Volume 10 (2009), Issue 2, Article 58, 10 pp.
ON MAXIMAL INEQUALITIES ARISING IN BEST APPROXIMATION
F. D. MAZZONE AND F. ZÓ
CONICET AND D EPARTAMENTO DE M ATEMÁTICA
U NIVERSIDAD NACIONAL DE R ÍO C UARTO
(5800) R ÍO C UARTO , A RGENTINA
fmazzone@exa.unrc.edu.ar
I NSTITUTO DE M ATEMÁTICA A PLICADA S AN L UIS
CONICET AND D EPARTAMENTO DE M ATEMÁTICA
U NIVERSIDAD NACIONAL DE S AN L UIS
(5700) S AN L UIS , A RGENTINA
fzo@unsl.edu.ar
Received 07 November, 2006; accepted 02 June, 2009
Communicated by S.S. Dragomir
A BSTRACT. Let f be a function in an Orlicz space LΦ and µ(f, L) be the set of all the best
Φ-approximants to f, given a σ−lattice L. Weak type inequalities are proved for the maximal
operator f ∗ = supn |fn |, where fn is any selection of functions in µ(f, Ln ), and Ln is an
increasing sequence of σ-lattices. Strong inequalities are proved in an abstract set up which can
be used for an operator as f ∗ .
Key words and phrases: Best approximants, Φ-approximants, σ-lattices, Maximal inequalities.
2000 Mathematics Subject Classification. Primary: 41A30; Secondary: 41A65.
1. I NTRODUCTION AND M AIN R ESULT
Let (Ω, A, µ) be a finite measure space and M = M(Ω, A, µ) the set of all A-measurable
real valued functions. Let Φ be a Young function, that is an even and convex function Φ : R →
R+ such that Φ(a) = 0 iff a = 0. We denote by LΦ the space of all the functions f ∈ M such
that
Z
(1.1)
Φ(tf )dµ < ∞,
Ω
for some t > 0.
We say that the function Φ satisfies the ∆2 condition (Φ ∈ ∆2 ) if there exists a positive
constant Λ = ΛΦ such that for all a ∈ R
Φ(2a) ≤ ΛΦ(a).
The first author was supported by CONICET and SECyT-UNRC. The second author was supported by CONICET and UNSL grants.
286-06
2
F. D. M AZZONE AND F. Z Ó
Under this condition, it is easy to check that f ∈ LΦ iff inequality (1.1) holds for every positive
number t.
The function Φ satisfies the ∇2 condition (Φ ∈ ∇2 ) if there exists a constant λ = λΦ > 2
such that
Φ(2a) ≥ λΦ(a).
A subset L ⊂ A is a σ-lattice iff ∅, Ω ∈ L and L is closed under countable unions and
intersections. Set LΦ (L) for the set of L-measurable functions in LΦ (Ω). Here, L-measurable
function means the class of functions f : Ω → R such that {f > a} ∈ L, for all a ∈ R.
A function g ∈ LΦ (L) is called a best Φ-approximation to f ∈ LΦ iff
Z
Z
Φ(f − g)dµ = min
Φ(f − h)dµ.
Ω
h∈LΦ (L)
Ω
We denote by µ(f, L) the set of all the best Φ-approximants to f . It is well known that µ(f, L) 6=
∅, for every f ∈ LΦ , see [9].
When L is a σ-field B ⊂ A and Φ(t) = t2 , the set µ(f, B) has exactly one element, namely
the conditional expectation EB (f ) relative to B, which is a linear operator in L2 and can be
continuously extended to all L1 . For Φ(t) = tp , 1 < p < ∞ we obtain the p-predictor PB (f )
in the sense of Ando and Amemiya [1], which coincides with the conditional expectation for
p = 2. The p-predictor operator PB (f ) is generally non-linear, and it is possible to extend it to
Lp−1 as a unique operator preserving a property of monotone continuity, see [10], where PL is
studied for the σ-lattice L. The operator PL (f ), when L is a σ-lattice and p = 2, falls within
what is called the theory of isotonic regression, first introduced by Brunk [4] (for applications
and further development, see [2, 14]). When Φ(x) = x and B is a σ-field, a function g in the
set µ(f, B) is a conditional median, see [15] and [11] for more recent results. All the situations
described above are dealt with by considering minimization problems using convex functions
and Orlicz Spaces LΦ . For other and more detailed applications, see [2, 14] and chapter 7 of
[13].
Rx
We adjust a Young function Φ to the origin by Φ̂(x) = 0 ϕ̂(t)dt with ϕ̂(x) = ϕ+ (x) −
ϕ+ (0)sign(x), where ϕ+ denotes the right continuous derivative of Φ. Now we can state our
principal result.
Theorem 1.1. Let Φ be a Young function such that Φ̂ ∈ ∆2 ∩ ∇2 . Suppose that Ln is an
increasing sequence of σ-lattices, i.e. Ln ⊂ Ln+1 for every n ∈ N. Let f be a nonnegative
function in LΦ , let fn be any selection of functions in µ(f, Ln ), and consider the maximal
function f ∗ = supn fn . Then there exists constants C and c such that f ∗ satisfies the following
weak type inequality:
Z
C
∗
ϕ+ (f )dµ,
(1.2)
µ({f > α}) ≤
ϕ+ (α) {f >cα}
for every α > 0.
The constant C only depends on ΛΦ̂ and c depends on ΛΦ̂ and λΦ̂ .
If ϕ+ (0) = 0 we can set c = 12 and we also have
Z
C
∗
(1.3)
µ({f > α}) ≤
ϕ+ (f )dµ,
ϕ+ (α)
{f ∗ >α}
for every α > 0.
The constants ΛΦ̂ and λΦ̂ are those used in the definitions of the conditions ∆2 and ∇2 respectively.
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M AXIMAL I NEQUALITIES
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Theorem 1.1 (in particular inequality (1.3) with ϕ+ (t) = tp−1 , 1 < p < ∞) is an Orlicz
version of the “martingale maximal theorem”, Theorem 5.1 given
in
[6]. The classical Doob
result is given by inequality (1.3) with ϕ+ (t) = t and fn = EBn f where Bn is a increasing
sequence of σ-fields in A.
We emphasise that our maximal operator f ∗ is built up with functions fn ∈ µ(f, Ln ) obtained
as a minimization problem in LΦ , though (1.2) and (1.3) can be seen as some sorts of weak type
inequalities in Lϕ+ for functions f ∈ LΦ , a strictly smaller subset of Lϕ+ . The extension of the
operator µ(f, L) to all Lϕ+ is not an easy task for general Φ and L, see [5] for some results in
this direction and Theorem 1.1 can be applied to the extension operator given there.
Since operators such as f ∗ as well as other operators obtained as a best approximation function are not linear or even not sublinear, and in many cases are not positive homogeneous operators, we will assume that the inequalities (1.2) or (1.3) hold for two fixed measurable functions
f and f ∗ and any a > 0. From this set up, we interpolate to obtain the so called strong inequalities. Now we state the interpolation problem as follows.
Let ϕ be a nondecreasing function from R+ into itself, and we consider two fixed measurable
functions f, g : Ω → R+ satisfying the following weak type inequality
Z
Cw
(1.4)
µ({f > a}) ≤
ϕ(g)dµ,
ϕ(a) {f >a}
for any a > 0.
We try to find functions Ψ such that the strong type inequality below holds:
Z
Z
(1.5)
Ψ (f )dµ ≤ Cs Ψ (g)dµ,
Ω
Ω
where Cs = Cs (ϕ, Ψ, Cw ). That is, Cs depends only on ϕ, Ψ and the constant Cw in inequality
(1.4).
An inequality closely related to (1.4) is the following one:
fw Z
C
ϕ(g)dµ,
(1.6)
µ({f > a}) ≤
ϕ(a) {g>ca}
for every a > 0, and c a constant less than one.
It is well known in harmonic analysis and classical differentiation theory that is possible to
obtain inequality (1.6) from inequality (1.4) when the functions f, g are related by f = T g and
the function T is a sublinear operator bounded from L∞ into itself (see [6] or [16], and the last
part of the proof of the Theorem 1.1). In this case we need to assume that inequality (1.4) holds
for any measurable function g in the domain ofRT and any a > 0. We see that inequality (1.4)
x
implies inequality (1.6) if the function Φ(x) = 0 ϕ(t)dt is ∇2 (see Lemma 2.2).
The strong inequality (1.5) will be a consequence of standard arguments in interpolation theory [16]. In Theorem 2.3 we introduce the notion of quasi-increasing functions which implicitly
appears in some theorems (see Theorem 1.2.1 in [8]). The notion of quasi-increasing functions
is used to define when a function Φ2 is “bigger” than a function Φ1 and we will write Φ1 ≺ Φ2
(see Definition 2.2). This notation is used to state interpolation results for Orlicz spaces in
Corollaries 2.4, 2.5 and 2.6. In [8] a condition related to x2 ≺ Φ(x) is used to obtain strong
inequalities. The relation x ≺ ϕ(x) is also named as a Dini condition, i.e.
Z x
ϕ(t)
dt ≤ Cφ(x),
t
0
for all x > 0 (see Theorem 1 and Proposition 3 in [3]). More on the relation Φ1 ≺ Φ2 is given
in Section 3 where we extend some results of [7].
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 58, 10 pp.
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4
F. D. M AZZONE AND F. Z Ó
The results of Sections 2 and 3 can be used to obtain the strong inequalities (1.5) for the
particular operator f ∗ given in Theorem 1.1.
It was proved in [7], in an abstract set up, that if two functions η and ξ are related by a weak
type inequality (1.4) with respect to the function Φ0 , that is,
Z
Cw
(1.7)
µ({η > a}) ≤ 0
Φ0 (ξ)dµ,
Φ (a) {η>a}
for any a > 0, then η and ξ satisfy the strong inequality
Z
Z
Ψ(η)dµ ≤ CΨ Ψ(ξ)dµ,
for the functions Ψ : Ψ = (Φ0p , 1 ≤ p, and Ψ = (Φ)p , for 1 ≤ p (also for some p in the range
0 < p < 1). In proving these results the conjugate function Φ∗ was heavily used. We recall that
Φ∗ (s) = sup{st − Φ(t)}.
t
As consequence of Sections 2 and 3 we obtain a result more general than those in [7] without
appealing to the conjugate function.
2. A S IMPLE T HEOREM
The following lemma is well known, see [12].
Lemma 2.1. For every a ∈ R+ we have Φ(a) ≤ aϕ+ (a). Moreover, Φ ∈ ∆2 iff there exists a
constant C > 0 such that aϕ+ (a) ≤ CΦ(a).
Lemma 2.2. Let ϕ be a nondecreasing function from R+ into itself such that ϕ(rx) ≤ 12 ϕ(x),
for a constant 0 < r < 1, and every x > 0. Suppose that f and g are nonnegative measurable
functions defined on Ω satisfying inequality (1.4). Then there exists a positive constant c =
c(r, Cw ) < 1 such that
Z
2Cw
(2.1)
µ({f > a}) ≤
ϕ(g)dµ,
ϕ(a) {g>ca}
for every a > 0.
Proof. By an inductive argument we get
(2.2)
2n ϕ(rn a) ≤ ϕ(a).
Let n ∈ N be such that C2nw < 12 , and set c = rn . Now, we split the integral on the right hand
side of (1.4) into the sets {g ≤ ca} and {g > ca}. By (2.2) we get
Z
Cw
1
µ({f > a}) ≤
ϕ(g)dµ + µ({f > a}).
ϕ(a) {g>ca}
2
Therefore inequality (2.1) follows.
Remark 1. It is not difficult to see that a Young function Ψ satisfies the ∇2 condition iff its right
derivative ψ+ fulfills the condition on Lemma 2.2. That is, ψ+ (rx) ≤ 21 ψ+ (x), for a constant
0 < r < 1, and every x > 0.
Rx
Proof. Since Ψ(x) = 0 ψ+ (t)dt, the condition on ψ+ implies that Ψ(rx) ≤ 21 Ψ(x), for every
x > 0, which is equivalent to the ∇2 condition given before, see [12]. Now, if we have this
condition for Ψ, it is readily seen that ψ+ ( 2r x) ≤ 12 ψ+ (x).
We note that if Φ ∈ ∇2 then ϕ+ (0) = ϕ− (0) = 0, see Remark 1.
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M AXIMAL I NEQUALITIES
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Definition 2.1. We say that the function η : R+ → R+ is a quasi-increasing function iff there
exists a constant ρ > 0 such that
Z
1 x
η(t)dt ≤ ρη(x),
(2.3)
x 0
for every x ∈ R+ .
Theorem 2.3. Let f and g be measurable and positive functions defined on Ω satisfying inequality (2.1). Let Ψ be a C 1 ([0, +∞)) Young function and let ψ be its derivative. Assume that
ψ
is a quasi-increasing function.Then
ϕ
Z
Z
2
(2.4)
Ψ (f )dµ ≤ 2Cw ρ Ψ
g dµ.
c
Ω
Ω
Proof. We have that
Z
∞
Z
Ψ (f )dµ =
ψ(a)µ({f > a})dµ
Z
Z ∞
ψ(a)
ϕ(g)dµ da
≤ 2Cw
ϕ(a)
{g>ca}
0
!
Z
Z c−1 g
ψ(a)
= 2Cw
ϕ(g)
da dµ.
ϕ(a)
Ω
0
Ω
0
(2.5)
Now, we get
Z
(2.6)
0
c−1 g
ψ(a)
ψ(c−1 g)
da ≤ ρc−1 g
ϕ(a)
ϕ(c−1 g)
Ψ (2c−1 g)
≤ρ
ϕ(c−1 g)
Therefore, from equations (2.5), (2.6) and since c < 1 in Lemma 2.2, we obtain Theorem
2.3.
Definition 2.2. Let ϕ1 , ϕ2 be two functions from R+ into R+ . We say that ϕ1 ≺ ϕ2 iff ϕ2 ϕ−1
1
is a quasi-increasing function.
The notation Φ1 ≺ Φ2 is also used if both Φ1 and Φ2 are Young functions, in this case
Definition 2.2 is applied for the restriction of these functions to R+ .
Remark 2. Let Φ1 and Φ2 be two Young functions and let ϕ1 + , ϕ2 + be their right derivatives.
If Φ1 , Φ2 ∈ ∆2 , using Lemma 2.1, we have Φ1 ≺ Φ2 ⇔ ϕ1 + ≺ ϕ2 + .
3
3
Remark 3. Despite the symbol used, ≺ is not an order relation. We have x2 ≺ x 2 and x 2 ≺ x,
but the relation x2 ≺ x is false. In fact, for two arbitrary powers we have xα ≺ xβ ⇔ α−1 < β.
We may define, and it is useful, the relation ϕ1 ≺ ϕ2 only for x near zero, say 0 < x ≤ 1,
and only for large values of x, i.e. 1 ≤ x. In the example given below, we will omit the rather
straightforward calculations.
Example 2.1. For 0 < x ≤ 1 we have xα ≺ ln(1 + x) if and only if 0 < α < 2, and for 1 ≤ x
the same relation is true only in the range 0 < α < 1. On the other hand ln(1 + x) ≺ xα for all
x and 0 < α. All the functions involved here are ∆2 functions, but (1 + x) ln(1 + x) − x is not
∇2 , so its derivative ln(1 + x) does not fulfill the condition on Lemma 2.2 (see Remark 1).
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 58, 10 pp.
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6
F. D. M AZZONE AND F. Z Ó
R xIn the following corollaries of Theorem 2.3 the Young function Φ is the one given by Φ(x) =
ϕ(t)dt. They are obtained using this theorem, Lemma 2.2, Remark 1 and Remark 2.
0
Corollary 2.4. Let f and g be measurable and positive functions defined on Ω satisfying inequality (1.4). Let Ψ be a C 1 ([0, +∞)) Young function and let ψ be its derivative. Assume that
ϕ ≺ ψ and the ∇2 condition for the function Φ holds. Then we have inequality (2.4).
Corollary 2.5. Let f and g be measurable and positive functions defined on Ω satisfying inequality (2.1), and assume Φ is a ∆2 function. Let Ψ be a C 1 ([0, +∞)) ∩ ∆2 Young function. If
Φ ≺ Ψ, then
Z
Z
(2.7)
Ψ (f )dµ ≤ C
Ψ (g)dµ,
Ω
Ω
where the constant C is independent of the functions f and g.
Corollary 2.6. Let f and g be measurable and positive functions defined on Ω satisfying inequality (1.4) , and assume Φ is a ∆2 ∩ ∇2 function. Let Ψ be a C 1 ([0, +∞)) ∩ ∆2 Young
function. If Φ ≺ Ψ, then
Z
Z
(2.8)
Ψ (f )dµ ≤ C
Ψ (g)dµ,
Ω
Ω
where the constant C is independent of the functions f and g.
Remark 4. By Corollary 2.6 we obtain inequality (1.5) for the following functions Ψ (all the
theorems quoted here belong to [7] and see that paper for a proof using conjugate functions). If
Ψ = Φ, clearly Φ ≺ Φ, that is Theorem 3.3. The case p > 1 of Theorem 3.8 follows by setting
Ψ = Φp . For Theorem 3.4, set Ψ = ϕp , p > 1 and observe that ϕ ≺ pϕp−1 ϕ0 .
The operator f ∗ introduced in [7] is a monotone operator and (f + c)∗ = f ∗ + c for any
constant c. We can use Corollary 2.5 to obtain
Z
Z
∗
(2.9)
Ψ (f )dµ ≤ C
Ψ (f )dµ,
Ω
Ω
Ψ
for every function f ∈ L , and all Ψ quoted in Remark 4. Now the condition ∇2 is dropped.
3. T HE R ELATION Φ ≺ Ψ
If η : R+ → R+ is a nondecreasing function then η is clearly a quasi-increasing function. On
the other hand, there are decreasing functions which are quasi-increasing functions. We note
that if η is a quasi-increasing and nonincreasing function then
Z x
Z x
2
x x
η(t)dt ≥ η
.
ρxη(x) ≥
η(t)dt ≥
2
2
0
0
Therefore, there exists a constant K such that
x
(3.1)
η
≤ Kη(x).
2
Lemma 3.1. Let η : R+ → R+ be a nonincreasing function. If η satisfies inequality (3.1) with
K < 2, then η is a quasi-increasing function.
Rx
Proof. In addition to the continuous average Aη(x) = x1 0 η(t)dt, is convenient to introduce
P 1 x
0
the discrete averages Ad η(x) = ∞
0 2k η( 2k ) and Ad η = Ad η − η.
As η is a nonincreasing function we have
1
(3.2)
Ad η ≤ Aη ≤ A0d η.
2
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 58, 10 pp.
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M AXIMAL I NEQUALITIES
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We estimate the discrete average A0d η,
(3.3)
A0d η(x)
n
∞ ∞
X
K
1 x X K
=
η n ≤
η(x) =
η(x).
n
2
2
2
2−K
1
1
Now the lemma follows by (3.2) and (3.3).
Corollary 3.2. Let Ψ Φ−1 be a nonincreasing function, Φ ∈ ∆2 and Ψ ∈ ∇2 . Moreover if we
assume that λ−1
Ψ ΛΦ < 2, then Φ ≺ Ψ .
The next corollary is a version of Theorem 3.8 in [7] for the case 0 < p < 1.
Corollary 3.3. Let ϕ : R+ → R+ be a nondecreasing function with the ∆2 condition 2ϕ(x) ≤
Kϕ( x2 ). Let f and g be measurable nonnegative functions defined on Ω satisfying inequality
(2.1). Then
Z
Z
p
(3.4)
Φ (f )dµ ≤ C
Φp (g)dµ,
Ω
−1
for any 1 ≥ p > ln(K/2)(ln K) and Φ(x) =
K 1−p )) as p → ln(K/2)(ln K)−1 .
Ω
Rx
0
ϕ(t)dt. Moreover the constant C is O(1/(2−
Proof. Since Φ(x) ≤ KΦ( x2 ) we have Φp−1 ( x2 ) ≤ K 1−p Φp−1 (x) for 0 < p < 1. Therefore, by
Lemma 3.1, Φ ≺ Φp whenever K 1−p < 2, and inequality (3.4) follows by Corollary 2.5.
Remark 5. It is possible to replace (2.1) by (1.4) to again obtain inequality (3.4) for the same
range of p if we place on ϕ the condition ϕ(rx) ≤ 12 ϕ(x) with a constant 0 < r < 1, and
2ϕ(x) ≤ Kϕ( x2 ), that is, if Φ ∈ ∆2 ∩ ∇2 (see Lemma 2.2).
Proposition 3.4. Let ΦRbe in C 1 ([0, +∞)) ∩ ∆2 and let Ψ be a quasi increasing function. For
x
Ψ1
the function Ψ1 (x) = 0 Ψ (t)dt, suppose that there exists a constant p > 1 such that [Φ]
p is
non-decreasing. Then Φ ≺ Ψ .
Proof. We have that log Ψ1 − p log Φ is a non-decreasing function in C 1 ((0, +∞)). Then
0
0
p ΦΦ , or (q − 1) ΨΦ ≥ q ΦΦΨ21 , with q = p/(p − 1). Therefore
Ψ Φ − Φ0 Ψ1
Ψ
q
≥ .
2
Φ
Φ
Ψ
Ψ1
≥
Integrating the above inequality on [, x] we get
Z x
Ψ1 (x)
Ψ
Ψ1 ()
(3.5)
q
≥
dt +
.
Φ(x)
Φ()
Φ
From the hypotheses we have that Ψ1 ()/Φ() → 0, when → 0. Therefore inequality (3.5)
implies that
Z x
Ψ1 (x)
Ψ
q
≥
dt.
Φ(x)
0 Φ
Taking into account that Ψ is a quasi-increasing function, it follows that Φ ≺ Ψ .
We can use Proposition 3.4 to prove a generalization of Theorem
R x 3.4 of [7] (see the end of
1
Remark 4). Indeed, given functions ϕ, θ ∈ C ∩ ∆2 set Φ(x) = 0 ϕ(t)dt and Ψ(x) = θ(ϕ(x)).
Then we have Φ ≺ Ψ if θ(x) ÷ xp is a nondecreasing function for some p > 1. In fact, Φ0 ≺ Ψ0
by Proposition 3.4.
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 58, 10 pp.
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8
F. D. M AZZONE AND F. Z Ó
4. P ROOF OF THE T HEOREM 1.1
We need some additional considerations.
Lemma 4.1. Let Φ be a convex function satisfying the ∆2 condition. Then there exists a constant
C > 0 such that for every a, x ≥ 0 we have that
ϕ+ (a) + C 2 ϕ+ (x − a) ≤ (C 2 + 1)ϕ+ (x).
Proof. If x ≥ a the assertion in the lemma is trivial. We suppose that x < a. Thus
max{x, a − x}. Then
a
ϕ+ (a) ≤ Kϕ+
2
≤ Kϕ+ (x) + Kϕ+ (a − x)
(4.1)
a−x
2
2
≤ K ϕ+ (x) + K ϕ+
2
≤ K 2 ϕ+ (x) + K 2 ϕ− (a − x).
The lemma follows using ϕ+ (y) = −ϕ− (−y) and (4.1).
a
2
≤
The following theorem was proved in [11]. We denote by L the σ-lattice of the sets D such
that Ω \ D ∈ L.
Theorem 4.2. Let f ∈ LΦ and L ⊂ A be a σ-lattice. Then g ∈ µ(f, L) iff for every C ∈ L,
D ∈ L and a ∈ R the following inequalities hold
Z
Z
(4.2)
ϕ± (f − a)dµ ≥ 0 and
ϕ± (f − a)dµ ≤ 0.
{g>a}∩D
{g<a}∩C
The set µ(f, L) admits a minimum and a maximum, i.e. there exist elements L(f, L) ∈
µ(f, L) and U (f, L) ∈ µ(f, L) such that for all g ∈ µ(f, L)
L(f, L) ≤ g ≤ U (f, L).
See [9, Theorem 14].
Now we prove Theorem 1.1.
Proof. We define An,1 = {f1 > α} and
Aj,n := {f1 ≤ α, . . . , fj−1 ≤ α, fj > α}
for j = 2, . . . , n.
Then we have that
An =
sup fj > α
= A1,n ∪ · · · ∪ An,n .
1≤j≤n
As a consequence of Theorem 4.2, we obtain
Z
ϕ+ (f − α)dµ ≥ 0.
Aj,n
Since Aj,n ∩ Ai,n = ∅ for i 6= j, it follows that
Z
Z
ϕ+ (f − α)dµ = lim
{f ∗ >α}
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 58, 10 pp.
n→∞
ϕ+ (f − α)dµ ≥ 0.
An
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M AXIMAL I NEQUALITIES
9
Therefore
(4.3)
ϕ+ (0)µ({f < α} ∩ {f ∗ > α})
Z
∗
≤ ϕ+ (0)µ({f ≥ α} ∩ {f > α}) +
ϕ̂(f − α)dµ.
{f ∗ >α}
Now, using Lemma 4.1 we have
Z
Z
ϕ̂(f − α)dµ ≤ C1
(4.4)
{f ∗ >α}
ϕ̂(f )dµ − C2 ϕ̂(α)µ({f ∗ > α})
{f ∗ >α}
with Ci , i = 1, 2, constants depending only on Λϕ̂ . Taking into account (4.3) and (4.4), we get
Z
∗
∗
(4.5)
ϕ+ (α)µ({f > α}) ≤ Cϕ+ (0)µ({f ≥ α} ∩ {f > α}) + C
ϕ̂(f )dµ,
{f ∗ >α}
where C = C(Λϕ̂ ). Thus we have proved inequality (1.3) of Theorem 1.1.
In order to prove inequality (1.2) of Theorem 1.1, we consider two cases.
Let us begin by assuming that ϕ+ (0) > 0. We then split the set {f ∗ > α} in the integral of
(4.5) in the two regions {f ∗ > α} ∩ {f > cα} and {f ≤ cα} ∩ {f ∗ > α}. Now we use the
fact that Φ̂ ∈ ∇2 and by Remark 1 there exist constants 0 < c < 1 and 0 < r small such that
ϕ̂(cx) ≤ rϕ̂(x). Then we have:
(4.6)
ϕ+ (α)µ({f ∗ > α}) ≤ Cϕ+ (0)µ({f ≥ α})
Z
ϕ̂(f )dµ + rCϕ+ (α)µ({f ∗ > α}).
+C
{f >cα}
We now use the Chebyshev inequality, rC < 12 and ϕ+ (0) ≤ ϕ+ (α) to obtain inequality (1.2)
with constant 4C.
The second case is ϕ+ (0) = 0. Now we have
Z
C
∗
µ({f > α}) ≤
ϕ+ (f )dµ
ϕ+ (α) {f ∗ >α}
for every f ∈ LΦ and α > 0.
Let f ∈ LΦ and define f1 = f χ{f ≥ α } . Thus f ≤ f1 + α/2. Then fn ≤ U (f1 , Ln ) + α/2 and
2
∗
{f > α} ⊂
α
sup U (f1 , Ln ) >
2
n
.
Therefore
α
µ ({f > α}) ≤ µ
sup U (f1 , Ln ) >
2
n
Z
C
≤
ϕ+ (f1 )dµ
ϕ+ (α) Ω
Z
C
=
ϕ+ (f )dµ.
ϕ+ (α) {f > α2 }
∗
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 58, 10 pp.
http://jipam.vu.edu.au/
10
F. D. M AZZONE AND F. Z Ó
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J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 58, 10 pp.
http://jipam.vu.edu.au/