Medians, Continuity, and Oscillation

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by  Jonathan Poelhuis

MEDIANS, CONTINUITY, AND VANISHING
OSCILLATION
JONATHAN POELHUIS AND ALBERTO TORCHINSKY
2 Abstract. In this paper we consider properties of medians as they
1 pertain to the continuity and vanishing oscillation of a function. Our
0 approach is based on the observation that medians are related to local
2 sharp maximal functions restricted to a cube of Rn.

l
u
J

1 In considering the problem of the resistance of materials to certain types
1
of deformations, F. John was led to the study of quasi-isometric mappings.

] The setting is essentially as follows. Let Q Rn be a cube and f a con-
A 0
⊂
C tinuous function on Q . Assume that to each subcube Q of Q with sides
0 0
h. parallel to those of Q0 there is assigned a constant cQ and let µQ be the
t
a function of the real variable M given by
m
y Q : f(y) c > M
[ µ (M) = |{ ∈ | − Q| }| .
Q
Q
1 | |
v Let φ(M) = sup µ (M), 0 < s < 1/2, and λ a number such that
0 Q⊂Q0 Q
1 φ(λ) s. Then under these assumptions
7 ≤
2 φ(M) Ae−BM/λ
. ≤
7
0 holds for all nonnegative M where A,B are universal functions of s and
2
1 the dimension n. Thus the space of functions of bounded mean oscillation
:
v (BMO) was introduced and the John-Nirenberg inequality established [5].
i
X Str¨omberg adopted this setting when studying spaces close to BMO and
r
a considered different ways of describing the oscillation of a function f on
cubes. He also incorporated the value s = 1/2 above, which corresponds to
the notion of median value m (Q) = m (1/2,Q) of f over Q [11]. The local
f f
maximalfunctionsofStr¨ombergareofparticularinterest becausetheyallow
for pointwise estimates for Caldero´n-Zygmund singular integral operators
[4, 6].
In this paper we consider properties of medians as they pertain to the
continuity and vanishing oscillation of a function. Our approach is based on
the observation that medians are related to local sharp maximal functions
restricted to a cube Q Rn with parameter 0 < s 1/2 by means of the
0
⊂ ≤
2010 Mathematics Subject Classification. Primary 42B25; Secondary 46E30.
Key words and phrases. Medians, Local sharp maximal function.
1
2 J. POELHUISAND A.TORCHINSKY
expression
m (1 s,Q)
M♯,φ f(x) = sup inf |f−c| −
0,s,Q0 x∈Q,Q⊂Q0 c φ(|Q|)
m (1 s,Q)
sup |f−mf(1−s,Q)| − ,
∼ φ( Q )
x∈Q,Q⊂Q0 | |
where φ = 1 in the case of functions with bounded median oscillation with
parameter s (bmo ), and satisfies appropriate conditions in the case of func-
s
tionswithvanishing median oscillationwithparameter s(vmo ).Str¨omberg
s
showed that bmo = BMO, and, similarly, we show here that for sufficiently
s
small s, vmo = VMO, the space of functions of vanishing mean oscillation.
s
Moreover, since VMO is known to contain bounded discontinuous functions
[10,8], we complete thepicture by giving criteria for continuity forfunctions
equivalent to a bounded function on a cube in terms of medians.
The paper is organized as follows. In Section 1 we introduce the notions
ofmedianandmaximalmedianwithrespecttoaparameter0 < s < 1;when
s = 1/2 we refer to these medians as biased with parameter s. In Section
6
2 we consider the a.e.convergence of maximal biased medians in the spirit
of Fujii’s results for s = 1/2 [3]. In Section 3, motivated by similar results
involving averages [9], we characterize continuity in terms of maximal bi-
ased medians with parameter > 1/2. In Section 4 we extend the Str¨omberg
decomposition of cubes to parameters > 1/2. Finally, in Section 5 we con-
sider the spaces of functions with vanishing median oscillation, establish a
John-Nirenberg type inequality they satisfy, and show that, as anticipated,
they coincide with VMO for sufficiently small s.
1. Medians and maximal medians
In what follows we restrict our attention to cubes with sides parallel to
the coordinate axis.
Definition 1.1. For a cube Q Rn, 0 < s < 1, and a real-valued measur-
⊂
able function f on Q we say that m (s,Q) is a median value of f over Q
f
with parameter s if
(1.1) y Q : f(y) < m (s,Q) s Q
f
|{ ∈ }| ≤ | |
and
(1.2) y Q : f(y) > m (s,Q) (1 s) Q .
f
|{ ∈ }| ≤ − | |
When s = 1/2, m (1/2,Q) = m (Q) corresponds to a median value of
f f
f over Q. The set of median values of f is one point or a closed interval as
the example f = χ on [0,1] shows. It is therefore convenient to work
[1/2,1)
MEDIANS, CONTINUITY, AND VANISHING OSCILLATION 3
with maximal medians, which are uniquely defined [1, 3]. More precisely,
we have
Definition 1.2. For a cube Q Rn, 0 < s < 1, and a real-valued measur-
⊂
able function f on Q, we say that M (s,Q) is the maximal median of f
f
over Q with parameter s if
M (s,Q) = sup M : y Q : f(y) < M s Q .
f
{ |{ ∈ }|} ≤ | |
The reader will have no difficulty in proving the sup above is assumed,
that is to say,
y Q : f(y) < M (s,Q) s Q .
f
|{ ∈ }| ≤ | |
To justify the nomenclature of maximal median we verify that M (s,Q)
f
satisfies the conditions that characterize medians. (1.1) is guaranteed since
M (s,Q) is the maximum value for which it holds. As for (1.2), let B =
f n
y Q : f(y) M (s,Q) + 1/n and note that B (1 s) Q , all n,
f n
{ ∈ ≥ } | | ≤ − | |
and y Q : f(y) > M (s,Q) liminf B . Then y Q : f(y) >
f n n
{ ∈ } ⊂ |{ ∈
M (s,Q) liminf B (1 s) Q .
f n n
}| ≤ | | ≤ − | |
Hereafter when considering a median we mean the maximal median and
denote it simply by m (s,Q). Clearly maximal medians satisfy
f
(1.3) y Q : f(y) m (s,Q) s Q ,
f
|{ ∈ ≤ }| ≥ | |
and
(1.4) y Q : f(y) m (s,Q) (1 s) Q .
f
|{ ∈ ≥ }| ≥ − | |
We summarize the basic properties of maximal medians that are of in-
terest to us in the following Proposition.
Proposition 1.1. Let Q Rn be a cube, 0 < s,t < 1, and f,g real-valued
⊂
measurable function on Q. Then the following properties hold:
(i) For s < t,
(1.5) m (s,Q) m (t,Q).
f f
≤
(ii) If f g a.e., then
≤
(1.6) m (s,Q) m (s,Q).
f g
≤
(iii) For a constant c,
(1.7) m (s,Q) c = m (s,Q).
f f−c
−
(iv) If f,g 0 a.e., 0 < s,s < 1, and 0 < t < s+s 1,
1 1
≥ −
(1.8) m (t,Q) m (s,Q)+m (s ,Q).
f+g f g 1
≤
4 J. POELHUISAND A.TORCHINSKY
(v) In general, m (s,Q) m (s,Q) m (s,Q). And, if m (s,Q)
−|f| f |f| f
≤ ≤ ≤
0,
(1.9) m (s,Q) m (1 s,Q).
f |f|
| | ≤ −
Thus for general f,
(1.10) m (s,Q) m (s,Q), 1/2 s < 1.
f |f|
| | ≤ ≤
(vi) If f 0 is locally integrable and f denotes the average of f over Q,
Q
≥
then
1
(1.11) m (s,Q) f .
f Q
≤ 1 s
−
Proof. (i) Since y Q : f(y) < m (s,Q) s Q < t Q , (1.5) holds.
f
|{ ∈ }| ≤ | | | |
(ii) Up to a set of measure zero y Q : g(y) < m (s,Q) y Q :
f
{ ∈ } ⊂ { ∈
f(y) < m (s,Q) . Therefore y Q : g(y) < m (s,Q) s Q , and so
f f
} |{ ∈ }| ≤ | |
m (s,Q) m (s,Q).
f g
≤
(iii)Since y Q : f(y) < m (s,Q) = y Q : f(y) c < m (s,Q) c
f f
{ ∈ } { ∈ − − }
it readily follows that m (s,Q) c m (s,Q). And since y Q :
f f−c
− ≤ { ∈
f(y) c < m (s,Q) = y Q : f(y) < m (s,Q)+c ,m (s,Q)+c
f−c f−c f−c
− } { ∈ } ≤
m (s,Q). Note that, in particular, m (s,Q) = c.
f c
(iv) For the sake of argument suppose that f,g are measurable functions
on Q such that m (t,Q) (m (s,Q)+m (s ,Q)) > 2η > 0. Then y
f+g f g 1
− { ∈
Q : f(y) < m (s,Q)+η y Q : g(y) < m (s ,Q)+η y Q : f(y)+
f g 1
}∩{ ∈ } ⊂ { ∈
g(y) < m (t,Q) . Now, since y Q : f(y) < m (s,Q) + η s Q ,
f+g f
} |{ ∈ }| ≥ | |
y Q : g(y) < m (s ,Q) + η s Q , and y Q : f(y) + g(y) <
g 1 1
|{ ∈ }| ≥ | | |{ ∈
m (t,Q) t Q , it readily follows that s + s 1 + t, which is not the
f 1
}| ≤ | | ≤
case.
(v) Since f f f , by (1.6), m (s,Q) m (s,Q) m (s,Q).
−|f| f |f|
−| | ≤ ≤ | | ≤ ≤
Now, if m (s,Q) 0 note that y Q : f(y) < m (s,Q) = y
f f
≤ { ∈ | | − } { ∈
Q : m (s,Q) < f(y) y Q : m (s,Q) < f(y) , and, therefore,
f f
−| |} ⊂ { ∈ }
y Q : f(y) < m (s,Q) (1 s) Q . Consequently, m (s,Q)
f f
|{ ∈ | | − }| ≤ − | | | | ≤
m (1 s,Q). And, since for s 1/2, (1 s) s, by (1.5) we have
|f|
− ≥ − ≤
m (s,Q) m (s,Q) for that range of s.
f |f|
| | ≤
(vi) We may assume that m (s,Q) = 0. Then by (1.4) and Chebychev’s
f
6
inequality,
1
(1 s) Q y Q : f(y) m (s,Q) f(y)dy,
− | | ≤ |{ ∈ ≥ f }| ≤ m (s,Q) Z
f Q
(cid:3)
and the conclusion follows.
The restriction 1/2 s < 1 is necessary for (1.10) to hold. Let Q = [0,1]
≤
and f(x) = 2χ (x)+χ (x); then for 0 < s < 1/2, m (s,Q) = 2
[0,1/2) [1/2,1) f
− −
MEDIANS, CONTINUITY, AND VANISHING OSCILLATION 5
but m (s,Q) = 1 < 2. And, in contrast to averages, the restriction 0 < t <
|f|
s+s 1 is necessary for (1.8) to hold. To see this let Q = [0,1], and pick
1
−
1/2 < s s < 1,andt = s+s 1 > 0.Iff = χ andg = χ ,
1 ≤ 1− [0,1−s] [1−s1,2(1−s1)]
y Q : f(y) + g(y) < 1 = (1 s,1 s ) (2(1 s ),1] has measure
1 1
{ ∈ } − − ∪ −
(s s)+1 2(1 s ) = t,and,therefore,althoughm (s,Q) = m (s ,Q) = 0,
1 1 f g 1
− − −
m (t,Q) = 1.
f+g
Finally, maximal medians can be expressed in terms of distribution func-
tions or nonincreasing rearrangements. Recall that the nonincreasing re-
arrangement f∗ of f at level λ > 0 is given by f∗(λ) = inf α > 0 : x
{ |{ ∈
Rn : f(x) > α λ and satisfies
| | }| ≤ }
(1.12) y Rn : f(y) > f∗(u) u, u > 0.
|{ ∈ | | }| ≤
We then have
Proposition 1.2. Let Q Rn be a cube, 0 < s < 1, and f a measurable
⊂
function on Q. Then
m (1 s,Q) = inf α > 0 : y Q : f(y) > α < s Q = (fχ )∗(s Q ).
|f| Q
− { |{ ∈ | | }| | |} | |
Proof. Let α = inf α > 0 : y Q : f(y) > α < s Q . Then for all
{ |{ ∈ | | }| | |}
ε > 0 it readily follows that y Q : f(y) α+ε > (1 s) Q , which
|{ ∈ | | ≤ }| − | |
together with (1.3) implies m (1 s,Q) α+ε. Thus m (1 s,Q) α.
|f| |f|
− ≤ − ≤
Next, by (1.12), y Q : f(y) > (fχ )∗(s Q ) s Q , which gives
Q
|{ ∈ | | | | }| ≤ | |
α (fχ )∗(s Q ).
Q
≤ | |
Finally, since for ε > 0, y Q : f(y) > (fχ )∗(s Q ) ε > s Q ,
Q
|{ ∈ | | | | − }| | |
by (1.2) it readily follows that (fχ )∗(s Q ) ε < m (1 s,Q), and,
Q |f|
| | − −
consequently, (fχ )∗(s Q ) m (1 s,Q). (cid:3)
Q |f|
| | ≤ −
Other equivalent expressions appearing in the literature include those in
[2].
2. Convergence of medians
The examples following Proposition 1.1 suggest that medians rely more
heavily on the distribution of the values of f than do averages. On the
other hand, averages and medians are not always at odds. In particular,
using (1.11) the reader should have no difficulty in verifying that the fol-
lowing version of the Lebesgue differentiation theorem holds: If f is a locally
integrable function on Rn and 1/2 s < 1, then
≤
lim m (s,Q) = f(x)
f
x∈Q,Q→x
at every Lebesgue point x of f.
6 J. POELHUISAND A.TORCHINSKY
Thus, in some sense m (s,Q) is a good substitute for f for small Q.
f Q
In fact, a more careful argument gives that the biased maximal medians
m (s,Q) of an arbitrary measurable function f converge to f a.e., a fact
f
observed by Fujii for the case s = 1/2 [3].
Theorem 2.1. Let f be a real-valued, finite a.e.measurable function on Rn,
and 0 < s < 1. Then
(2.1) lim m (s,Q) = f(x) a.e.
f
x∈Q,Q→x
In particular, (2.1) holds at every point of continuity x of f.
Proof. For k 1 and an integer j, let E = x Rn : (j 1)/2k
k,j
≥ { ∈ − ≤
f(x) < j/2k , a = (j 1)/2k, and put S (x) = ∞ a χ (x).
} k,j − k j=−∞ k,j Ek,j
Note that since f is finite a.e., Rn = E excepPt possibly for a set
k,j k,j
of measure 0, and when f(x) is finite wSe have 0 f(x) S (x) 2−k,
k
≤ − ≤
which gives m (s,Q) m (s,Q) m (s,Q) + 2−k for all cubes Q. Let
Sk ≤ f ≤ Sk
A = x E : x is a point of density for E , A = ∞ A . Since
k,j { ∈ k,j k,j} k j=−∞ k,j
f is finite a.e., Rn A = 0 for all k, and if A = ∞ A ,Salso Rn A = 0.
| \ k| k=1 k | \ |
We claim that the limit in question exists forSx A. Given ε > 0, pick
∈
k such that 2−k+1 < ε. Then x A for some j, and
k,j
∈
A Q
k,j
lim | ∩ | = 1.
x∈Q,Q→x Q
| |
Let δ = max s,1 s and note that for all cubes Q with small enough
{ − }
measure containing x,
A Q
k,j
| ∩ | > δ.
Q
| |
We restrict our attention to such small cubes Q containing x. Note that
for these cubes m (s,Q) = a . Indeed, onthe one hand, since S (y) = a
Sk k,j k k,j
for y A , y Q : S (y) < a Ac Q < s Q , and, therefore,
∈ k,j |{ ∈ k k,j}| ≤ | k,j ∩ | | |
a m (s,Q). And, on the other, since for ε > 0, y Q : S (y) <
k,j ≤ Sk { ∈ k
a +ε A Q, it follows that y Q : S (y) < a +ε A
k,j k,j k k,j k,j
} ⊃ ∩ |{ ∈ }| ≥ | ∩
Q (s + η) Q . Hence, m (s,Q) a + ε, and since ε is arbitrary,
| ≥ | | Sk ≤ k,j
m (s,Q) a .
Sk ≤ k,j
Then, since a = m (s,Q) = S (x) for x A ,
k,j Sk k ∈ k,j
m (s,Q) f(x) m (s,Q) m (s,Q) + m (s,Q) f(x)
| f − | ≤ | f − Sk | | Sk − |
2−k +(f(x) S (x)) 2−k+1 < ε.
k
≤ − ≤
In other words, m (s,Q) f(x) < ε for x A and all Q with small enough
f
| − | ∈
measure containing x.
MEDIANS, CONTINUITY, AND VANISHING OSCILLATION 7
Now, at a point of continuity x of f, given ε > 0, let δ > 0 be such
that f(y) f(x) ε for y B(x,δ). Then for y in a cube Q containing x
| − | ≤ ∈
and contained in B(x,δ) we have ε f(y) f(x) ε, and, consequently,
− ≤ − ≤
ε = m (s,Q) m (s,Q) = m (s,Q) f(x) m (s,Q) = ε, and so
−ε f−f(x) f ε
− ≤ − ≤
m (s,Q) f(x) ε. (cid:3)
f
| − | ≤
3. A median characterization for continuity
We say that a measurable function f on a cube Q Rn is equivalent
0
⊂
to a continuous function on Q if the values of f can be modified on a
0
set of Lebesgue measure 0 so as to coincide with a continuous function on
Q ; similarly for f equivalent to a bounded function on a cube. In this
0
section we characterize those measurable functions equivalent to a bounded
function ona cube that areequivalent to a continuous function onthat cube
in terms of medians, keeping in mind that in the case of locally integrable
functions the condition involves the consideration of oscillations involving
two nonoverlapping cubes [9].
Definition 3.1. For 0 < s < 1 and nonoverlapping cubes Q ,Q Rn, let
1 2
⊂
Q Q
1 2
Ψ (f,Q ,Q ) = | | m (s,Q )+ | | m (s,Q ),
s 1 2 f 1 f 2
Q Q Q Q
1 2 1 2
| ∪ | | ∪ |
and
Ω(f,s,δ) = sup inf Ψ ( f c ,Q ,Q ).
s 1 2
c | − |
diam(Q1∪Q2)≤δ
Ψ isaweighted averageofmaximal mediansoff inthespirit ofaverages
s
andΩ(f,s,δ)isrelatedtotheoscillationofameasurable functiononacube,
as shown by the following result.
Theorem 3.1. Let Q Rn be a cube, 1/2 < s < 1, f a measurable
0
⊂
function on Q that is equivalent to a bounded function there, and ω(f,δ),
0
δ > 0, the essential modulus of continuity of f defined by
ω(f,δ) = sup ess sup f(x+h) f(x) .
x,x+h∈Q0| − |
|h|≤δ(cid:0) (cid:1)
Then we have Ω(f,s,δ) = ω(f,δ)/2.
Proof. For nonoverlapping cubes Q ,Q Q , let
1 2 0
⊂
Θ = ess osc(f,Q Q ) = ess sup f ess inf f .
1 ∪ 2 Q1∪Q2 − Q1∪Q2
Let δ > 0. If x Q Q Q is such that x+h Q where h δ, since
1 2 0 0
∈ ∪ ⊂ ∈ | | ≤
ess sup f(x+h) f(x) ess sup f ess inf f ,
x,x+h∈Q0| − | ≥ Q1∪Q2 − Q1∪Q2
8 J. POELHUISAND A.TORCHINSKY
taking the sup over h δ it readily follows that ω(f,δ) Θ. Moreover,
| | ≤ ≥
since for y Q and an arbitrary constant c,
1
∈
f(y) c max ess sup f c,c ess inf f ,
| − | ≤ { Q1∪Q2 − − Q1∪Q2 }
picking c = (sup inf f +ess inf f)/2, it follows that f(y) c
Q1∪Q2 Q1∪Q2 | − | ≤
Θ/2, and, consequently, m (s,Q ) m (s,Q ) = Θ/2; similarly we
|f−c| 1 Θ/2 1
≤
have m (s,Q ) Θ/2. Therefore,
|f−c| 2
≤
Q Θ Q Θ Θ
1 2
infΨ ( f c ,Q ,Q ) | | + | | = ,
s 1 2
c | − | ≤ Q1 Q2 2 Q1 Q2 2 2
| ∪ | | ∪ |
and, consequently, Ω(f,s,δ) Θ/2 ω(f,δ)/2.
≤ ≤
Conversely, let 0 < t < 2s 1. Then for fixed δ > 0, given ε > 0, pick h
−
with h < δ such that ess sup f(x+h) f(x) ω(f,δ) ε. Then
| | x,x+h∈Q0| − | ≥ −
E = x Q : x+h Q and f(x+h) f(x) ω(f,δ) ε has positive
0 0
{ ∈ ∈ | − | ≥ − }
measure. Let x E be a point of density of E and a small enough so that
∈
Q(x,a),Q(x+h,a) are nonoverlapping and
E Q(x,a)
| ∩ | > 1 t.
Q(x,a) −
| |
Now, since y Q(x+h,a) : g(y) < M = y Q(x,a) : g(y +h) < M
{ ∈ } { ∈ }
and Q(x,a) = Q(x+h,a) , it readily follows that m (s,Q(x+h,a)) =
|f−c|
| | | |
m (s,Q(x,a)),and,consequently, since f(y+h) f(y) f(y+h)
|f(·+h)−c|
| − | ≤ | −
c + f(y) c , by (1.8) and (1.6), m (s,Q(x,a))+m (s,Q(x+h,a))
|f−c| |f−c|
| | − | ≥
m (t,Q(x,a)) m (t,Q(x,a)). Therefore,
|f−c|+|f(·+h)−c| |f(·+h)−f|
≥
Ψ ( f c ,Q(x,a),Q(x+h,a))
s
| − |
1 1
m (s,Q(x,a))+ m (s,Q(x+h,a))
|f−c| |f−c|
≥ 2 2
1
m (t,Q(x,a)).
|f(·+h)−f|
≥ 2
Finally, since E Q(x,a) = y Q(x,a) : f(y + h) f(y)
| ∩ | |{ ∈ | − | ≥
ω(f,δ) ε > (1 t) Q(x,a) ,itreadilyfollowsthatm (t,Q(x,a))
|f(·+h)−f|
− }| − | | ≥
ω(f,δ) ε, which, since ε is arbitrary, implies Ψ ( f c ,Q(x,a),Q(x +
s
− | − |
h,a)) ω(f,δ)/2. Thus Ω(f,s,δ + (√2a)n) ω(f,δ), and letting a 0,
≥ ≥ →
Ω(f,s,δ) ω(f,δ)/2. (cid:3)
≥
Theorem 3.2. Let Q Rn be a cube, 1/2 < s < 1, and f a measurable
0
⊂
function on Q that is equivalent to a bounded function there. Then, f is
0
equivalent to a continuous function on Q iff lim Ω(f,s,η) = 0.
0 η→0+
The proof follows at once from Theorem 3.1. Note that by Proposition
1.2 the conclusion can also be stated in terms of rearrangements.
MEDIANS, CONTINUITY, AND VANISHING OSCILLATION 9
4. A decomposition of cubes
Str¨omberg’s essential tool in dealing with the oscillation of functions and
local maximal functions is a decomposition of cubes [11]. In this section we
extend the results to biased medians with parameters > 1/2.
We begin by introducing the local sharp maximal function restricted to
a cube.
Definition 4.1. Let Q Rn and 0 < s 1/2. For a measurable function
0
⊂ ≤
f on Q , M♯ f(x), the local sharp maximal function restricted to Q of
0 0,s,Q0 0
f is defined at x Q as
0
∈
(4.1)
M♯ f(x) = sup infinf α 0 : y Q : f(y) c > α < s Q .
0,s,Q0 x∈Q,Q⊂Q0 c { ≥ |{ ∈ | − | }| | |}
When Q = Rn, M♯ f(x) = M♯ f(x) denotes the local sharp maximal
0 0,s,Rn 0,s
function of f at x Rn.
∈
The range 0 < s 1/2 is necessary since for s > 1/2, M♯ f(x) = 0 for
≤ 0,s,Q
a function f that takes two different values.
Local maximal functions, as well as maximal functions defined in terms
in rearrangements, can be expressed in terms of medians. Let ω (f,Q) =
s
inf ((f c)χ )∗(s Q ). Then by Proposition 1.2,
c Q
− | |
M♯ f(x) = sup ω (f,Q) = sup infm (1 s,Q).
0,s,Q0 x∈Q,Q⊂Q0 s x∈Q,Q⊂Q0 c |f−c| −
The first expression above is used by Lerner [6, 7].
An efficient choice for c in the infimum above is m (1 s,Q).
|f−mf(1−s,Q)| −
Indeed, for Q Q and a constant c, since 1 s 1/2, by (1.7) and (1.10),
0
⊂ − ≥
(4.2) m (1 s,Q) c m (1 s,Q) .
f |f−c|
| − − | ≤ − |
Then, since f(y) m (1 s,Q) f(y) c + c m (1 s,Q) , by (1.5),
f f
| − − | ≤ | − | | − − |
(1.7), and (4.2),
m (1 s,Q) m (1 s,Q)+ c m (1 s,Q)
|f−mf(1−s,Q)| − ≤ |f−c| − | − f − |
m (1 s,Q)+m (1 s,Q) = 2m (1 s,Q),
|f−c| |f−c| |f−c|
≤ − − −
and, consequently,
(4.3) infm (1 s,Q) m (1 s,Q) 2 infm (1 s,Q).
c |f−c| − ≤ |f−mf(1−s,Q)| − ≤ c |f−c| −
The decomposition of cubes relies on three lemmas which we prove next.
Lemma 4.1. Let Q Rn be a cube, 0 < s 1/2, 1/2 t 1 s, and f
⊂ ≤ ≤ ≤ −
a measurable function on Q. Then for any η > 0,
(4.4) y Q : f(y) m (t,Q) 2 inf M♯ f(x)+η < s Q .
(cid:12){ ∈ | − f | ≥ x∈Q 0,s,Q }(cid:12) | |
(cid:12) (cid:12)
10 J. POELHUISAND A.TORCHINSKY
Proof. For fixed c, let α(c) = m (1 s,Q). Then by (4.2) and (1.5),
|f−c|
−
(4.5) m (t,Q) c m (t,Q) m (1 s,Q) = α(c),
f |f−c| |f−c|
| − | ≤ ≤ −
and by (1.4)
(4.6) y Q : f(y) c α(c)+ε < s Q , ε > 0.
|{ ∈ | − | ≥ }| | |
Let m = inf α(c) and pick c such that m α(c ) m+1/k, all k.
c k k
{ } ≤ ≤
Then by (4.5),
f(y) c f(y) m (t,Q) m (t,Q) c
k f f k
| − | ≥ | − |−| − |
f(y) m (t,Q) α(c ),
f k
≥ | − |−
and, consequently, since 2m + η α(c ) + (η 2/k), y Q : f(y)
k
≥ − { ∈ | −
m (t,Q) 2m + η y Q : f(y) c α(c ) + ε , where we
f k k k
| ≥ } ⊂ { ∈ | − | ≥ }
have chosen k sufficiently large so that ε = η (2/k) > 0. Then by (4.6),
k
−
y Q : f(y) m (t,Q) > 2m+η < s Q .Finally, since M# f(x) m
|{ ∈ | − f | }| | | 0,s,Q ≥
for all x Q, (4.4) holds. (cid:3)
∈
Lemma 4.2. Let Q Rn be a cube, 0 < s 1/2, 1/2 t 1 s, η > 0,
⊂ ≤ ≤ ≤ −
and f a measurable function on an open cube containing Q. Then for any
family of cubes Q with (1 ε)Q Q (1+ε)Q,
ε ε
{ } − ⊂ ⊂
limsup m (t,Q) m (t,Q ) 2 inf M♯ f(x)+η.
ε→0+ | f − f ε | ≤ x∈Q 0,s,Q
Proof. Let A = inf M♯ f(x). For the sake of argument assume there
x∈Q 0,s,Q
is a sequence ε 0 such that m (t,Q) m (t,Q ) > 2A+η for all k.
k → | f − f εk |
Then by (1.10),
2A+η < m (t,Q) m (t,Q ) m (t,Q ),
| f − f εk | ≤ |f−mf(t,Q)| εk
and, consequently, by (1.4),
(4.7) y Q : f(y) m (t,Q) > 2A+η (1 t) Q .
|{ ∈ εk | − f | }| ≥ − | εk|
Since Q (1+ε )Q, the left-hand side of (4.7) is bounded above by
εk ⊂ k
y (1+ε )Q : f(y) m (t,Q) > 2A+η
k f
|{ ∈ | − | }|
((1+ε )n 1) Q + y Q : f(y) m (t,Q) > 2A+η ,
k f
≤ − | | |{ ∈ | − | }|
and since (1 ε )Q Q , the right-hand side of (4.7) is bounded below by
− k ⊂ εk
(1 t)(1 ε )n Q .
k
− − | |
Whence combining these estimates it follows that
y Q : f(y) m (t,Q) > 2A+η (1 t)(1 ε )n ((1+ε )n 1) Q .
f k k
|{ ∈ | − | }| ≥ − − − − | |
(cid:0) (cid:1)