Abstract We study weak-type modular inequalities for the Hardy operator restrictedto non-increasing functions on weighted Lp(·) spaces, where p(·) is a variable ex-ponent. These new estimates complete the results of Boza and Soria (J. Math. Anal.Appl. 348:383–388, 2008) where we showed some necessary and sufficient condi-tions on the exponent p(·) and on the weights to obtain weighted modular inequal-ities with variable exponents. For this purpose, we introduced the class of weightsBp(·). We prove that, for exponents p(x) > 1, this is also the class of weights forwhich the weak modular inequality holds, and a characterization is also given in thecase p(x) ≤ 1. Finally, we compare our theory with the results in Neugebauer (Stud.Math. 192(1):51–60, 2009), giving examples for very concrete and simple exponentswhich show that inequalities in norm hold true in a very general context.
In this way, for exponents p(x) ≥ 1, we obtain a Banach space (see [12] for furtherproperties).
In recent years, spaces of variable integrability have received a lot of attention,not only from a theoretical point of view but also with motivations to study its ap-plications. In particular, properties about the behaviour of any interesting operator inharmonic analysis can be also considered in the variable exponent setting. For ex-ample, in relation with the Hardy-Littlewood maximal operator M , it is shown in [8]that for w ≡ 1
‖Mf ‖p(x) ≤ C‖f ‖p(x),
provided that the exponent p(x) verifies some regularity properties and 1 < p(x) <
∞ (for more details, see [8] and also [9]). A characterization of the weights, with thesame restrictions on p(x), so that
‖Mf ‖p(x),w ≤ C‖f ‖p(x),w,
has been studied in [10], where the class Ap(·) of Muckenhoupt weights, in the con-text of variable exponents, is introduced.
Concerning the Hardy operator
(Sf )(x) = 1
x
∫ x
0f (t) dt, x > 0,
and other related integral operators, we have to mention the work [11], where theauthors study boundedness in norm in variable Lp spaces with weights, and [1],where weighted weak type inequalities with variable exponents are considered. In[14], Hardy’s inequalities are studied in this context for weights of power-type.
In [3], and motivated by the study of estimates on rearrangement invariant spaces,we were interested in characterizing the set of weights w defined in (0,+∞) forwhich a modular inequality holds for the Hardy operator restricted to non-increasingfunctions; that is,
∫ +∞
0
(Sf (x)
)p(x)w(x)dx ≤ C
∫ +∞
0
(f (x)
)p(x)w(x)dx, (2)
for some positive constant C and f decreasing.
Weighted weak modular and norm inequalities for the Hardy 461
If p is a constant exponent, the theory of Ariño and Muckenhoupt (see [2]) provesthat (2) is equivalent to the so called Bp-condition for the weight w; that is, theexistence of some constant C > 0 such that for any r > 0,
∫ +∞
r
(r
x
)p
w(x)dx ≤ C
∫ r
0w(x)dx.
For other related results concerning the class of Bp weights, see [6, 7, 17].By testing inequality (2) on characteristic functions of the form fs,r (x) =
sχ(0,r)(x), s, r > 0 we obtain, analogously to the constant case, the so called Bp(·)condition, namely
∫ +∞
r
(sr
x
)p(x)
w(x)dx ≤ C
∫ r
0sp(x)w(x)dx. (3)
This definition for the Bp(·) class of weights in the variable exponent setting shouldbe compared with the one obtained by Neugebauer in his recent paper [16] wherethe parameter s > 0 is omitted. Note that, in order to get a necessary condition toensure inequality (2), this parameter is needed due to the lack of homogeneity of thiscondition.
The main result in [3] shows that, for exponents p(·) whose oscillation around theorigin tends to zero (see definitions below), the existence of weights satisfying (2) im-plies that p(·) is essentially constant and hence these weights belong to the classicalBp class. Nevertheless, we also showed that (2) holds for weights w satisfying a Bp(·)condition corresponding to non-constant exponents p(x), whose oscillation aroundthe origin is maximum. That is, for (2) to hold, either p is constant or extremelyoscillatory near the origin.
This modular inequality, for general functions f , was considered in [18], wherethe author proved (see Theorem 2.2) that the only possibility for (2) to hold is that,essentially, p has to be a constant function. Also, we should mention here the resultdue to Lerner in [13] who proved that a modular inequality without weights holdsfor the Hardy-Littlewood maximal operator or Calderón-Zygmund singular integrals,if and only the exponent p is equal to a constant almost everywhere. In contrast tothese facts, in the decreasing case, as shown in [3], there are non-constant exponentssatisfying (2).
In Sect. 2, we shall deal with modular weak type inequalities. Our main goal isto study conditions on the exponent p(·) and on the weight w to ensure that theinequality
supr>0
∫ r
0
(Sf (r)
)p(x)w(x)dx ≤ C
∫ +∞
0
(f (x)
)p(x)w(x)dx (4)
holds for every non-increasing function defined on [0,∞).When p(·) is constant, inequality (4) takes the form
supr>0
(Sf (r)
)p∫ r
0w(x)dx ≤ C
∫ +∞
0
(f (x)
)pw(x)dx, (5)
462 S. Boza, J. Soria
which is equivalent to S : Lp
dec(w) −→ Lp,∞(w). It was proved in [15] that the classof weights for which such an inequality holds is exactly Bp in the case 1 < p < ∞.In the case 0 < p ≤ 1 (see [4]), the class of weights that characterizes (5) are thosesatisfying the Rp condition: there exists a constant C > 0 such that for 0 < r < t <
+∞,
1
tp
∫ t
0w(x)dx ≤ C
1
rp
∫ r
0w(x)dx. (6)
In the variable setting, a two-weighted weak type inequality of the form
∫{x∈A: |Tf (x)|>λ}
u(x)dx ≤∫
A
(K |f (x)|
λ
)p(x)
v(x) dx (7)
was studied in [1], where T is a Hardy type operator, A a subset of the real line,K > 0 and f any arbitrary measurable function. Note the difference between (7) and(4), specially the role played by the constants K and C, respectively, due to the lackof homogeneity, although they both agree for constant exponents. Also in [8], similarinequalities are studied in the non-weighted case for the Hardy-Littlewood maximaloperator.
C.J. Neugebauer, in [16], considers weights in a similar Bp(·) class, when theparameter s is omitted in condition (3). His main result shows that if 1 ≤ p(x) ≤supp(x) < ∞ and the exponent p(x) is non-decreasing, w belongs to this new Bp(·)class if and only if inequality (2) holds for f non-increasing satisfying f (0+) ≤ 1.Moreover, these two conditions are shown to be equivalent to the following inequalityinvolving the Lp(x) norm: for each 0 < γ ≤ 1, there exists 1 ≤ cγ < ∞ such that
‖Sf ‖p(x),σw ≤ cγ ‖f ‖p(x),σw,
for every f non-increasing with f (0+) ≤ 1, and every 0 < σ < ∞, for which‖f ‖p(x),σw ≥ γ. However, in Sect. 3, we prove that modular and norm inequalitiesare not equivalent in general since we will show, in the case of two explicit examplesof increasing and decreasing exponents, that we can characterize those weights forwhich a norm inequality holds for any non-increasing f . But, as a consequence ofthe results obtained in [3], for the same exponents there are no weights for which (2)is true.
For a given variable exponent p : R+ −→ R
+ and a weight w in (0,+∞), we willdenote by p−
w and p+w , respectively, the essential infimum and supremum of p on the
support of w. For the infimum and supremum of p(·) in (0,+∞), we use p− andp+, respectively.
The symbol f g will indicate the existence of a universal positive constant C
(independent of all parameters involved) so that (1/C)f ≤ g ≤ Cf . Also f � g willdenote the existence of a constant C such that f ≤ Cg. Whenever C is explicitlywritten, its value may change from one occurrence to the next.
Weighted weak modular and norm inequalities for the Hardy 463
2 Modular weak type inequalities
Definition 2.1 Given p : R+ −→ R
+ such that 0 < p− ≤ p+ < +∞ and a weightw in (0,+∞), let us define the following local oscillation of p:
ϕp(·),w(δ) = supx∈(0,δ)∩suppw
p(x) − infx∈(0,δ)∩suppw
p(x).
We observe that ϕp(·),w is an increasing and positive function such that
limδ→∞ϕp(·),w(δ) = p+
w − p−w. (8)
Theorem 2.2 Let w be a weight on (0,∞) and p : R+ −→ R
+ such that 0 < p− ≤p+ ≤ 1, and assume that ϕp(·),w(0+) = 0. The following facts are equivalent:
(a) There exists a positive constant C such that, for any positive and nonincreasingfunction f :
supr>0
∫ r
0(Sf (r))p(x)w(x)dx ≤ C
∫ +∞
0(f (x))p(x)w(x)dx.
(b) For any 0 < t < r , s > 0
∫ r
0
(st
r
)p(x)
w(x)dx ≤ C
∫ t
0sp(x)w(x)dx. (9)
(c) p|suppw ≡ p0 a.e. and w ∈ Rp0 .
Proof To check that condition (a) implies (b) it is enough to test the weak modularinequality (4) for the functions fs,t (x) = sχ(0,t)(x), s, t > 0, for which Sfs,t (x) =sχ(0,t)(x) + st
xχ[t,∞)(x). Hence, if r ≥ t and s > 0, (4) gives
∫ t
0sp(x) w(x) dx +
∫ r
t
(st
r
)p(x)
w(x)dx ≤ C
∫ t
0sp(x)w(x)dx,
which is equivalent to (9).To obtain (c) from (b), we are going to prove that, as in the Bp(·) case, condition
(b) implies that ϕp(·),w is a constant function, namely ϕp(·),w ≡ p+w − p−
w . This factand the hypothesis on ϕp(·),w imply ϕp(·),w ≡ 0, and hence, due to (8), p|suppw ≡p+
w = p−w = p0 a.e. and, finally, (9) means w ∈ Rp0 .
Let us suppose that ϕp(·),w is not constant a.e. Then, one of the following twoconditions must hold:
(i) there exists δ > 0 such that
α := supx∈(0,δ)∩suppw
p(x) < p+w < +∞, (10)
and hence, there exists ε > 0 such that
|{x ≥ δ : p(x) ≥ α + ε} ∩ suppw| > 0,
464 S. Boza, J. Soria
or(ii) there exists δ > 0 such that
β := infx∈(0,δ)∩suppw
p(x) > p−w > 0, (11)
and then, for some ε > 0,
|{x ≥ δ : p(x) ≤ β − ε} ∩ suppw| > 0.
In case (i), we observe that condition (b), for t = δ, implies that
supr≥δ
∫ r
0
(sδ
r
)p(x)
w(x)dx ≤ C
∫ δ
0sp(x)w(x)dx.
And then, using (10), we obtain that there exists N > δ such that, for s > 1 andr = N ,
sα+ε
∫{x∈[δ,N ]:p(x)≥α+ε}
(δ
N
)p(x)
w(x)dx
≤∫
{x∈[δ,N ]:p(x)≥α+ε}
(sδ
N
)p(x)
w(x)dx
≤ C
∫ δ
0sp(x)w(x)dx ≤ Csα
∫ δ
0w(x)dx,
which is clearly a contradiction if we let s ↑ +∞.Similarly, in case (ii), let us consider the same condition (b) for t = δ, and now
take s < 1. Taking into account (11), there exists M > δ such that:
sβ−ε
∫{x∈[δ,M]: p(x)≤β−ε}
(δ
M
)p(x)
w(x)dx ≤ Csβ
∫ δ
0w(x)dx,
which is a contradiction if we let s ↓ 0.Finally, the fact that condition (c) implies (a) is a consequence of [4, Theo-
rem 3.3]. �
Remark 2.3 Looking at the proof of the previous theorem, we observe that condi-tion ϕp(·),w(0+) = 0 implies, with no restriction on the exponent, that if w verifiescondition (9), the exponent p(·) must be a constant function. However, this is not anecessary condition for (4) to be true.
Indeed, let us consider the following exponent p(x) restricted to the interval (0,1]:
p(x) =⎧⎨⎩
p+, for x ∈ A := ⋃∞k=0(
122k+1 , 1
22k ],p−, for x ∈ B := ⋃∞
k=1(1
22k , 122k−1 ],
and the weight w(x) = χ(0,1)(x). We observe that, in this case, the function p(x)
oscillates on each neighbourhood of zero between its maximum and minimum and,
Weighted weak modular and norm inequalities for the Hardy 465
as a consequence, the corresponding function ϕp(·),w becomes a non zero constantfunction. As shown in [3, p. 386] if 1 < p− < p+ < +∞ the modular inequality (2)holds and, then, also (4).
It can also be shown that (4) holds for the oscillatory exponent p defined abovewith 1 = p− < p+ < +∞ and w(x) = χ(0,1)(x). In order to show this, we writef = fA + fB := f χA + f χB and obtain
∫ r
0
(Sf (r)
)p(x)dx �
∫A∩[0,r)
(SfA(r)
)p+dx +
∫A∩[0,r)
(SfB(r)
)p+dx
+∫
B∩[0,r)
(SfA)(r) dx +∫
B∩[0,r)
(SfB)(r) dx
:= (I) + (II) + (III) + (IV).
To estimate (I) and (IV), we use Hölder’s inequality and obtain, for 0 < r < 1,
(I) ≤ r(SfA(r))p+ = 1
rp+−1
(∫ r
0fA(x)dx
)p+
≤∫ 1
0(fA(x))p
+dx,
and
(IV) ≤ r(SfB)(r) =∫ r
0fB(x)dx ≤
∫ 1
0fB(x)dx.
To estimate (II) and (III), we proceed as above,
(II) ≤ r(SfB(r))p+ ≤
∫ 1
0(fB(x))p
+dx ≤ 2
∫ 1
0(fA(x))p
+dx,
where the last inequality follows from the fact that f is decreasing (see [3, Exam-ple 2.4]) and, similarly,
(III) ≤ r(SfA(r)) ≤∫ 1
0fA(x)dx ≤ 2
∫ 1
0fB(x)dx.
Collecting terms we finally obtain
sup0<r<1
∫ r
0(Sf (r))p(x) dx ≤ C
∫ 1
0(f (x))p(x) dx.
Theorem 2.2 solves the characterization of weights for which a weak modularinequality holds for the Hardy operator restricted to non increasing functions for ex-ponents p(·) such that p+ ≤ 1 and vanishing oscillation at zero. Weights satisfyinginequality (4) are defined to be in the Wp(·) class and those satisfying condition (9)are defined to be in Rp(·). To clarify what happens in the case p− > 1, the followinglemmae are needed. The corresponding proofs in the case p(·) a constant exponentcan be found in [15].
466 S. Boza, J. Soria
Lemma 2.4 Let p : R+ −→ R
+ be an exponent such that 0 < p− ≤ p+ < +∞ andw a weight in R
+. If there exist constants a > 1 and 0 < C < p−/p+, dependingon a, such that ∫ at
0 ( sa)p(x)w(x)dx∫ t
0 sp(x)w(x)dx≤ C
for all s, t > 0, then w ∈ Bp(·).
Proof For 0 < r < +∞, let us write, using Fubini’s theorem,
L ≡∫ ∞
r
∫ at
0
(s
a
)p(x) 1
tp(x)+1w(x)dx dt
=∫ ar
0
1
p(x)
(s
a
)p(x) 1
rp(x)w(x)dx +
∫ ∞
ar
1
p(x)
(s
x
)p(x)
w(x)dx.
Then, this equality and the hypothesis imply that
1
p+
∫ ∞
ar
(s
x
)p(x)
w(x)dx ≤ L ≤ C
∫ ∞
r
∫ t
0sp(x) 1
tp(x)+1w(x)dx dt.
Again, using the conditions on p(·) and since a > 1,∫ ∞
r
∫ t
0sp(x) 1
tp(x)+1w(x)dx dt
=∫ r
0
∫ ∞
r
dt
tp(x)+1sp(x)w(x)dx +
∫ ∞
r
∫ ∞
x
dt
tp(x)+1sp(x)w(x)dx
≤ 1
p−
[∫ r
0
(s
r
)p(x)
w(x) dx +∫ ∞
r
(s
x
)p(x)
w(x)dx
]
≤ 1
p−
[∫ ar
0
(s
r
)p(x)
w(x)dx +∫ ∞
ar
(s
x
)p(x)
w(x)dx
].
Combining the last two estimates, we obtain
(1
p+ − C
p−
)∫ ∞
ar
(s
x
)p(x)
w(x)dx ≤ C
p−
∫ ar
0
(s
r
)p(x)
w(x)dx,
or ∫ ∞
ar
(s
x
)p(x)
w(x)dx ≤ C
p− − Cp+
∫ ar
0
(s
r
)p(x)
w(x)dx.
Changing the variables t := ar and σ := s/t , we obtain the inequality
∫ ∞
t
(σ t
x
)p(x)
w(x)dx ≤ Cap+
p− − Cp+
∫ t
0σp(x)w(x)dx,
which is the Bp(·) condition. �
Weighted weak modular and norm inequalities for the Hardy 467
Lemma 2.5 If w ∈ Rp(·) then, for all a > 1 and t > 0,
∫ at
t
(st
x
)p(x)
w(x)dx ≤ C(1 + loga)
∫ t
0sp(x)w(x)dx.
Proof Since w ∈ Rp(·), we have that for a, r > 1 and s, t > 0,
∫ a
1
∫ rt
0
(s
r
)p(x)
w(x)dxdr
r≤ C
∫ a
1
∫ t
0sp(x)w(x)dx
dr
r
= C loga
∫ t
0sp(x)w(x)dx.
Interchanging the order of integration, we obtain that the integral on the left is equalto
∫ at
t
sp(x)
p(x)w(x)
[(t
x
)p(x)
−(
1
a
)p(x)]dx.
Hence, since w ∈ Rp(·) we can easily obtain
1
p+
∫ at
t
(st
x
)p(x)
w(x)dx ≤ C loga
∫ t
0sp(x)w(x)dx + 1
p−
∫ at
t
(s
a
)p(x)
w(x)dx
≤ C(1 + loga)
∫ t
0sp(x)w(x)dx. �
The following theorem shows that weights verifying the weak type inequality (4)for an exponent p(·) must satisfy a Bp(·) condition (see [15, Theorem 7.2] for theproof in the constant case).
Theorem 2.6 Let w be a weight belonging to Wp(·) and p(·) an exponent such thatp− > 1. Then w ∈ Bp(·).
Proof Using Lemma 2.4 above it is enough to prove that there exist constants a > 1and C < p−/p+ such that
∫ at
0 ( sa)p(x)w(x)dx∫ t
0 sp(x)w(x)dx≤ C, (12)
for all s, t > 0. Indeed, let us fix s, t > 0 and a > 1, and consider
f (x) =
⎧⎪⎨⎪⎩
s for 0 < x < t,stx
for t ≤ x ≤ at,
0 for x > at ,
468 S. Boza, J. Soria
whose maximal function is
(Sf )(x) =
⎧⎪⎪⎨⎪⎪⎩
s for 0 < x < t,
1x(st + st log(x/t)) for t ≤ x ≤ at,
1x(st + st loga) for x > at.
Since w ∈ Wp(·), taking into account Lemma 2.5,
∫ at
0
(s
a(1 + loga)
)p(x)
w(x)dx
=∫ at
0
(Sf (at)
)p(x)w(x)dx
≤ C
∫ ∞
0
(f (x)
)p(x)w(x)dx
= C
∫ t
0sp(x)w(x)dx + C
∫ at
t
(st
x
)p(x)
w(x)dx
≤ C(1 + loga)
∫ t
0sp(x)w(x)dx.
This inequality implies that
∫ at
0
(s
a
)p(x)
w(x)dx ≤ C(1 + loga)1−p−∫ t
0sp(x)w(x)dx.
Taking a > 1 small enough, the last inequality gives (12) for an appropriate con-stant C. �
As a consequence of the previous theorem and [3, Theorem 2.1], we obtain thefollowing corollary.
Corollary 2.7 Let w be a weight in the class Wp(·), and p(·) and exponent verifyingp− > 1 and ϕp(·)(0+) = 0. Then p(·) ≡ p0 and w ∈ Bp0 .
Remark 2.8 This last corollary together with Remark 2.3 lead us to conclude that,for exponents with vanishing oscillation at zero and weights w ∈ Wp(·), necessarilyp(·) must be identically constant p0. Then, if p0 ≤ 1, this condition is equivalent tow ∈ Rp0 and for p0 > 1 it is equivalent to w ∈ Bp0 .
The following proposition shows a connection between the Rp(·) and the Bp(·)class of weights:
Proposition 2.9 If w ∈ Rp(·) then, for δ > 0, w ∈ Bp(·)+δ .
Weighted weak modular and norm inequalities for the Hardy 469
Proof It suffices to prove it for 0 < δ < 1. The Rp(·) condition implies that, for r < 1and s, t > 0,
∫ t/r
0(sr)p(x)w(x)dx ≤ C
∫ t
0sp(x)w(x)dx.
Multiply both members of the inequality by (sr)−ε , 0 < ε = 1 − δ < 1, and integratein 0 < r < 1 to obtain
∫ 1
0
∫ t/r
0(sr)p(x)−εw(x)dx dr ≤ Cε
∫ t
0sp(x)−εw(x)dx.
Applying Fubini’s theorem, we get that the integral of the left hand side is greaterthan or equal to
∫ ∞
t
∫ t/x
0(sr)p(x)−εw(x)dr dx ≥ 1
p+ + 1 − ε
∫ ∞
t
sp(x)−ε
(t
x
)p(x)+1−ε
w(x)dx.
Collecting terms, we obtain
∫ ∞
t
(st
x
)p(x)+1−ε
w(x)dx ≤ Cp(·),ε∫ t
0sp(x)+1−εw(x)dx,
which means that w ∈ Bp(·)+1−ε = Bp(·)+δ . �
For completeness, we now formulate the following result which says that the con-verse of Lemma 2.4 is true.
Theorem 2.10 Let w be a locally integrable function. Then w ∈ Bp(·) if and only if,for any given ε ∈ (0,1), there exists aε > 1 such that, for every t > 0,
∫ at
0
(s
a
)p(x)
w(x)dx ≤ ε
∫ t
0sp(x)w(x)dx.
Proof Due to Lemma 2.4 we just need to prove the necessity of the condition. In [3,Proposition 2.5] it is proved that there exists η > 0 such that w ∈ Bp(·)−η and hence,since also w ∈ Rp(·)−η, for any a > 1 we get that
∫ at
0 ( sa)p(x)w(x)dx∫ t
0 sp(x)w(x)dx=
∫ at
0 ( sa)p(x)−ηw(x)dx∫ t
0 sp(x)−ηw(x)dx
(1
a
)η
≤ C
(1
a
)η
= ε.�
3 Weighted norm inequalities
As it was mentioned in the introduction, the work of Neugebauer [16] deals with thequestion of whether modular and norm inequalities are both equivalent to a certainBp(·) condition. His results are, in some aspects, partial since only bounded functions
470 S. Boza, J. Soria
are considered and also the exponents p(x) for which the results are obtained mustbe increasing.
In this section we show that, with no restriction on the functions, other than mono-tonicity, and for very simple exponents (e.g., a two step function which can be in-creasing or decreasing), a characterization of those weights for which of a norm in-equality holds can be obtained for the Hardy operator. This result tells us that a weightverifying a norm inequality for this exponent must also satisfy a local Bp condition.This characterization should be compared with the results obtained in [3], which im-ply that for these particular exponents there are no weights for which a modular in-equality holds. We are going to formulate the proposition for the decreasing exponentp(x) = 2p0χ(0,1)(x)+p0χ(1,+∞)(x), 1 < p0 < +∞, but it is easy to see that an ana-logue result can be obtained, with the obvious changes, for the increasing exponentp(x) = p0χ(0,1)(x) + 2p0χ(1,+∞)(x), 1 < p0 < +∞.
More precisely, we can formulate the result as follows:
holds for C > 0 independent of the non-increasing function f , if and only if:
(i) wχ(0,1) ∈ B2p0
(ii)∫ +∞r
( rx)p0w(x)dx ≤ C
∫ r
0 w(x)dx, for some C > 0 and all r > 1.
Proof Using (1) and some technical calculations, one can prove that
‖f ‖p0p(·),w
∫ +∞
1f p0(x)w(x)dx
+√(∫ +∞
1f p0(x)w(x)dx
)2
+∫ 1
0f 2p0(x)w(x)dx.
Hence, to obtain inequality (13) it is necessary and sufficient to prove that, for anydecreasing function f ,(∫ +∞
1(Sf )p0(x)w(x)dx
)2
�(∫ +∞
1f p0(x)w(x)dx
)2
+∫ 1
0f 2p0(x)w(x)dx
(14)and∫ 1
0(Sf )2p0(x)w(x)dx �
(∫ +∞
1f p0(x)w(x)dx
)2
+∫ 1
0f 2p0(x)w(x)dx. (15)
Since f is decreasing, (15) is equivalent to∫ 1
0(Sf )2p0(x)w(x)dx �
∫ 1
0f 2p0(x)w(x)dx,
which turns out to be equivalent to the condition wχ(0,1) ∈ B2p0 .
Weighted weak modular and norm inequalities for the Hardy 471
On the other hand, writing explicitly Sf in (14), we can see that it is equivalent tothe inequalities
(∫ 1
0f (x)dx
)2p0(∫ +∞
1
w(x)
xp0dx
)2
�(∫ +∞
1f p0(x)w(x)dx
)2
+∫ 1
0f 2p0(x)w(x)dx, (16)
and(∫ +∞
1
(∫ x
1f (s) ds
)p0 w(x)
xp0dx
)2
�(∫ +∞
1f p0(x)w(x)dx
)2
+∫ 1
0f 2p0(x)w(x)dx. (17)
Since we are restricting to the case of non-increasing f , provided that the integral∫ +∞1
w(x)xp0 dx is finite, inequality (16) can be expressed as an inclusion between the
Lorentz spaces 2p0(wχ(0,1)) ↪→ 1(χ(0,1)) which is satisfied due to the conditionwχ(0,1) ∈ B2p0 (see [17, Theorems 1 and 4]).
Finally, to ensure that (17) holds for every non-increasing f , it is equivalent torestrict the inequality to a decreasing function f with f (x) ≡ 1 for 0 < x ≤ 1, that is
(∫ +∞
1
(∫ x
1f (s)ds
)p0 w(x)
xp0dx
)2
�(∫ +∞
1f p0(x)w(x) dx
)2
+∫ 1
0w(x)dx.
Let us observe that this inequality is equivalent to
∫ +∞
1
(∫ x
1f (s) ds
)p0 w(x)
xp0dx ≤ C1 + C2
(∫ +∞
1f p0(x)w(x)dx
), (18)
for some positive constants C1 and C2 independent of the decreasing function f .Testing this inequality on a characteristic function in (0, r) with r > 1, and taking into
account the necessary condition that has already appeared∫ +∞
1
w(x)
xp0dx < +∞, it
is easy to see that a necessary condition for (18) to hold is that
∫ +∞
r
(r
x
)p0
w(x)dx �∫ r
0w(x)dx, r > 1. (19)
Let us see that this local Bp0 condition is also enough to have (18). Indeed, the distri-bution formula (see [5]) applied to a positive and decreasing function f in (0,+∞)
with f (x) ≡ 1 in 0 < x < 1, implies that
∫ +∞
1f p(t)w(t) dt = p
∫ 1
0yp−1
(∫ λf (y)
1w(t) dt
)dy, (20)
where λf (y) = |{x ∈ (0,+∞) : f (x) > y}|. Then, we write
472 S. Boza, J. Soria
I :=∫ +∞
1
(∫ x
1f (s) ds
)p0 w(x)
xp0dx
= p0
∫ +∞
1
[∫ x
1
(1
t − 1
∫ t
1f (s) ds
)p0−1
f (t)(t − 1)p0−1 dt
]w(x)
xp0dx.
Let us define the nonincreasing function g(t) = f (t)( 1t−1
∫ t
1 f (s) ds)p0−1, for t > 1,and g(t) ≡ 1, if 0 < t ≤ 1. Since g is a nonincreasing function, we get by (20) thatthe last integral above can be written as
p0
∫ +∞
1
∫ +∞
1g(t)(t − 1)p0−1χ(1,x)(t) dt
w(x)
xp0dx
= p0
∫ +∞
1
∫ 1
0
(∫ λg(y)
1(t − 1)p0−1χ(1,x)(t) dt
)dy
w(x)
xp0dx
=∫ +∞
1
∫ 1
0
[min
(λg(y), x
) − 1]p0 dy
w(x)
xp0dx
=∫ +∞
1
[∫ g(x)
0(x − 1)p0 dy +
∫ 1
g(x)
(λg(y) − 1
)p0 dy
]w(x)
xp0dx.
Using Fubini’s theorem, hypothesis (19) and again the distribution formula, we finallyget that
I =∫ +∞
1g(x)(x − 1)p0
w(x)
xp0dx +
∫ 1
0
(λg(y) − 1
)p0
∫ +∞
λg(y)
w(x)
xp0dx dy
≤∫ +∞
1g(x)w(x)dx +
∫ 1
0
(λg(y) − 1
)p0 1
λg(y)p0
(∫ λg(y)
0w(x)dx
)dy
≤∫ +∞
1g(x)w(x)dx +
∫ +∞
0g(x)w(x)dx ≤ 2
∫ +∞
0g(x)w(x)dx
= 2∫ +∞
0h(x) f (x)w(x)dx,
where we have defined the function h as h(t) = ( 1t−1
∫ t
1 f (s) ds)p0−1, for t > 1, andh(t) ≡ 1 if 0 < t ≤ 1. Therefore, using Hölder’s inequality in this last expression, weobtain
I ≤ ‖h‖L
p′0 (w)
‖f ‖Lp0 (w)
=(∫ 1
0w(x) dx +
∫ +∞
1
(1
x − 1
∫ x
1f (s) ds
)p0
w(x)dx
)(p0−1)/p0
×(∫ 1
0w(x) dx +
∫ +∞
1f p0(x)w(x)dx
)1/p0
Weighted weak modular and norm inequalities for the Hardy 473
�(∫ 2
0w(x)dx +
∫ +∞
1
(1
x
∫ x
1f (s) ds
)p0
w(x)dx
)(p0−1)/p0
×(∫ 2
0w(x)dx +
∫ +∞
1f p0(x)w(x)dx
)1/p0
·
Setting A := ∫ 20 w(x)dx and N := ∫ +∞
1 f p0(x)w(x)dx, we observe that the lastinequality obtained above can be written as
Ip0
(A + I )p0−1� (A + N),
which easily turns out to give us the inequality I � (A + N), and this is (18). �
Remark 3.2 It is important to observe that there are weights satisfying the conditionsof Proposition 3.1. For example, let us consider the exponent p(·) defined there in thecase p0 = 2, and the weights
wα,β(x) ={
tα, 0 < t ≤ 1,
tβ, t > 1.
Then, (i) and (ii) of Proposition 3.1 hold, if and only if −1 < α < 3 and −1 < β < 1.We also remark that none of these weights can verify the Bp(·) condition (3), for
p(·) as above, since p(·) does not have a constant oscillation on (0,∞). Moreoverif, in addition α > 1, wα,β also fails to satisfy the Bp(·) condition (3) restricted to theparameter s = 1, which is the class of weights considered in [16, Theorem 6] to studythe inequality (13) on a certain subclass of monotone functions.
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