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L ERNER & C ARLOS P E´ REZ

A BSTRACT
...
This new index is defined by means of the local maximal operator mλ f
...

As an application it is shown a new characterization of the Muckenhoupt Ap class of weights: u ∈ Ap if and only if for any ε > 0 there
is a constant c such that for any cube Q and any measurable subset
E ⊂ Q,
!
|E|
logε
|Q|



|Q|
|E|



≤c

u(E)
u(Q)

1/p


...

Other applications are given, in particular within the context of the
variable Lp spaces
...
I NTRODUCTION
The main purpose of this paper is to provide a new way of defining the upper
Boyd index for general function quasi-Banach spaces X over Rn with respect to the
Lebesgue measure but not necessarily rearrangement-invariant
...
This problem is characterized in Theorem 1
...

2697
c , Vol
...
6 (2007)
Indiana University Mathematics Journal

2698

A NDREI K
...
The most interesting is a
new characterization of the Ap class of weights
...
Recall that a weight u
satisfies the Ap,1 condition if there is a constant c such that for any cube Q and
any measurable subset E ⊂ Q,
|E|
≤c
|Q|

(1
...
This class of weights is interesting because it
E
characterizes the weights for which M is of restricted weak type (p, p), namely,
sup t p u{x : MχE (x) > t} ≤ cu(E),
t>0

1
p,∞
which, as it is well known, is equivalent to M : Lp,
u → Lu
...
We will show
that u ∈ Ap if and only if condition (1
...
As an
special case we show that u ∈ Ap if and only if for any ε > 0 there is a constant c
such that for any cube Q and any subset E ⊂ Q,

|E|
logε
|Q|



|Q|
|E|


≤c

u(E)
u(Q)

!1/p

...
See Theorem 2
...

In the classical setting, if X is any rearrangement-invariant Banach function
space, then the well-known result due to Lorentz [19] and Shimogaki [25] about
¯ X
...
This result was
It establishes that M is bounded on X if and only if αa
extended by Montgomery-Smith [20] to the case of any rearrangement-invariant
quasi-Banach function space
...
Then, since X is rearrangement-invariant, the
0

problem is reduced to the study of the boundedness of the Hardy operator
...

However, in Analysis there are lots of important spaces that are not rearrangement-invariant in general
...
For some particular spaces different criteria of the
boundedness of M are well known
...


Lorentz-Shimogaki Theorem

2699

To pursue this direction we introduce a generalized definition of the upper
Boyd index
...

We give the following generalization of the upper Boyd index
...
1
...


kf kX ≤1

We define the generalized upper Boyd index as
(1
...

log(1/λ)

We observe that ΦX (λ) ≥ 1, 0 < λ < 1, since |f | ≤ mλ f a
...
(see [18,
Lemma 6])
...

Our main result is the following theorem, which can be regarded as an extension of the Lorentz-Shimogaki theorem
...
2
...
The following
statements are equivalent:
(i) M is bounded on X ;
(ii) αX < 1;
(iii) ΦX ∈ L1 (0, 1);
(iv) limλ→0 λΦX (λ) = 0
...

As a consequence of this result, we can show that if X satisfies any of the condition of the theorem, then X has a certain kind of self-improving property
...
This property is well known for weighted Lebesgue
spaces [6, 21], and also for Lorentz spaces [2, 5] and for variable Lp spaces [10]
...
For instance, in the case of weighted

A NDREI K
...
Theorem
1
...

Corollary 1
...
Let X(Rn ) be any quasi-Banach function space
...

The paper is organized as follows
...
2 to weighted Lebesgue and Lorentz spaces, and also to variable
Lp spaces
...
The remaining sections of
the paper provide the proofs of the results stated in Sections 1 and 2
...
2 and Corollary 1
...

2
...
1
...
Given a weight u, we denote by Lpu ,
p > 0, the space of all measurable f for which
Z
kf k

p
Lu

≡

p

Rn

|f (x)| u(x) dx

1/p

< ∞
...


Given a locally integrable function f on Rn , the Hardy-Littlewood maximal operator M is defined by
Mf (x) = sup
Q3x

1
|Q|

Z
Q

|f (y)| dy,

where the supremum is taken over all cubes Q containing x
...

p

Theorem 2
...
Let 1 < p < ∞
...


As we mentioned in the Introduction, u ∈ Ap,1 is equivalent to the condition
that M be of restricted weak type (p, p) with respect to u
...
233], Ap,1 ⊂ Ap+ε
for any ε > 0
...

Proposition 2
...
Let 1 < p < ∞
...

Observe that Ap 6≡ Ap,1 ; for example, u(x) = |x|n(p−1) satisfies Ap,1 but not
Ap
...
2 yields several characterizations of the Ap condition
...
2; however its proof is much simpler, without
the use of interpolation and the property Ap ⇒ Ap−ε
...

First of all, we calculate the generalized upper Boyd index of Lpu
...


Theorem 2
...
For any p > 0 we have
(2
...
2)

αLpu =

1
p

lim

λ→0

log(1/νu (λ))

...
We have the following application of Theorem 1
...

Theorem 2
...
Let 1 < p < ∞
...

(i) M is bounded on Lpu ;
(ii) limλ→0 νu (λ)/λp = +∞;
(iii) limλ→0 log(1/νu (λ))/ log(1/λ) < p ;
(iv) if ψ ∈ A, then for any cube Q and any subset E ⊂ Q,
(2
...


2702

A NDREI K
...

The proof of this theorem completely bypasses the Ap condition
...

However, it is interesting to stress the following corollary
...
5
...
The Ap condition is equivalent to any of the
conditions above
...
2
...
We also remark
that the proofs of both Theorems 2
...
4 use only basic properties of M such
as the weak type and the reverse weak type inequalities
...
3)
...
6
...
3) with
1−1/p
ψ(t) = log
(1 + t)
...
Let u and w be weights
defined on Rn and R+ respectively
...


A full characterization of the boundedness of M on Λpu (w) for arbitrary u
and w was obtained recently by Carro, Raposo, and Soria [5]; we also refer to [5]
for a complete account of related results in this area
...
On the other hand, the
case of Λp (w) (i
...
, when u = 1) was characterized by Ari˜no and Muckenhoupt
[2]; in this case w must satisfy the so-called Bp condition
...
In [5, Theorem 3
...
5], among others, the fol-

lowing characterization was obtained
...
7 ([5])
...
M is bounded on Λpu (w) if and only if
there exists q < p such that for some constant c and for every finite family of cubes
and sets (Qj , Ej )j with Ej ⊂ Qj
...
4)

S

j

|Qj |
|Ej |

!q

...
4) is equivalent to the same condition but with a unique Q and
E ⊂ Q
...
7 represents a generalized version of Proposition 2
...

We show that Theorems 2
...
4 and their proofs can be generalized with
minor changes to the spaces Λpu (w)
...

Theorem 2
...
For any p > 0 we have
ΦΛpu (w) (λ) 

(2
...
6)

1
p

lim

λ→0

log(1/νu,w (λ))

...
9
...
Given weights u and w , the following statements
are equivalent
...


2704

A NDREI K
...
4, item (iii) here is a reformulation of Theorem 2
...

2
...
Variable Lp spaces Let p : Rn → [1, ∞) be a measurable function
...

 λ 
Rn

The spaces Lp(·) (Rn ) are a special case of Musielak-Orlicz spaces (cf
...

The behavior of some classical operators in harmonic analysis on Lp(·) (Rn ) has
been intensively investigated in recent years
...
Given any measurable function p , let
p− = ess infx∈Rn p(x)

and

p+ = ess supx∈Rn p(x)
...
It has been proved by Diening [9] that if
p satisfies the following uniform continuity condition:
(2
...
After that, the second condition on p has been improved independently by Cruz-Uribe, Fiorenza,
and Neugebauer [8] and Nekvinda [23]
...
7)
and
|p(x) − p∞ | ≤

(2
...
In [23], the boundedness of M is deduced
from (2
...
8) : there exist
constants c , p∞ such that 0 < c < 1, p∞ > 1, and
Z

(2
...


We make several remarks about (2
...
First, since p is bounded, it is clear that
(2
...
Next, (2
...
Indeed, for any c1 one can take c2 < c1 and a constant k depending on c1 ,
1/x
1/x
≤ kxc1 for any x ≥ 0
...
9) is equivalent to saying
that there exist α, p∞ such that 0 < α < 1, p∞ > 1, and
c2 such that c2

Z

(2
...


It is easy to see that (2
...
10) with α < e−nc
...
2, we give a different approach to Nekvinda’s theorem
...

Theorem 2
...
Let p be a bounded positive function with p− > 0 satisfying
(2
...
10) for some α ∈ (0, 1) and p∞ > 0
...

Since (2
...
10 coincide with the conditions of Nekvinda’s theorem
...
10 clearly yields αLp(·) ≤ 1/p− < 1, and thus, by Theorem 1
...

We should mention that when proving Theorem 2
...

3
...
1
...


Recall that the local maximal operator mλ f is defined for any measurable
function f by
mλ f (x) = sup (f χQ )∗ (λ|Q|)

(0 < λ < 1),

Q3x

where the supremum is taken over all cubes Q containing x
...
1)

mλ f (x) > α ⇐⇒ Mχ{|f |>α} (x) > λ
...
2)

mλ (χE )(x) = χ{M(χ

E )>λ}

(x)
...
L ERNER & C ARLOS P E´ REZ

We will use the following simple properties of mλ :
(3
...
4)

(mλ f (x))δ = mλ (|f |δ )(x)

(δ > 0) ,

mλ (f + g)(x) ≤ mλ/2 f (x) + mλ/2 g(x)
...
41])
...
1
...
For any measurable function f ,
(3
...
6)

(x ∈ Rn , ξ <

m2n λξ f (x) ≤ mξ (mλ f )(x)

1
)
...
By (3
...
Therefore, setting α = ((mλ f )χQ )∗ (t)
we obtain
(3
...


In particular, when Q = Rn , (3
...
5)
...
7) t = ξ|Q|, we immediately get (3
...

Similarly, by (3
...
5)
...
2
...
Let M0 be the set of all real-valued measurable functions on Rn
...
e
...
e
...
e
...


Lorentz-Shimogaki Theorem

2707

We will essentially use a version of the Aoki-Rolewicz theorem (see [1, 24] or
[15, p
...
, fk one has
(3
...

We recall that two functions f and g from M0 are said to be equimeasurable
if they have the same distribution function
...

3
...
The upper Boyd index
...
149])
...

We briefly recall how the upper Boyd index is defined in [3, 4]
...
62] says that for any r-i Banach function
space X over Rn there is a r-i Banach function space X¯ over (0, ∞) such that
kf kX = kf ∗ kX¯
...


kϕkX¯ ≤1

Finally, the upper Boyd index α¯ X of X is defined by
(3
...

1t→∞
log t
log t

¯ X = inf
α

Observe that the function hX (t) can be defined more naturally, without the use of
the space X¯
...
Indeed, given any ϕ on (0, ∞), one can consider the function Aϕ (x) =
ϕ∗ (vn |x|n ) on Rn , where vn is the volume of the unit ball
...
Also, we use
that (Da f )∗ (t) = f ∗ (an t)
...


On the other hand,

kD(1/t)1/n f kX = kE1/t f ∗ kX¯
...
L ERNER & C ARLOS P E´ REZ

2708

From the last two identities we easily have that
hX (t) = sup kD(1/t)1/n f kX

(3
...


kf kX ≤1

Consider now the case of the quasi-Banach r-i space X
...

Given any quasi-Banach r-i space X(Rn ), we define its upper Boyd index α¯ X by
equality (3
...
10)
...
4
...
Recall that the generalized upper index
given in Definition 1
...


kf kX ≤1

In this section we show that this function is essentially equivalent to a submultiplicative function
...
2
...
A non-negative function ϕ on
E is said to be submultiplicative if
ϕ(ξλ) ≤ ϕ(ξ)ϕ(λ)

(λ, ξ ∈ E)
...
2 ([3, p
...
Let ψ be any non-decreasing submultiplicative
function on [1, ∞) with ψ(1) = 1
...

t>1
log t
log t

Proposition 3
...
Let ϕ be any non-increasing submultiplicative function on
(0, 1] with ϕ(1) = 1
...

log(1/λ) λ<1 log(1/λ)

Lorentz-Shimogaki Theorem

2709

The first equivalence follows from the previous proposition, and the second
one is trivial
...

This is enough to give meaning to the limit in Definition 1
...
3, the limit defining αX exists:
αX = lim

λ→0

log ΦX (λ)

...
4
...
If ΦX (λ0 ) < ∞,
for some λ0 ∈ (0, 1/4n ] then there is a non-increasing, submultiplicative on (0, 1]
function Φ˜ X such that Φ˜ X (1) = 1, and
(3
...

Proof
...
6) that
km2n λξ f kX ≤ kmξ (mλ f )kX ≤ ΦX (ξ)kmλ f kX ≤ ΦX (ξ)ΦX (λ)kf kX ,

and thus,
(3
...

2n

(0 < λ ≤ 1)
...
Next, Φ˜ X is
non-increasing because ΦX is so
...
11) holds trivially with c1 = 1/ΦX (1−)
...
12) that
ΦX (ξλ)
ΦX (ξ/2n )
≤
Φ(λ) ≤ ΦX (1/4n )Φ(λ),
ΦX (ξ)
ΦX (ξ)

❐

which proves the right-hand inequality in (3
...
Observe
that c2 is finite since ΦX (λ0 ) < ∞, 0 < λ0 ≤ 1/4n
...
P ROOF OF THE M AIN R ESULTS
Denote M 2 f = MMf
...


A NDREI K
...
1
...
1)

c
M 2 f (x)
λ log(1/λ)

Proof
...


f ∗ (τ) dτ
...
122], we get
(f χQ )∗ (λ|Q|) ≤
≤
≤

1

Z λ|Q|

λ|Q| 0

(f χQ )∗ (τ) dτ

1
λ log(1/λ)|Q|
1
λ log(1/λ)|Q|

Z λ|Q|
0

Z |Q|
0

(f χQ )∗ (τ) log

|Q|
dτ
τ

(f χQ )∗∗ (τ) dτ

Z |Q|
c
≤
(M(f χQ ))∗ (τ) dτ
...
1)
...
2
...
4 combined with Propositions 3
...
We will show that (i) ⇒ (iv) and
(ii) ⇒ (i)
...
Thus, by Lemma 4
...

λ log(1/λ)

Lorentz-Shimogaki Theorem

2711

Therefore limλ→0 λΦX (λ) = 0, which proves (i) ⇒ (iv)
...
This means that there are constants c > 0 and
δ < 1 such that for any f ,
kmλ f kX ≤ cλ−δ kf kX
...
2)

We next observe that for any cube Q,
1

Z

|Q|

and hence,
Mf (x) ≤

Q

|f | =

Z1
0

Z1
0

(f χQ )∗ (λ|Q|) dλ,

mλ f (x) dλ ≤

∞
X

2−i m2−i f (x)
...
8) along with (4
...


i=1

This completes the proof of (ii) ⇒ (i)
...
Consider the
spherically symmetric rearrangement of f defined by
f ? (x) = f ∗ (vn |x|n ),

where vn is the volume of the unit ball
...
It follows from (3
...


Therefore,
kD(2n λ)1/n f kX ≤ kmλ f kX ≤ kD(λ/3n )1/n f kX

and
hX

 1 
 3n 
≤ ΦX (λ) ≤ hX

...
2) and (3
...

αX = α

A NDREI K
...
3
...

Xr
X

From (3
...


❐

By Theorem 1
...
Applying Theorem
1
...

5
...
1
...
In order to prove Theorem 2
...

Lemma 5
...
Let p > 0
...
1)

u{x : MχE(x) > λ}
ΦLpu (λ) = sup
u(E)
E

!1/p
(0 < λ < 1),

where the supremum is taken over all measurable sets E with 0 < u(E) < ∞
...
Denote the function on the right-hand side of (5
...
It
follows from the definition of ΦLpu and from (3
...


Therefore, taking the supremum over all such E , we obtain
(5
...


On the other hand, by (3
...


Multiplying this inequality by pαp−1 and then integrating with respect to α ∈
(0, ∞), we get
Thus, ΦLpu (λ) ≤ ψp (λ), which, along with (5
...
1)
...


Lorentz-Shimogaki Theorem

2713

Lemma 5
...
Let ϕ be any non-increasing positive function on (0, 1)
...

(i) There is a positive constant c such that for any measurable set E ,
u{x : MχE (x) > λ} ≤ cϕ(λ)u(E)

(5
...
4)

ϕ(|E|/|Q|)

u(E)

...
In the particular case of ϕ(t) = t p , this lemma was proved in [16]
...
We briefly outline the
details
...
3) holds
...
Then, setting in (5
...
4)
...
4)
...
e
...
Next, it follows from
(5
...
Since u is doubling, Mu is of weak
type (1, 1) with respect to u, and hence,
(
u{x : M(χE ) > λ} ≤ u x : Mu (χE )(x) >

1
cϕ(λ)

)
≤ c 0 ϕ(λ)u(E),

❐

proving (5
...

Proof of Theorem 2
...
We have to prove that
ΦLpu (λ) 

1
1/p

νu (λ)

,

where we recall that
νu (λ) = inf
Q

inf

E⊂Q:|E|=λ|Q|

u(E)
u(Q)

(0 < λ < 1)
...
1 and 5
...
L ERNER & C ARLOS P E´ REZ

2714
and therefore,

1

(5
...


On the other hand, by the definition of νu ,
νu

|E|
|Q|

!
≤

u(E)
u(Q)

(E ⊂ Q)
...
Hence, by Lemma 5
...


From this and from Lemma 5
...
5), yields (2
...
Next, from (2
...
2)
...
4
...
2 and 2
...
To prove that (i)–(iii) are equivalent to (iv), we show that
(iv) ⇒ (ii) and (iii) ⇒ (iv)
...
3) that λψ(1/λ) ≤ cνu1/p (λ) or, in other words,
ψ(1/λ)p ≤ c

νu (λ)

...

It follows from (iii) that there is δ < 1 such that λδ ≤ cνu1/p (λ)
...


Since ψ(t) = O(t ε ) for all ε > 0, we have

which completes the proof
...
6
...
The
first one is that the Ap condition is trivially equivalent (see, e
...
, [27, p
...
6)

|Q|

Q

|f | dx ≤ c

!1/p

Z

1

p

u(Q)

Q

|f | u dx


...
6))
...
g
...
175] for the unweighted case; the proof readily works for any doubling
weight)
Z

(5
...

By (5
...


Therefore, setting in (5
...
7), we obtain
(5
...
On the other hand, it is easy to see that
B −1 (B(t)p ) ∼ t p log(1 + 1/t)p−1

(0 < t < 1),

and therefore (5
...
3) with ψ(t) = log1−1/p (1 + t)
...
L ERNER & C ARLOS P E´ REZ

2716

The proofs of Theorems 2
...
9 are almost identical to the proofs of Theorems
2
...
4
...
8, we will need two lemmas similar to
Lemmas 5
...
2
...
Then we give a brief proof of
Theorem 2
...
We omit the proof of Theorem 2
...
8 exactly in the same way as Theorem 2
...
3
...


Lemma 5
...
Let p > 0
...
9)

Φ

p
Λu (w)

W (u{x : MχE (x) > λ})
sup
W (u(E))
E

(λ) =

!1/p
(0 < λ < 1),

where the supremum is taken over all measurable sets E with 0 < u(E) < ∞
...
Denote the function on the right-hand side of (5
...
Observe that (χE )∗u (t) = χ(0,u(E)) (t)
...
2) we easily obtain
that
ψp (λ) ≤ ΦΛpu (w) (λ)
...
1),
(5
...


Since

Z
W (u{x : |f (x)| > α}) =

{t :fu∗ (t)>α}

w(t) dt,

we obtain from (5
...
9)
...
4
...
The following statements are equivalent
...
11)

W (u{x : MχE (x) > λ}) ≤ cϕ(λ)W (u(E))

(0 < λ < 1);

(ii) there is a positive constant c such that for any finite family of cubes {Qj } and
any family of sets {Ej } of positive measure with Ej ⊂ Qj ,
(5
...


Lorentz-Shimogaki Theorem

2717

Proof
...
2,
since it follows directly from the definitions
...
Let Ej ⊂
S Qj , j = 1,
...
Denote E = j Ej
and λ = minj |Ej |/|Qj |
...
From this and
from (5
...
12)
...
Hence, by (5
...


❐

From this, by a limiting argument, we get (5
...

Proof of Theorem 2
...
Recall that νu,w is defined by
νu,w (λ) = inf

W u

inf

W u

{Qj } {Ej }:Ej ⊂Qj , minj |Ej |/|Qj |=λ

S
S





j

Ej

j

Qj



,

where the infimum is taken over all finite families of cubes {Qj } and over all
families of sets {Ej } such that Ej ⊂ Qj with minj |Ej |/|Qj | = λ
...
3 and 5
...
13)

p

ΦΛp (w) (λ)

≤ cνu,w (λ)
...


A NDREI K
...
3 and 5
...
13),
which proves the theorem
...
2
...

Lemma 5
...
Let r , q and ϕ be non-negative functions such that r− > 0 and
0 ≤ ϕ ≤ 1
...


Proof
...


Then
r (x)

ϕ(x)

r (x)

= ϕ(x)

χE2 +

≤ α1/(q(x)−r (x)) χE2

1

!q(x)−r (x)

ϕ(x)q(x) χE3 + ϕ(x)r (x) χE1c
ϕ(x)
 1/r (x)
1
+
ϕ(x)q(x) χE3 + ϕ(x)q(x) χE1c ,
α

❐

proving the lemma
...
10
...
14)

Rn


p(x)
λ1/p− mλ f (x)
dx ≤ c

Z

whenever

Rn

|f (x)|p(x) dx ≤ 1
...


Let us show that for any x ,
(5
...
16)

p(·)

(λ1/p− mλ/2 f1 (x))p(x) ≤ cλmλ/2 (f1 )(x) ,


p(·)
(λ1/p− mλ/2 f2 (x))p(x) ≤ c ψ(x) + λmλ/4 (f2 )(x) ,

with ψ ∈ L1 , where c and kψkL1 depend only on p and n
...
15) and (5
...
14)
...
5) implies
(5
...
17) shows that the L1 -norms of the right-

hand sides of (5
...
16) are bounded by the constants depending only on
p and n
...
4),


(λ1/p− mλ f (x))p(x) ≤ 2p+ (λ1/p− mλ/2 f1 (x))p(x) + (λ1/p− mλ/2 f2 (x))p(x) ,

we have that (5
...
16) imply (5
...

To prove (5
...
We claim that



|Q| p(x)−p− (Q)
1/p−
∗
F (Q, x) ≡ λ
(f1 χQ ) λ
≤ c,

(5
...
Indeed, by Chebyshev’s inequality,
λ

1/p−

∗



(f1 χQ )

λ

|Q|

2


≤
≤
≤

2
|Q|

2
|Q|

2

!1/p−
kf1 kp−
!1/p−  Z
Rn

!1/p−

|Q|

|f1 (x)|

p(x)

1/p−

dx


...
18)
...
7)
yields
F (Q, x) ≤

2

!c/p− log(1/ diam Q)

|Q|

√
≤ (2e n)nc/p− log 2 ,

proving (5
...
Now, it follows from (5
...
3) that







|Q| p(x)
|Q| p− (Q)
≤ c λ1/p− (f1 χQ )∗ λ
λ1/p− (f1 χQ )∗ λ

2

2

= cλ

p− (Q)/p−

p− (Q)

(f1

p(·)

≤ cλmλ/2 (f1

proving (5
...


∗

χQ )

)(x),


λ

|Q|

2



2720

A NDREI K
...
16), we apply Lemma 5
...
3) and (3
...
This proves (5
...
17) and (2
...

Thus, we have proved (5
...
16) which proves the theorem
...


Acknowledgments We are very grateful to Michael Cwikel for his useful comments and remarks
...
K
...
Both authors are
also supported by the same institution under research grant MTM2006-05622
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E- MAIL: carlosperez@us
...

2000 M ATHEMATICS S UBJECT C LASSIFICATION: 42B20, 42B25, 42B35
...

Article electronically published on November 13th, 2007

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Title: Functions the basics
Description: 28 pages of Algebra notes about functions. Definitions, theorems with proofs and examples

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