋆ g ∈ S0 (Rn ). By Theorem 1.1 and Lemma 3.2,
∥M (M⋆ g)∥X ≤ c∥(M⋆ g)# ∥X ≤ c∥M g∥X .
Therefore,
∫
Rn
M l+1 f (x)g(x)dx ≤ c′ Al ∥f ∥X ′ ∥M g∥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 maximal
operator M c (where the supremum is taken over all balls centered at x)
has a fixed point f ∈ Lp (Rn ) (that is, M c f = f ) if and only if n ≥ 3
and p > n/(n − 2). Then Martı́n and Soria [21] extended this result
to more general rearrangement-invariant (r.i.) spaces. Also, it was
remarked in [21] that the usual uncentered maximal operator M has
no fixed points in any r.i. space, regardless of the dimension.
We consider a related question about the lower operator norm of M
defined by
∥M ∥lX = inf ∥M f ∥X .
∥f ∥X =1
∥M ∥lX
It is clear that
≥ 1 for any X and any dimension. Taking into
account the above mentioned results about fixed points, it is natural
FEFFERMAN-STEIN INEQUALITY
9
to ask whether there exists a Banach function space X different from
L∞ such that ∥M ∥lX = 1.
If X = L∞ , we trivially have ∥M ∥lL∞ = 1. However, the existence
of X different from L∞ for which ∥M ∥lX = 1 is not an obvious fact.
For example, by Riesz’s sunrise lemma [8, p. 93],
∫
1
|f |.
|{x ∈ R : M f (x) > α}| ≥
α {x∈R:|f (x)|>α}
Integrating this inequality gives
∥M f ∥
Lp (R)
( p )1/p
≥
∥f ∥Lp (R) ,
p−1
and hence for any p > 1,
∥M ∥lLp (R)
( p )1/p
≥
> 1.
p−1
Using a non-standard proof of the Fefferman-Stein inequality combined 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 the
lower and upper Boyd indices, respectively [2, p. 149]. In general,
0 ≤ αX ≤ βX ≤ 1. By the Lorentz-Shimogaki theorem [2, p. 154], M
is 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 on
X, 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 two
results.
Theorem 4.4. For any f ∈ L1loc (Rn ) and for a.e. x ∈ Rn ,
(4.1)
M M f (x) ≤ cM f # (x) + M f (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 bounded
and compactly supported measurable functions {fj } such that for a.e.
x ∈ Rn ,
(4.2)
lim fj (x) = f (x)
j→∞
and
(4.3)
(fj )# (x) ≤ cf # (x),
where the constant c depends only on n.
Before proving Theorem 4.4 and Lemma 4.5 let us show how the
proof of Theorem 4.1 follows.
Proof of Theorem 4.1. We shall prove an equivalent statement saying
that if A ≡ ∥M ∥lX > 1, then M is bounded on X ′ . By Corollary 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∥M f ∥X ≤ ∥M M f ∥X ≤ c∥M f # ∥X + ∥M f ∥X .
Suppose that ∥f ∥X < ∞. Then ∥M f ∥X < ∞, and we obtain
c
(4.4)
∥f ∥X ≤ ∥M f ∥X ≤
∥M f # ∥X ≤ c′ ∥f # ∥X .
A−1
Take now an arbitrary f ∈ S0 (Rn ) ∩ L∞ . By Lemma 4.5, there is
a sequence {fj } satisfying (4.2) and (4.3). Since each fj is bounded
and compactly supported, we have that ∥fj ∥X < ∞ (we have used here
that 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 inequality 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 onedimensional 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)
M M f (x) ≤ 3M f # (x) + M m1/2 f (x).
FEFFERMAN-STEIN INEQUALITY
11
Let x, y ∈ Q and let Q′ be an arbitrary cube containing y. We have
that either Q ⊂ 3Q′ or Q′ ⊂ 3Q. If Q ⊂ 3Q′ , then
(f χQ′ )∗ (|Q′ |/2) ≤ ((f − f3Q′ )χQ′ )∗ (|Q′ |/2) + |f |3Q′
∫
2
|f − f3Q′ | + |f |3Q′
≤
|Q′ | Q′
≤ 2 · 3n f # (x) + M f (x).
If Q′ ⊂ 3Q, then
(f χQ′ )∗ (|Q′ |/2) ≤ ((f − f3Q )χQ′ )∗ (|Q′ |/2) + |f |3Q
≤ m1/2 ((f − f3Q )χ3Q )(y) + M f (x).
Therefore,
m1/2 f (y) ≤ m1/2 ((f − f3Q )χ3Q )(y) + 2 · 3n f # (x) + M f (x).
From this, using Lemma 2.4 with g ≡ 1, we get
∫
1
1
m1/2 f (y)dy ≤
∥m1/2 ((f − f3Q )χ3Q )∥L1
|Q| Q
|Q|
+ 2 · 3n f # (x) + M f (x)
∫
c
|f − f3Q | + 2 · 3n f # (x) + M f (x)
≤
|3Q| 3Q
≤ cf # (x) + M f (x),
and hence
M m1/2 f (x) ≤ cf # (x) + M f (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 value
of f over Q we mean a, possibly nonunique, real number mf (Q) such
that
|{x ∈ Q : f (x) > mf (Q)}| ≤ |Q|/2
and
|{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 cube
centered at x such that ℓ(Qx ) = dist(∂Q0 , Qx ), where ∂Q and ℓ(Q) are
the boundary and the side length of Q, respectively. Set
(
)
AQ0 f (x) = f (x) − mf (Qx ) χQ0 (x).
12
ANDREI K. LERNER
Proposition 4.6. For all x ∈ Rn ,
(AQ0 f )# (x) ≤ cf # (x),
(4.7)
where the constant c depends only on n.
Proof. Take an arbitrary cube Q containing x, and consider
∫
1
Ω(Q) ≡ inf
|AQ0 f (y) − c|dy.
c |Q| Q
If Q ∩ Q0 = ∅, we trivially have Ω(Q) = 0. Therefore, assume that
Q ∩ 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 observation 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 any
y ∈ Q,
(
)∗ (
)
|mf (Qy ) − c| ≤ (f − c)χQy |Qy |/2
∫
∫
2
2 · 15n
≤
|f − c| ≤
|f − c|.
|Qy | Qy
|5Qy0 | 5Qy0
Thus
∫
1
|f (y) − c|dy + inf
|mf (Qy ) − c|dy
c |Q| Q
Q
∫
2 · 15n
#
≤ f (x) + inf
|f − c| ≤ (2 · 15n + 1)f # (x).
c |5Qy0 | 5Q
y0
1
Ω(Q) ≤ inf
c |Q|
∫
Case 2. Assume now that ℓ(Qy ) < 2ℓ(Q) for any y ∈ Q ∩ Q0 . Then
Qy ⊂ 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
∫
1
Ω(Q) ≤
|AQ0 f (y)|dy
|Q| Q
∫
∫
1
1
|f (y) − f3Q |dy +
|f3Q − mf (Qy )|dy
≤
|Q| Q
|Q| Q
(
)
1
≤ 3n f # (x) +
∥m1/2 (f − f3Q )χ3Q ∥L1 ≤ cf # (x).
|Q|
Combining both cases yields
∫
1
|AQ0 f (y) − (AQ0 f )Q |dy ≤ 2Ω(Q) ≤ cf # (x),
|Q| Q
FEFFERMAN-STEIN INEQUALITY
13
proving (4.7).
Proof of Lemma 4.5. Set Qj = (−j, j)n and fj = AQj f . It is clear that
fj 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 with
ℓ(Qx ) = dist(∂Qj , Qx ) we have |Qx | ≥ (j/3)n . Hence, by (4.6), for
x ∈ 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 w
is a weight, that is, a non-negative locally integrable function. In this
1
′
case X ′ = Lp (σ), where 1/p′ + 1/p = 1, and σ = w− p−1 .
Conditions on a weight w for which the weighted Fefferman-Stein
inequality
∥f ∥Lp (w) ≤ c∥f # ∥Lp (w)
(5.1)
(1 < p < ∞)
holds are discussed in the next section. Theorems 1.1 and 1.2 provide
several reformulations of (5.1). Here we obtain yet another inequalities
equivalent 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 ∈ L1loc (Rn ),
∫
∫
(
)
p−1
Mr (M f ) w |f | dx ≤ c
(M f )p w dx;
Rn
Rn
(iii) there exist c > 0 and r > 1 such that for any f ∈ L1loc (Rn ),
∫
∫
Mp,r (f, w)|f | dx ≤ c
(M f )p w dx,
Rn
Rn
where
(
Mp,r (f, w)(x) = sup
Q∋x
1
|Q|
)p−1 (
)1/r
∫
1
r
.
|f |
w
|Q| Q
Q
∫
Proof. By Theorems 1.1 and 1.2, if (i) holds, then
∫
Mr φ|f | dx ≤ c∥φ∥Lp′ (σ) ∥M f ∥Lp (w) .
Rn
14
ANDREI K. LERNER
Setting here φ = (M f )p−1 w, we get (ii).
Next, (ii) implies (iii) trivially since
(
)
Mp,r (f, w)(x) ≤ Mr (M f )p−1 w (x).
It remains to prove that (iii) ⇒ (i). By Remark 3.1, it suffices to
show that
∫
(M ∆ φ)|f | dx ≤ c∥φ∥Lp′ (σ) ∥M f ∥Lp (w) .
(5.2)
Rn
We can suppose f, φ ≥ 0. By the Calderón-Zygmund decomposition,
Ωk = {x ∈ Rn : M ∆ φ(x) > 2k } = ∪j Qkj ,
where Qkj are pairwise disjoint dyadic cubes such that
1
2 < k
|Qj |
∫
k
Qkj
φ ≤ 2n+k .
From this, setting Ek = Ωk \ Ωk+1 and Eik = Ek ∩ Qki , we have
∫
∫
∑∫
∑
∆
∆
(5.3)
f
φQkj
(M φ)f =
(M φ)f ≤ 2
Rn
k
∫
Ek
)
∑( 1
f
χQkj
k
k
|Q
|
E
Rn
j
j
k,j
)
∑( 1 ∫
≤ 2∥φ∥Lp′ (σ) ∥
f
χQkj ∥Lp (w) .
|Qkj | Ejk
k,j
= 2
φ
Next we note that
|Qkj
Ejk
k,j
∫
∩ Ωk+l | =
∑
|Qk+l
i |
Qk+l
⊂Qkj
i
≤
1
2(k+l)
∫
Qkj
<
∑ ∫
1
2(k+l)
Qik+l ⊂Qkj
φ ≤ 2n−l |Qkj |.
Therefore, setting
)
∑( 1 ∫
f χQkj ∩Ek+l
Tl f (x) =
|Qkj | Ejk
k,j
Qik+l
φ
FEFFERMAN-STEIN INEQUALITY
15
and using (iii) and Hölder’s inequality, we obtain
∫
)p
∑( 1 ∫
p
(Tl f ) w =
f
w(Qkj ∩ Ek+l )
k
k
|Q
|
n
R
Ej
j
k,j
(
)p−1 ( 1 ∫
)1/r ∫
∑ 1 ∫
−l/r ′
r
≤ c2
f
w
f
|Qkj | Ejk
|Qkj | Qkj
Ejk
k,j
∑∫
−l/r ′
Mp,r (f, w)f
≤ c2
k,j
−l/r ′
≤ c2
Hence,
∫
Ejk
−l/r ′
Rn
Mp,r (f, w)f ≤ c2
∫
(M f )p w.
Rn
∞
)
∑( 1 ∫
∑
f χQkj ∥Lp (w) ≤
∥
∥Tl f ∥Lp (w)
|Qkj | Ejk
k,j
l=0
≤c
∞
∑
′
2−l/pr ∥M f ∥Lp (w) ≤ c∥M f ∥Lp (w) ,
l=0
which combined with (5.3) yields (5.2), and therefore the theorem is
proved.
6. On the Cp condition
∫
Given a measurable set E ⊂ Rn , let w(E) = E w(x)dx. We say that
w satisfies the A∞ condition if there are positive constants c, δ such
that for any cube Q and any subset E ⊂ Q,
(6.1)
w(E) ≤ c(|E|/|Q|)δ w(Q).
Given r > 0, denote by rQ the cube concentric with Q having diameter r times that of Q. If w(Q) on the right-hand side of (6.1) is
replaced by w(2Q), then the corresponding condition is called the weak
A∞ condition.
We say that w satisfies the Cp condition if there are positive constants
c, δ such that for any cube Q and any subset E ⊂ Q,
∫
δ
(6.2)
w(E) ≤ c(|E|/|Q|)
(M χQ )p w.
Rn
It is easy to see that
A∞ ⊂ weak A∞ ⊂ Cp .
Also, Cp+ε ⊂ Cp if ε > 0. An interesting analysis of the Cp condition
can be found in [12]. In particular, it was shown there that Cp ̸=
∪ε>0 Cp+ε .
16
ANDREI K. LERNER
Let T be a Calderón-Zygmund singular integral operator, that is,
T = p.v.f ∗ K with kernel K satisfying the standard conditions
b L∞ ≤ c,
∥K∥
|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 CalderónZygmund operators as well.
The weighted theory of the Fefferman-Stein inequality has been developed in parallel to the one of Coifman’s inequality relating singular
integrals 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 ∥Lp (w) ≤ c∥M f ∥Lp (w)
(1 < p < ∞).
This result was based on a good-λ inequality related T f and M f .
However, the Fefferman-Stein inequality originally was also proved with
the help of a good-λ inequality related f and f # . Therefore, it has
been 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 the
Hilbert transform, the Cp condition is necessary for (6.3), and he conjectured 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 sufficient for (6.3). Using almost the same arguments, Yabuta [28] showed
that Cp is necessary for (5.1) and Cp+ε is sufficient.
Here we give a completely different proof of a slightly improved version 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
dt
φp (t) p+1 < ∞.
t
0
ep condition if there are positive
We say that a weight w satisfies the C
constants c, δ such that for any cube Q and any subset E ⊂ Q,
∫
δ
φp (M χQ )w.
w(E) ≤ c(|E|/|Q|)
Rn
Theorem 6.1. The Cp condition is necessary for
∫
∫
(6.4)
Mp,r (f, w)|f | dx ≤ c
(M f )p w dx,
Rn
ep is sufficient.
and the C
Rn
FEFFERMAN-STEIN INEQUALITY
17
ep ⊂ Cp . On
Remark 6.2. It is easy to see that φp (t) ≤ ctp , and hence C
p+ε
the other hand, taking φp (t) such that t
≤ cφp (t) for any ε > 0 (for
−2
p
ep . Hence, by
example, φp (t) = t log (1 + 1/t)), we get ∪ε>0 Cp+ε ⊂ C
Theorem 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) are
very closely related in view of the following pointwise inequality [1]:
(|T f |α )# (x) ≤ c(M f )α (x) (0 < α < 1).
(6.5)
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
(
)1/r
∫
∫
1
1
r
(6.6)
w
≤c
(M χQ )p w.
|Q| Q
|Q| Rn
From this, by Hölder’s inequality we get the Cp condition with δ = 1/r′ .
ep . Then for 0 < t < |Q| (cf. [2, p. 53]),
Suppose now that w ∈ C
∫ t
∫
∗
δ
(wχQ ) (τ )dτ = sup w(E) ≤ c(t/|Q|)
φp (M χQ )w.
Rn
E⊂Q,|E|=t
0
From this,
1
(wχQ ) (t) ≤
t
∗
∫
0
t
1
(wχQ ) (τ )dτ ≤ 1−δ
t |Q|δ
Hence, fixing some 1 < r <
∫
∫
w
r
|Q|
=
Q
∫
c
∗
1
,
1−δ
φp (M χQ )w.
(wχQ )∗ (t)r dt
λ|Q|
∫
∗
r
|Q|
(wχQ ) (t) dt +
0
≤ cλ
Rn
for 0 < λ < 1 we get
0
=
∫
(
1−r(1−δ)
|Q|
1
|Q|
∫
Rn
(wχQ )∗ (t)r dt
λ|Q|
φp (M χQ )w
)r
+ |Q|(wχQ )∗ (λ|Q|)r .
Therefore,
(
)1/r
∫
∫
λ1/r−(1−δ)
1
r
≤c
(6.7)
w
φp (M χQ )w + (wχQ )∗ (λ|Q|).
|Q| Q
|Q|
n
R
18
ANDREI K. LERNER
Further, if x ∈ Q, then
)p−1
(
∫
∫
1
1
|f |
φp (M χQ )w
|Q| Q
|Q| Rn
)
(
)p−1 (
∫
∫
∫
∞
1
1
1 ∑
≤c
|f |
w+
φp (2−kn )
w
|Q| Q
|Q| Q
|Q| k=1
2k Q\2k−1 Q
)p−1 (
)
(
∫
∫
∞
∑
1
1
kpn
−kn
≤c
|f |
w
2 φp (2 )
k Q|
k Q|
|2
|2
kQ
kQ
2
2
k=1
(∫ 1
dt )
φp (t) p+1 Mp,1 (f, w)(x).
≤c
t
0
We now observe that it is enough to prove (6.4) for compactly supported f . Also, one can assume that the right-hand side of (6.4) is
finite, otherwise there is nothing to prove. This means, in particular,
that
∫
w(x)
dx < ∞.
pn
Rn 1 + |x|
It follows from this that
∫
∫
1
w(x)
sup
w≤c
dx < ∞,
p
pn
0∈Q,|Q|≥1 |Q|
Q
Rn 1 + |x|
which
easily implies that Mp,1 (f, w)(x) < ∞ a.e. Since (wχQ )∗ (λ|Q|) ≤
∫
1
w, we obtain also that
λ|Q| Q
)p−1
(
∫
1
|f |
(wχQ )∗ (λ|Q|) < ∞ a.e.
sup
|Q| Q
Q∋x
Therefore, (6.7) shows that Mp,r (f, w)(x) < ∞ a.e.
Hence, applying (6.7) again and using Hölder’s inequality, we get
(
)
Mp,r (f, w)(x) ≤ cλ1/r−(1−δ) Mp,1 (f, w)(x) + mλ (M f )p−1 w (x)
(
)
≤ cλ1/r−(1−δ) Mp,r (f, w)(x) + mλ (M f )p−1 w (x).
From this, taking λ small enough, we obtain
(
)
(6.8)
Mp,r (f, w)(x) ≤ cmλ (M f )p−1 w (x).
This inequality combined with Lemma 2.4 yields
∫
∫
(
)
Mp,r (f, w)|f | dx ≤ c
mλ (M f )p−1 w |f | dx
n
Rn
∫R
≤ c
(M f )p w dx,
Rn
and therefore the theorem is proved.
FEFFERMAN-STEIN INEQUALITY
19
We make several concluding remarks. The question about a necessary and sufficient condition on w for which (6.4) (or, equivalently, the
ep condiFefferman-Stein inequality (5.1)) holds remains open. The C
tion is probably not a necessary condition for (6.4). Indeed, the proof
ep condition implies (6.8). This along
of Theorem 6.1 shows that the C
with Lemma 2.4 gives that for all suitable f and g,
∫
∫
(M f )p−1 (M g)w dx,
Mp,r (f, w)|g| dx ≤ c
Rn
Rn
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 is
equivalent to the following statement: there exist c > 0 and r > 1 such
that for each cube Q and any g ∈ L1loc (Rn ),
(
)p (
)1/r
∫
∫
∫
(
)p
1
1
1
r
(6.9)
|g|
w
≤c
M (gχQ ) w.
|Q| Q
|Q| Q
|Q| Rn
Indeed, (6.9) with g ≡ 1 gives (6.6). On the other hand, if xQ is the
center of Q, then
)p
∫
(
)p ∫
∫
∫ (
|g|
1
Q
p
|g|
w
(M χQ ) w ≤ c
|Q| Q
|x − xQ |n + |Q|
Rn
Rn
∫
(
)p
≤ c
M (gχQ ) w,
Rn
which along with (6.6) implies (6.9).
Denote by Mp the class of weights w for which the following FeffermanStein-type inequality holds (cf. [6]): there is c > 0 such that for any
sequence of functions {fj } with pairwise disjoint supports,
∫ ( ∑ )p
∑∫
p
(M fj ) w ≤ c
M(
fj ) w.
j
Rn
Rn
j
Then the Cp ∩ Mp condition yields (5.1). To show this, we keep the
same notation as in the proof of Theorem 5.1. Using (6.9) along with
the Mp condition, we get
∫
)p ( 1 ∫
)1/r
∑( 1 ∫
r
p
−l/r′
f
w
|Qkj |
(Tl f ) w ≤ c2
k
k
k
k
|Q
|
|Q
|
n
Ej
Qj
R
j
j
k,j
∫
∫
∑
(
)p
−l/r′
−l/r ′
≤ c2
M (f χEjk ) w ≤ c2
(M f )p w,
k,j
Rn
Rn
20
ANDREI K. LERNER
and now we can follow the proof of Theorem 5.1. The above argument
raises a natural question whether Cp ⇒ Mp . Observe that the sharp
function estimate of the vector-valued maximal operator [5] shows that
Cp+ε ⇒ Mp for any ε > 0.
Acknowledgment. I am grateful to the referee for useful remarks and
corrections.
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Department of Mathematics, Bar-Ilan University, 52900 Ramat Gan,
Israel
E-mail address: [email protected]