ON WEIGHTED ESTIMATES OF NON-INCREASING
REARRANGEMENTS
A. K. Lerner
Department of Mathematics, Odessa State University
2, Dvoryanskaya str. 270 000 Odessa, UKRAINE
Dedicated to Professor P. L. Ul’yanov on the occasion of his 70th birthday
Abstract
Let ω be a weight satisfing Muckenhoupt’s condition A∞ . In present
paper the estimate of rearrangement fω∗ (t) was obtained
fω∗ (t) ≤ 2(Mλ# f )∗ω (2t) + fω∗ (2t)
(0 < t < ∞),
where f is any measurable function, Mλ# f is local sharp maximal function due to John [12] and Strömberg [18]. Before (Bennett, DeVore and
Sharpley [3], Bagby and Kurtz [1]) the similar estimates were expressed
in terms of Fefferman-Stein sharp-function f # which is sufficiently larger
then Mλ# f .
In paper several applications of this estimate were pointed out.
1
Introduction
A non-negative locally integrable on Rn function ω(x) is called
a weight.
R
With any weight function we associate the measure ω(E) = E ω(x) dx. In
case ω(x) ≡ 1, ω(E) = |E| is the Lebesgue measure. We shall assume that
ω(Rn ) = +∞.
Let f be a measurable function on Rn . The distribution function for f
with respect to the measure ω is defined by the equality
µf (λ) = ω{x ∈ Rn : |f (x)| > λ} (0 < λ < ∞).
Assume that µf (λ) < ∞, λ > 0. We say that the function fω∗ (t) is a nonincreasing rearrangement of f with respect to the measure ω if it is nonincreasing on (0, +∞) and equimeasurable with |f (x)|, that is,
|{t ∈ (0, +∞) : fω∗ (t) > λ}| = µf (λ)
for any λ > 0. We shall assume that the rearrangement is continuous from
the left. Then it is uniquely determined and can be defined by the equality
fω∗ (t) = inf{λ > 0 : µf (λ) < t}
1
or by (see [5, p. 32], [14]):
fω∗ (t) = sup inf |f (x)| (0 < t < ∞)
ω(E)=t x∈E
(by virtue of the conditions
on ω, for any t > 0, a set E exists with ω(E) = t).
R
Denote fω∗∗ (t) = t−1 0t fω∗ (τ ) dτ . If ω is the Lebesgue measure we use the
notations f ∗ (t), f ∗∗ (t).
We say that the weight function ω satisfies Muckenhoupt’s condition A∞ if
there exist constants c, δ > 0 such that for any cube Q, and for any measurable
subset E ⊂ Q, holds the inequality
ω(E) ≤ c (|E|/|Q|)δ ω(Q).
Many other equivalent definitions of A∞ can be found in the paper [6] of
Coifman and Fefferman.
In this paper an estimate for the rearrangement fω∗ (t) (ω ∈ A∞ ) in terms
of the maximal function Mλ# f is obtained.
For a given measurable function f the maximal function Mλ# f is defined
as
(1.1)
Mλ# f (x) = sup inf ((f − c)χQ )∗ (λ|Q|), 0 < λ ≤ 1,
Q3x c∈R
(the supremum is taken over all cubes Q containing the point x; χQ denotes
the characteristic function of Q).
The definition (1.1) was given by John [12] in 1965. Later, in 1972, Fefferman and Stein [10] introduce the maximal function f # which measure the
mean oscillations. It is defined as follows
1
Q3x |Q|
f # (x) = sup
Z
Q
Z
|f (y) − fQ | dy
(fQ = |Q|−1
Q
f ).
The space BM O consists of all locally integrable functions such that
f # ∈ L∞ and the BM O-semi-norm is defined as kf k∗ = kf # k∞ .
Since tf ∗ (t) ≤ tf ∗∗ (t) ≤ kf k1 , it is clear that λMλ# f (x) ≤ f # (x)
(0 < λ ≤ 1). Thus
λkMλ# f k∞ ≤ kf k∗
(0 < λ ≤ 1).
John [12] and Strömberg [18] proved that, in case λ ≤ 1/2, the converse
statement is also true. The following theorem holds.
Theorem [12, 18]. If 0 < λ ≤ 1/2 then
kf k∗ ≤ ckMλ# f k∞
2
where c depends only on the dimension.
Bennett, DeVore and Sharpley proved in [3] the inequality: For any function f ∈ L1loc (Rn ) and any cube Q ⊂ Rn ,
(1.2)
(f · χQ )∗∗ (t) − (f · χQ )∗ (t) ≤ c(f # )∗ (t) (0 < t < |Q|/6).
This result implies, in particular, the theorem of Fefferman and Stein [10]:
kM f kp ³ kf # kp
(1 < p < ∞)
(M f is the standard maximal operator of Hardy – Littlewood).
Afterwards Bagby and Kurtz [1] established the weighted inequality: If
ω ∈ A∞ , then for any function f ∈ L1loc (Rn ),
(1.3)
fω∗ (t) ≤ c(f # )∗ω (2t) + fω∗ (2t)
(t > 0).
There and in the consequent paper [2] similar estimates connecting the rearrangements of the functions M f and f # , as well as of T f and M f , are obtained (T f is the singular integral operator of Calderón – Zygmund). These
estimates are ”rearrangemental” analogies of earlier known ”good λ inequalities” [10, 6].
The main result of the present paper is an estimate which is more precise
than (1.3). Namely, we show that if ω ∈ A∞ , f is a measurable function, and
Q is an arbitrary cube, then
(f · χQ )∗ω (t) ≤ 2(Mλ# f )∗ω (2t) + (f · χQ )∗ω (2t) (0 < t < cω(Q)),
fω∗ (t) ≤ 2(Mλ# f )∗ω (2t) + fω∗ (2t)
(0 < t < ∞).
Moreover, with the help of these inequalities certain known results concerning estimates of the Lpω - and BM O-norms of maximal and singular integral operators can be obtained.
2
Auxiliary propositions
A measure ω is said to satisfy the doubling condition (ω ∈ D), if for any
cube Q holds ω(2Q) ≤ cω(Q). The condition A∞ yields D, but the inverse is
not true (see [9]).
Lemma 2.1. Assume that the measure ω satisfies the doubling condition
D and let 0 < λ < 1.
3
(i) If Q0 is a cube and E ⊂ Q0 is an arbitrary measurable set of positive measure with ω(E) ≤ λω(Q0 ), then there exist mutually disjoint cubes
{Qi } ⊂ Q0 covering E and such that
(2.1)
λω(Qi ) < ω(Qi ∩ E) ≤ cω λω(Qi ).
(ii) If E ⊂ Rn is an arbitrary measurable set of finite positive ω-measure,
then there exist mutually disjoint cubes {Qi } covering E and satisfying inequality (2.1).
Proof. Item (i) follows
from the Calderón – Zygmund ”weight” lemma
Z
1
(see [15]): If
|f (y)|ω(y) dy ≤ λ then there exist mutually disjoint
ω(Q0 ) Q0
Z
1
|f (y)|ω(y) dy ≤ cω λ and |f (x)| ≤ λ
cubes {Qi } ⊂ Q0 such that λ <
ω(Qi ) Qi
S
for almost all x ∈ Q0 \ i Qi (the proof is the same as that of the ordinary
lemma of Calderón – Zygmund [16, p. 27]). Letting f (y) = χE (y) we get (i).
For the proof of item (ii), it is necessary to decompose Rn into sufficiently big
non-overlapping cubes Qk in such a way that ω(E ∩ Qk ) ≤ λω(Qk ). We can
do this because the set E has a finite ω-measure. Then it remains to apply
item (i) to each cube Qk . The lemma is proved.
Remark 2.1. It is meant in the formulation of Lemma 2.1 that the cubes
Qi cover the set E almost everywhere, i.e., |E \ ∪i Qi | = 0. Note else that
in the lemma cited in [15] the family {Qi } of cubes may be empty in case
|f (x)| ≤ λ almost everywhere. But, in our case, this is impossible since we
take λ < 1 and f (x) = χE (x).
Lemma 2.2. Let ω ∈ D. Then, for any measurable function f and for
every cube Q, the following inequality holds
³
(f · χQ )∗ω (λω(Q)) ≤ 2 inf (f − c)χQ
´∗
c∈R
ω
(λω(Q)) + (f · χQ )∗ω ((1 − λ)ω(Q))
(0 < λ < 1).
Proof. If 1/2 ≤ λ < 1, then the lemma is trivial.
0 < λ < 1/2. For any constant c we have
³
|c| ≤
inf (|c − f (x)| + |f (x)|) ≤ (|c − f | + |f |)χQ
x∈Q
³
≤
Therefore
(f − c)χQ
´∗
ω
Assume that
´∗
ω
(ω(Q))
(λω(Q)) + (f · χQ )∗ω ((1 − λ)ω(Q)).
³
(f · χQ )∗ω (λω(Q)) ≤ (f − c)χQ
4
´∗
ω
(λω(Q)) + |c|
³
≤ 2 (f − c)χQ
´∗
(λω(Q)) + (f · χQ )∗ω ((1 − λ)ω(Q)).
ω
The proof is completed.
Remark 2.2. It is clear that the last lemma is valid under even more
general conditions on ω.
Consider now the weighted analogy of the function Mλ# f :
³
#
Mλ,ω
f (x) = sup inf (f − c)χQ
Q3x c∈R
´∗
ω
(λω(Q))
(0 < λ ≤ 1).
Lemma 2.3. If ω satisfies the condition A∞ , then for every λ ≤ 1 there
exists λ0 , λ00 ≤ 1 such that
#
f (x) ≤ Mλ#0 f (x)
Mλ#00 f (x) ≤ Mλ,ω
(2.2)
for all x ∈ Rn .
Proof. Let E ⊂ Q and ω(E) = λω(Q). Then it follows from the definition
of A∞ that for certain c, δ > 0 we have
|E| ≥
Thus
λ1/δ
|Q|.
c
³
inf |f (x) − ξ| ≤ (f − ξ)χQ
´∗ µ λ1/δ
x∈E
c
¶
|Q| ,
ξ ∈ R.
Taking a supremum over all E ⊂ Q with ω(E) = λω(Q), we get
³
(f − ξ)χQ
´∗
ω
³
(λω(Q)) ≤ (f − ξ)χQ
´∗ µ λ1/δ
c
¶
|Q| ,
ξ ∈ R.
This inequality yields the right-hand side of (2.2) with λ0 = λ1/δ /c.
Assume that E ⊂ Q and |E| = λ00 |Q|. Then |Q \ E| = (1 − λ00 )|Q| and
hence ω(Q \ E) ≤ c(1 − λ00 )δ ω(Q). Therefore
ω(E) ≥ (1 − c(1 − λ00 )δ )ω(Q)
and
inf |f (x) − ξ| ≤ ((f − ξ)χQ )∗ω ((1 − c(1 − λ00 )δ )ω(Q)).
x∈E
Letting λ = 1 − c(1 − λ00 )δ we get λ00 = 1 − (1 − λ)1/δ /c. Inequality (2.2) is
proved with λ00 = 1 − (1 − λ)1/δ /c, λ0 = λ1/δ /c.
5
Denote by Lpω (Rn ) the space of all functions f for which
µZ
kf kp,ω ≡
Rn
¶1/p
|f (x)|p ω(x) dx
< ∞.
By virtue of the equimeasurability of the rearrangement the following equality
holds:
Z
Z ∞
p
(2.3)
|f (x)| ω(x) dx =
(fω∗ (t))p dt (p > 0).
Rn
0
Lemma 2.4. Let the functions f, g satisfy the inequality
fω∗ (t) ≤ cgω∗ (γt) + fω∗ (2t)
(2.4)
(0 < t < ∞, γ > 0).
Assume that fω∗ (+∞) = 0. Then
kf kp,ω ≤ cp,γ kgkp,ω
(0 < p < ∞).
Proof. Applying consecutively (2.4) we get
fω∗ (t)
≤c
∞
X
gω∗ (2k γt)
k=0
∞
c X
≤
log 2 k=0
Z 2k γt ∗
g (s)
2k−1 γt
s
c
ds =
log 2
Z ∞ ∗
g (s)
γt/2
s
ds.
Hence, from Hardy inequation [16, p. 319] and (2.3), it follows that
kf kp,ω = kfω∗ kp ≤
°Z ∞ ∗
g (s)
2c 1 °
· °
log 2 γ °
t
s
°
°
ds°
° ≤
p
2c 1
· pkgkp,ω
log 2 γ
(1 ≤ p < ∞).
If 0 < p < 1 the arguments are similar since in this case (2.4) implies that
(fω∗ (t))p ≤ (cgω∗ (γt))p + (fω∗ (2t))p .
The lemma is proved.
3
The basic inequality
Theorem 3.1. Let ω satisfy the doubling condition. Then, for any measurable function f and for every cube Q, the following inequalities hold
#
(3.1) (f · χQ )∗ω (t) ≤ 2(Mλ,ω
f )∗ω (2t) + (f · χQ )∗ω (2t)
(3.2)
#
fω∗ (t) ≤ 2(Mλ,ω
f )∗ω (2t) + fω∗ (2t)
(0 < t < ∞),
where cω is the constant from (2.1), 0 < λ ≤ 1/5cω .
6
(0 < t ≤ ω(Q)/5cω ),
Proof. Fix an arbitrary cube Q ⊂ Rn . Let λ ≤ 1/5cω , t ≤ ω(Q)/5cω
and let E ⊂ Q be an arbitrary measurable set with ω(E) = t. According to
Lemma 2.1 (i) there exist mutually disjoint cubes {Qi } ⊂ Q covering E and
such that
1
(3.3)
ω(Qi ∩ E) >
ω(Qi ),
5cω
X
(3.4)
ω(Qi ) ≥ 5
X
i
ω(Qi ∩ E) = 5ω(E) = 5t.
i
Select from them the cubes {Q0i } which are contained in the set
#
#
E ∗ = {x ∈ Rn : Mλ,ω
f (x) > (Mλ,ω
f )∗ω (2t)},
In other words, for Q0i the following inequality holds
#
inf ((f − c)χQ0 )∗ω (λω(Q0i )) ≤ (Mλ,ω
f )∗ω (2t).
c
i
This relation and Lemma 2.2 yield
(f · χQ0i )∗ω (ω(Q0i )/5cω ) ≤ 2 inf c ((f − c) · χQ0 )∗ω (λω(Q0i ))
i
+ (f · χQ0 )∗ω ((1 − 1/5cω )ω(Q0i ))
(3.5)
i
#
≤ 2(Mλ,ω
f )∗ω (2t)
+ (f · χQ0 )∗ω ((1 − 1/5cω )ω(Q0i )).
i
Further, since ω(E ∗ ) ≤ 2t and the cubes Qi are mutually disjoint, taking into
P
account (3.4), we get i ω(Q0i ) ≥ 3t. Thus
inf (f · χQ0 )∗ω ((1 − 1/5cω )ω(Q0i )) ≤ (f · χQ )∗ω ((1 − 1/5cω )3t)) ≤ (f · χQ )∗ω (2t).
i
i
Hence, by virtue of (3.3) and (3.5), we have
inf |f (x)| ≤ inf
x∈E
inf
i x∈E∩Q0i
|f (x)| ≤ inf (f · χQ0 )∗ω (ω(Q0i ∩ E))
i
i
#
≤ inf (f · χQ0 )∗ω (ω(Q0i )/5cω ) ≤ 2(Mλ,ω
f )∗ω (2t) + (f · χQ )∗ω (2t).
i
i
Taking supremum in this inequality over all sets E ⊂ Q with ω(E) = t, we
get (3.1). Inequality (3.2) may be proved either in the same manner using
Lemma 3.1 (ii), or from (3.1) passing to the limit as Q → Rn . The theorem
is proved
Note that in the non-weighted case a ”good λ” inequality was proved
in [11] connecting the functions f and Mλ# f . Thus inequalities (3.1) and
7
(3.2) extend the idea of transition from ”good λ inequalities” to estimates of
rearrangements.
Lemma 2.3 and inequality (3.2) yields the following statement.
Corollary 3.1. If ω ∈ A∞ , then for every measurable function f we have
(3.6)
fω∗ (t) ≤ 2(Mλ# f )∗ω (2t) + fω∗ (2t) (0 < t < ∞, 0 < λ < λ0 (ω)).
Inequality (3.6) and Lemma 2.4 imply immediately
Theorem 3.2. Let ω ∈ A∞ . Then, for any f satisfying fω∗ (+∞) = 0,
holds the inequality
(3.7)
kf kLpω ≤ cp,ω kMλ# f kLpω
(0 < p < ∞, 0 < λ < λ0 (ω)).
In the non-weighted case this result was proved in [11].
4
Certain estimates for operators
The right-hand side of inequality (3.7) contains the Lpω -norm of the function Mλ# f . As mentioned already in the introduction
(4.1)
kf k∗ ≤ cn kMλ# f k∞
for 0 < λ ≤ 1/2.
It is shown in this section that application of our basic inequality (3.6)
and John – Strömberg inequality (4.1), allow us to give short proofs of certain
known results.
To this end consider first how the operator Mλ# acts on the operators M f
and T f . Set
Z
1
M f (x) = sup
|f (y)| dy.
Q3x |Q| Q
Theorem 4.1. Assume that f ∈ L1loc (Rn ). If M f (x) < ∞ almost everywhere, then
(4.2)
Mλ# (M f )(x) ≤ cn,λ f # (x) (0 < λ ≤ 1)
for all x ∈ Rn .
Proof. Fix a cube Q. Let x ∈ Q and let Q0 be an arbitrary cube containing
x. If Q0 ⊂ 3Q, then
(4.3) |f |Q0 ≤ |f − f3Q |Q0 + |f |3Q ≤ M ((f − f3Q )χ3Q )(x) + inf M f (ξ).
ξ∈Q
8
Assume that Q0 6⊂ 3Q. Then Q ⊂ 3Q0 and in this case
|f |Q0 ≤ |f − f3Q0 |Q0 + |f |3Q0 ≤ 3n inf f # (ξ) + inf M f (ξ).
ξ∈Q
ξ∈Q
Hence, using (4.3), we get
(4.4)
M f (x) ≤ M ((f − f3Q )χ3Q )(x) + 3n inf f # (ξ) + inf M f (ξ)
ξ∈Q
ξ∈Q
for all x ∈ Q. Making use of the fact that the operator M is of weak type
(1,1) [16, p. 15] and (4.4), we get
inf ((M f − c)χQ )∗ (λ|Q|) ≤ ((M f − inf M f )χQ )∗ (λ|Q|)
c
Q
≤ (M ((f − f3Q )χ3Q ))∗ (λ|Q|) + 3n inf f # (ξ)
≤
cn
λ|Q|
ξ∈Q
Z
3Q
|f (y) − f3Q | dy + 3n inf f # (ξ).
ξ∈Q
Taking supremum over all Q containing x we get (4.2). The theorem is proved.
Note that a weaker variant of inequality (4.2) for the local case was given
without proof in [11].
Corollary 4.1.
(i) If f ∈ BM O(Rn ) and M f (x) < ∞ a.e., then M f ∈ BM O(Rn ) and
kM f k∗ ≤ cn kf k∗ .
(ii) If ω ∈ A∞ , then
(4.5)
(M f )∗ω (t) ≤ cn,ω (f # )∗ω (2t) + (M f )∗ω (2t)
(t > 0)
for any function f ∈ L1loc (Rn ).
(iii) If ω ∈ A∞ , then
(4.6)
kM f kLpω ≤ cn,p,ω kf # kLpω
(0 < p < ∞)
for all functions f with (M f )∗ω (+∞) = 0.
Item (i) follows from inequalities (4.1), (4.2). This result was proved in
[3]. Inequality (4.5) was obtained in [1]. It follows immediately from (3.6)
and (4.2). Inequality (4.6) is the weighted Fefferman – Stein theorem [10]. It
follows either from (3.7) or (4.5).
9
Suppose that the kernel k(x) satisfies the standard conditions:
|k(x)| ≤
Z
c
,
|x|n
k(x) dx = 0
(0 < R1 < R2 < ∞),
R1 <|x|<R2
(4.7)
|k(x) − k(x − y)| ≤
c|y|α
|x|n+α
(|y| ≤ |x|/2, α > 0).
Set, for f ∈ Lp (Rn ) (1 ≤ p < ∞),
Z
T f (x) = P.V.
¯
¯
Rn
f (y)k(x − y) dy,
¯
¯
Z
T ∗ f (x) = sup ¯¯
ε>0
f (y)k(x − y) dy ¯¯.
|x−y|>ε
Theorem 4.2. For all f ∈ Lp (Rn )
(1 ≤ p < ∞) we have
(4.8)
Mλ# (T f )(x) ≤ cn,λ M f (x),
(4.9)
Mλ# (T ∗ f )(x) ≤ cn,λ M f (x).
This theorem was proved in [11]. To prove it, decompose the function into
two parts. Let x be the center of the cube Q. Set f1 = f · χ3Q , f2 = f · χRn \3Q .
By virtue of the weak type (1,1) of the operator T (see [16, p. 42]), we have
c
(T f1 ) (λ|Q|) ≤
λ|Q|
Z
∗
3Q
|f (y)| dy ≤ c0 M f (x).
On the other hand, using the conditions (4.7) on the kernel, it is not difficult
to show that for all z 0 , z 00 ∈ Q holds the estimate
|T f2 (z 0 ) − T f2 (z 00 )| ≤ cM f (x).
This yields (4.8). Inequality (4.9) can be proved in a similar way.
The above theorem and inequalities (3.6), (4.1) imply
Corollary 4.2.
(i) If f ∈ Lp ∩ L∞ (Rn ), then T f, T ∗ f ∈ BM O(Rn ). Moreover,
kT f k∗ ≤ cn kf k∞ ,
kT ∗ f k∗ ≤ cn kf k∞ .
(ii) If ω ∈ A∞ , then
(T f )∗ω (t) ≤ cn,ω (M f )∗ω (2t) + (T f )∗ω (2t)
10
(t > 0),
(T ∗ f )∗ω (t) ≤ cn,ω (M f )∗ω (2t) + (T ∗ f )∗ω (2t)
(t > 0)
for all f ∈ Lp (R)n (1 ≤ p < ∞).
(iii) If ω ∈ A∞ , then
(4.10)
kT ∗ f kLpω ≤ cn,p,ω kM f kLpω
(0 < p < ∞)
for all functions f with (T ∗ f )∗ω (+∞) = 0.
Item (i) goes back to Stein [17]. Item (ii) was obtained in [2]. More
precisely, in the paper of Bagby and Kurtz [2], it was proved that for any
γ ∈ (0, 1) there exists a constant c(γ) such that
(T ∗ f )∗ω (t) ≤ c(γ)(M f )∗ω (γt) + (T ∗ f )∗ω (2t)
(t > 0),
and hence, their method does not allow to take (M f )∗ω (2t). Inequality (4.10)
was proved in [6].
Cordoba and Fefferman proved in [8] the inequality
(4.11)
(T f )# (x) ≤ cp Mp f (x) (1 < p < ∞)
where Mp f (x) = (M |f |p )1/p (x). We shall show now that this inequality can
be improved. To this end, we use the following estimate obtained in Jawerth
and Torchinsky [11]: For all f ∈ L1loc (Rn ) and all x ∈ Rn ,
(4.12)
c1 M Mλ# f (x) ≤ f # (x) ≤ c2 M Mλ# f (x)
where 0 < λ < λ0 (n), c1 and c2 depends on n and λ. The relations (4.8) and
(4.12) imply the inequality:
(T f )# (x) ≤ cM M f (x).
This inequality is more precise than (4.11) since, as shown in [7],
M Mp f (x) ≤ cMp f (x) (p > 1)
and thus
M M f (x) ≤ M Mp f (x) ≤ cMp f (x)
for p > 1.
Let us note some consequences from inequality (3.1). Denote by BM O(ω)
the space of all functions f such that
kf k∗,ω ≡ sup
Q⊂Rn
1
ω(Q)
Z
Q
|f (y) − fQ,ω |ω(y) dy < ∞
11
µ
fQ,ω =
1
ω(Q)
¶
Z
Q
f (y)ω(y) dy .
Let us start with the John – Nirenberg theorem [13] in the weight case. The
claim that this theorem can be extended to the case of weights satisfying the
doubling condition was mentioned already in [15].
Theorem 4.3 [John – Nirenberg]. Let ω ∈ D. Then, for every function
f ∈ BM O(ω) and every cube Q holds the inequality:
((f − fQ,ω )χQ )∗ω (t) ≤ ckf k∗,ω log+
2ω(Q)
t
(0 < t < ∞),
or, equivalently,
µ
ω{x ∈ Q : |f (x) − fQ,ω | > α} ≤ 2ω(Q) exp
α
−
ckf k∗,ω
¶
(0 < α < ∞)
where c depends only on ω.
Proof. It follows from inequality (3.1) that
(4.13)
((f − fQ,ω )χQ )∗ω (t) ≤ 10cω kf k∗,ω + ((f − fQ,ω )χQ )∗ω (2t)
for t < ω(Q)/5cω . But
((f −
fQ,ω )χQ )∗ω (t)
5cω
≤
ω(Q)
Z
Q
|f (x) − fQ,ω |ω(x) dx ≤ 5cω kf k∗,ω
provided t ≥ ω(Q)/5cω . Thus inequality (4.13) is valid for all t > 0.
Suppose now that ω(Q)/2k+1 < t ≤ ω(Q)/2k . Applying (4.13) k times we
get
2ω(Q)
10cω
kf k∗,ω log+
.
log 2
t
((f − fQ,ω )χQ )∗ω (t) ≤ 10cω kf k∗,ω (k + 1) ≤
The theorem is proved.
Remark 4.1. Just in the same way the weighted analogy of John and
Strömberg theorem follows from (3.1) : For any cube Q there exists a constant
aQ,ω such that
#
((f − aQ,ω )χQ )∗ω (t) ≤ ckMλ,ω
f k∞ log+
2ω(Q)
t
(0 < t < ∞, 0 < λ ≤ 1/5cω ).
Hence
1
ω(Q)
Z
Q
|f (y) − fQ,ω |ω(y) dy ≤ 2
12
1
ω(Q)
Z
Q
|f (y) − aQ,ω |ω(y) dy
≤
#
ckMλ,ω
f k∞ Z
ω(Q)
Thus
(4.14)
ω(Q)
0
log+
2ω(Q)
#
dt ≤ c0 kMλ,ω
f k∞ .
t
#
kf k∗,ω ³ kMλ,ω
f k∞ .
Since for λ ≤ 1/2 we have kf k∗ ³ kMλ# f k∞ , the inequality (4.14) and
Lemma 2.3 imply the result of Muckenhoupt and Wheeden [15], namely: If
ω ∈ A∞ , then BM O(ω) = BM O.
We shall show next that (3.1) implies the inequality of Bennett, DeVore
and Sharpley (1.2). Indeed, integrating inequality (3.1) we get
(4.15)
∗
∗∗
∗∗
(f · χQ )∗∗
ω (t) − (f · χQ )ω (t) ≤ 2((f · χQ )ω (t) − (f · χQ )ω (2t))
#
f )∗∗
≤ 4(Mλ,ω
ω (2t) (0 < t < ω(Q)/5cω , 0 < λ ≤ 1/5cω ).
Let ω be the Lebesgue measure. Then, according to Hardy – Littlewood –
Hertz inequality (see [4]), we have
c1 f ∗∗ (t) ≤ (M f )∗ (t) ≤ c2 f ∗∗ (t)
(0 < t < ∞).
The left-hand side of this inequality, (4.12) and (4.15) yield (1.2).
Acknowledgement. The author is grateful to V. I. Kolyada under the
guidance of whom this work was completed.
References
[1] R. J. Bagby and D. S. Kurtz, Covering lemmas and the sharp function, Proc. Amer. Math. Soc. 93 (1985), 291–296.
[2] R. J. Bagby and D. S. Kurtz, A rearranged good λ inequality, Trans.
Amer. Math. Soc. 293 (1986), 71–81.
[3] C. Bennett, R. DeVore and R. Sharpley, Weak-L∞ and BM O,
Ann. of Math. 113 (1981), 601–611.
[4] C. Bennett and R. Sharpley, Weak-type inequalities for H p and
BM O, Proc. Symp. Pure Math. 35 (1979), 201–229.
[5] K. M. Chong and N. M. Rice, Equimeasurable rearrangements of
functions. In: Queen’s Papers in Pure and Appl. Math. 28, Queen’s
University, Kingston, Ont., 1971.
[6] R. R. Coifman and C. Fefferman, Weighted norm inequalities for
maximal functions and singular integrals, Stud. Math. (PRL) 15 (1974),
241–250.
13
[7] R. R. Coifman and R. Rochberg, Another characterization of BM O,
Proc. Amer. Math. Soc. 79 (1980), 249–254.
[8] A. Cordoba and C. Fefferman, A weighted norm inequality for singular integrals, Stud. Math. (PRL) 57 (1976), 97–101.
[9] C. Fefferman and B. Muckenhoupt, Two non-equivalent conditions
for weight functions, Proc. Amer. Math. Soc. 45 (1974), 99–104.
[10] C. Fefferman and E. M. Stein, H p spaces of several variables, Acta
Math. 129 (1972), 137–193.
[11] B. Jawerth and A. Torchinsky, Local sharp maximal functions, J.
Approx. Theory 43 (1985), 231–270.
[12] F. John, Quasi-isometric mappings, Seminari 1962 – 1963 di Analisi,
Algebra, Geometria e Topologia, Rome, 1965.
[13] F. John and L. Nirenberg, On functions of bounded mean oscillation,
Comm. Pure Appl. Math. 14 (1961), 415–426.
[14] V.I. Kolyada, On the imbedding of certain classes of functions of several
variables, Sybir. Mat. J. 14 (1973), 766–790 (in Russian).
[15] B. Muckenhoupt and R. L. Wheeden, Weighted bounded mean oscillation and the Hilbert transform, Stud. Math. (PRL) 55 (1976), 279–
294.
[16] E. Stein, Singular Integrals and differentiability properties of functions,
Mir, Moscow, 1973 (in Russian).
[17] E. M. Stein, Singular integrals, harmonic functions, and differentiability properties of functions of several variables, Proc. Symp. Pure. Math.
10 (1967), 316–335.
[18] J.-O. Strömberg, Bounded mean oscillation with Orlicz norms and
duality of Hardy spaces, Indiana Univ. Math. J. 28 (1979), 511–544.
14