Nonlinear Analysis 85 (2013) 125–131

Contents lists available at SciVerse ScienceDirect

Nonlinear Analysis

journal homepage: www.elsevier.com/locate/na

Grand Lebesgue spaces with respect to measurable functionsClaudia Capone a, Maria Rosaria Formica b, Raffaella Giova b,∗

a C.N.R. Istituto per le Applicazioni del Calcolo ‘‘Mauro Picone’’, (sez. Napoli) Via P. Castellino 111 - 80131 Napoli, Italyb Dipartimento di Statistica eMatematica per la Ricerca Economica, Università degli Studi di Napoli ‘‘Parthenope’’, ViaMedina 40 - 80133Napoli, Italy

a r t i c l e i n f o

Article history:Received 8 August 2012Accepted 21 February 2013Communicated by Enzo Mitidieri

MSC:46E3042B25

Keywords:Grand Lebesgue spacesBanach function spacesRearrangement-invariant spacesFunction normEmbedding resultsHardy inequality

a b s t r a c t

Let 1 < p < ∞. Given Ω ⊂ Rn a measurable set of finite Lebesgue measure, the norm ofthe grand Lebesgue spaces Lp)(Ω) is given by

|f |Lp)(Ω) = sup0<ε<p−1

ε1

p−ε

1

|Ω|

Ω

|f |p−εdx 1

p−ε

.

In this paper we consider the norm |f |Lp),δ (Ω) obtained replacing ε1

p−ε by a genericnonnegative measurable function δ(ε). We find necessary and sufficient conditions on δin order to get a functional equivalent to a Banach function norm, and we determine the‘‘interesting’’ class Bp of functions δ, with the property that every generalized functionnorm is equivalent to a function norm built with δ ∈ Bp. We then define the Lp),δ(Ω)spaces, prove some embedding results and conclude with the proof of the generalizedHardy inequality.

© 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Let Ω ⊂ Rn, n ≥ 2, be a measurable set of Lebesgue measure |Ω| < +∞. In 1992 Iwaniec and Sbordone [1] studied theintegrability properties of the Jacobian determinant, and introduced the grand Lebesgue space Ln)(Ω) as a space such that

|Df | ∈ Ln)(Ω) ⇒ |Jf | ∈ L1loc(Ω)

for all Sobolev mappings f : Ω → Rn, f = (f 1, . . . , f n).Since then the grand Lebesgue spaces play an important role in PDEs theory (see e.g. [2–8]) and in Function Spaces Theory

(see e.g. [9–12] and references therein). It turns out that such spaces are Banach Function Spaces in the sense of [13]: namely(here and in the following we will use the letter p instead of n, assuming 1 < p < ∞)

Lp)(Ω) =

f ∈ Mo : ∥f ∥p) = ρ(|f |) = sup

0<ε<p−1ε

1p−ε

1

|Ω|

Ω

|f |p−εdx

1p−ε

< +∞

,

where Mo is the set of all real valued measurable functions on Ω , and, denoting by M+o the subset of Mo of the nonnegative

functions, ρ : M+o → [0, +∞] is such that for all f , g, fn(n = 1, 2, 3, . . .) in M+

o , for all constants λ ≥ 0, and for allmeasurable subsets E ⊂ Ω , the following properties hold:

∗ Corresponding author. Tel.: +39 081 5474934; fax: +39 081 5474904.E-mail addresses: c.capone@na.iac.cnr.it (C. Capone), mara.formica@uniparthenope.it (M.R. Formica), raffaella.giova@uniparthenope.it,

rgiova@unina.it (R. Giova).

0362-546X/$ – see front matter© 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.na.2013.02.021

126 C. Capone et al. / Nonlinear Analysis 85 (2013) 125–131

(1) ρ(f ) = 0 ⇔ f = 0 a.e. inΩ(2) ρ(λf ) = λρ(f )(3) ρ(f + g) ≤ ρ(f ) + ρ(g)(4) 0 ≤ g ≤ f a.e. inΩ ⇒ ρ(g) ≤ ρ(f )(5) 0 ≤ fn ↑ f a.e. inΩ ⇒ ρ(fn) ↑ ρ(f )(6) E ⊂ Ω ⇒ ρ(χE) < +∞

(7) E ⊂ Ω ⇒E fdx ≤ CEρ(f )

for some constant CE, 0 < CE < ∞, depending on E and ρ, but independent of f .Grand Lebesgue spaces belong to a special category of Banach Function Spaces: they are rearrangement-invariant,

namely, setting

µf (λ) = |x ∈ Ω : |f (x)| > λ| ∀ λ ≥ 0 (1.1)

it is ρ(f ) = ρ(g) whenever µf = µg .A generalization of the grand Lebesgue spaces are the spaces Lp),θ , θ ≥ 0, defined by (see e.g. [14])

∥f ∥Lp),θ (Ω) = sup0<ε<p−1

εθ 1

|Ω|

Ω

|f |p−εdx

1p−ε

.

When θ = 0 the spaces Lp),0(Ω) reduce to Lebesgue spaces Lp(Ω) and when θ = 1 the spaces Lp),1(Ω) reduce to grandLebesgue spaces Lp)(Ω).

A useful property of the norm, used in [15,16] is the fact that the supremum over (0, p− 1) in the norm of Lp)(Ω) can becomputed also in any smaller interval (0, ε0): the result is an equivalent expression of the norm (i.e. each expression can bemajorized by the other, multiplied by a constant not depending on f ). Obviously, the constants involved in the equivalencewill depend on p and ε0. This phenomenon has been clarified also in a more general context in [17].

We recall also the continuous embeddings, easy consequence from the definition,

Lp(Ω) ⊂ Lp),θ (Ω) ⊂ Lp−ϵ(Ω), 0 < ϵ ≤ p − 1 θ > 0.

2. The main results

Let δ : (0, p − 1) → [0, +∞[ be a measurable function, and for all f ∈ M+o set

ρp),δ(f ) = ess sup0<ε<p−1

δ(ε)1

p−ε

?Ωf p−εdx

1p−ε

, (2.1)

where>Ω

stands for 1|Ω|

Ω. For 1 ≤ r < ∞, we will also write ∥f ∥r to denote the normalized norm of f in Lr(Ω):

∥f ∥r =

?Ωf rdx

1r

.

By convention, we establish that the right hand side of (2.1) is ∞ if for some 0 < ε < p − 1 the function f ∈ Lp−ε(Ω): thisposition gives always a meaning to the ess sup, also when the indeterminate form 0 ·∞ appears. The case δ(ε) = εθ , θ > 0,gives back the norm of the Lp),θ (Ω) spaces.

The first goal of this paper is to find a necessary and sufficient condition on δ such that ρp),δ is equivalent to a Banachfunction norm, i.e. equivalent to a functional satisfying all the properties (1)–(7) listed in the previous section.

It is clear that the first way to prove that ρp),δ is equivalent to a Banach function norm is to try to reproduce the analogousproof, valid for grand Lp spaces. This latter proof is an easy consequence of the classical properties of the norm of Lebesguespaces, and it seems, for this reason, almost absent in literature. The problem when considering the functional ρp),δ is thatδ is defined almost everywhere, and the expression δ(ε) does not have the meaning of value attained in ε. Moreover, theestimate of ρp),δ looksmuch less evident when, for instance, δ attains the value zero infinite times in a neighborhood of zero.

Besides solving completely the problems above, we will show in particular that for any measurable bounded δ, ρp),δ isequivalent to a Banach function norm, and that the same resulting space can be obtained by using a new function δ, definedeverywhere, whose expression is explicitly shown. After this step, also in the case of bounded measurable functions, theproof of being equivalent to a Banach function norm can be considered equally trivial as in the case of grand Lebesguespaces.

Going back to our first goal, an immediate necessary condition is suggested by property (6), when E = Ω: since ρp),δ(χΩ)must be finite, it must be δ ∈ L∞(0, p − 1). The Theorem we will prove is that this condition is actually also sufficient.

Theorem 2.1. Let 1 < p < ∞ and let δ : (0, p − 1) → [0, +∞[ be a measurable function, not identically zero. The mappingρp),δ is equivalent to a Banach function norm if and only if δ ∈ L∞(0, p − 1).

C. Capone et al. / Nonlinear Analysis 85 (2013) 125–131 127

The proof of Theorem 2.1 requires some intermediate results of independent interest. As a byproduct, we will determinethe ‘‘interesting’’ classBp of functions δ ∈ L∞(0, p−1), with the property that every ρp),δ , obtained from a generic boundedmeasurable δ, is equivalent to a function norm built with δ ∈ Bp.

Lemma 2.2. If δ1, δ2 : (0, p − 1) → [0, +∞[ are measurable functions such that

ess sup0<ε<σ

δ1(ε)1

p−ε = ess sup0<ε<σ

δ2(ε)1

p−ε , σ ∈]0, p − 1], (2.2)

then ρp),δ1 = ρp),δ2 .

Proof. Since we may exchange the roles of ρp),δ1 and ρp),δ2 , it is sufficient to prove that for all f ∈ M+o

ρp),δ1(f ) ≤ ρp),δ2(f ). (2.3)

If f is identically zero, (2.3) is trivially true, therefore we may work with functions f having positive Lebesgue norm. If forsome ε it is f ∈ Lp−ε(Ω), then both sides of (2.3) are ∞, therefore we may consider the functions f such that

0 < ∥f ∥p−ε < ∞ for all 0 < ε < p − 1. (2.4)

If ρp),δ1(f ) or ρp),δ2(f ) is zero, then the other one is also zero: in fact, if for instance ρp),δ1(f ) = 0, from (2.4) we get thatδ1 = 0 a.e. in (0, p − 1). By (2.2) used with σ = p − 1, we deduce that δ2 = 0 also, and our claim is proved.

Consider the case 0 < ρp),δ1(f ) < ∞ and fix η > 0. By the definition of ρp),δ1(f ) there exists a set of positive measureTη ⊂ (0, p − 1) such that

δ1(ε)1

p−ε ∥f ∥p−ε > ρp),δ1(f ) − η, ε ∈ Tη

from which

δ1(ε)1

p−ε >ρp),δ1(f ) − η

∥f ∥p−ε

, ε ∈ Tη. (2.5)

Now set

e′

η = ess infTη

x ∈ [0, p − 1[

e′′

η = ess supTη

x ∈]0, p − 1]

and fix σ , e′η < σ < e′′

η . From (2.5) and themonotonicity of the (normalized) norm ∥f ∥r with respect to r (due to theHölder’sinequality),

ess sup0<ζ<σ

δ1(ζ )1

p−ζ >ρp),δ1(f ) − η

∥f ∥p−ε

, ε ∈]0, σ [

and by (2.2)

ess sup0<ζ<σ

δ2(ζ )1

p−ζ >ρp),δ1(f ) − η

∥f ∥p−ε

, ε ∈]0, σ [.

We deduce the existence of a set T ′η ⊂ (0, σ ) of positive measure such that

δ2(ζ )1

p−ζ >ρp),δ1(f ) − η

∥f ∥p−ε

, ζ ∈ T ′

η, ε ∈]0, σ [

from which

δ2(ε)1

p−ε >ρp),δ1(f ) − η

∥f ∥p−ε

, ε ∈ T ′

η

ess sup0<ε<σ

δ2(ε)1

p−ε ∥f ∥p−ε > ρp),δ1(f ) − η.

Since η can be arbitrarily small, we get the assertion.Finally, if ρp),δ1(f ) = ∞, we may follow the same argument as before, replacing ρp),δ1(f ) − η by anyM > 0. The lemma

is therefore proved.

The next proposition shows that monotone functions play an important role in our study. The term increasing for afunction δ means that if ε1 < ε2, then δ(ε1) ≤ δ(ε2).

128 C. Capone et al. / Nonlinear Analysis 85 (2013) 125–131

Proposition 2.3. If δ ∈ L∞(0, p − 1), 0 ≤ δ ≤ 1, then there exists δ ∈ L∞(0, p − 1) such that:

(i) 0 ≤ δ ≤ 1

(ii) δ(ε)1

p−ε increasing in ε and left continuous(iii) ρp),δ = ρp),δ.

Proof. For any given δ ∈ L∞(0, p − 1), 0 ≤ δ ≤ 1, the function

δ(ε) =

ess sup0<ζ<ε

δ(ζ )1

p−ζ

p−ε

, ε ∈ (0, p − 1)

has trivially the property (i). Property (ii) follows by a well known characterization of increasing, left continuous functions,see e.g. [18, Theorem 8.19]. We have to prove only that ρp),δ = ρp),δ .

On one hand, the definition of δ gives immediately that

δ(ε)1

p−ε = ess sup0<ζ<ε

δ(ζ )1

p−ζ , ε ∈ (0, p − 1) (2.6)

and, on the other hand, by (ii),

δ(ε)1

p−ε = ess sup0<ζ<ε

δ(ζ )1

p−ζ , ε ∈ (0, p − 1). (2.7)

Combining (2.6) and (2.7), we get

ess sup0<ζ<ε

δ(ζ )1

p−ζ = ess sup0<ζ<ε

δ(ζ )1

p−ζ , ε ∈ (0, p − 1).

By Lemma 2.2, we get (iii).

The following definition plays a crucial role in the study of the generalization of the grand Lebesgue spaces with respectto measurable functions.

Definition 2.4. Let 1 < p < ∞. A function δ, left continuous on (0, p − 1), is said to be in the class Bp if

(j) δ(0+) = 0(jj) 0 < δ ≤ 1

(jjj) δ(ε)1

p−ε is increasing in ε.

It is easy to check that functions in Bp are increasing. Moreover, the left continuity of its functions permit us to writemore simply sup instead of ess sup in the expressions related to ρp),δ .

We have now the prerequisites for the

Proof of Theorem 2.1. Let δ ∈ L∞(0, p − 1) be nonnegative. Of course we may think to divide δ by its (positive) L∞(Ω)norm, therefore without loss of generality we may assume that 0 ≤ δ ≤ 1. Moreover, by Proposition 2.3, without loss ofgenerality we may assume that (ii) holds true, therefore, in particular, δ increasing and left continuous. Therefore it makessense to compute δ(0+).

If δ(0+) > 0, for any measurable function f on Ω , possibly not in Lp(Ω), it is

δ(0+)∥f ∥p = δ(0+) sup0<ε<p−1

∥f ∥p−ε ≤ ρp),δ(f ) ≤ ∥f ∥p.

If δ(0+) = 0, since δ is increasing, there are two possibilities: there exists, or not, 0 < ε < p − 1 such that δ(ε) = 0. Inthe first case let ε0 = maxε : 0 < ε < p − 1, δ(ε) = 0. It is 0 < ε0 < p − 1 and

ρp),δ(f ) = sup0<ε<p−1

δ(ε)1

p−ε ∥f ∥p−ε = supε0<ε<p−1

δ(ε)1

p−ε ∥f ∥p−ε.

After a change of variable in the sup,

ρp),δ(f ) = sup0<ε<p−ε0−1

δ(ε + ε0)1

p−ε0−ε ∥f ∥p−ε0−ε.

Setting δ(ε) = δ(ε + ε0) and r = p − ε0, we get

ρp),δ(f ) = ρr),δ(f ).

C. Capone et al. / Nonlinear Analysis 85 (2013) 125–131 129

Since, by the maximality of ε0, it is δ(ε) > 0 when ε > 0, the first case we are studying will be concluded after theexamination of the second case.

The assumptions we have now on δ imply that δ ∈ Bp. At this point, all the axioms of the Banach function norms arestraightforward to prove.

As a byproduct of the proof of Theorem 2.1, we have the following

Theorem 2.5. Let 1 < p < ∞ and let δ ∈ L∞(0, p − 1), δ ≥ 0, δ ≡ 0. The mapping ρp),δ is equivalent to ρp),δ , where δ is theincreasing, left continuous function defined by

δ(ε) =

ess sup0<ζ<ε

δ(ζ )

∥δ∥∞

1p−ζ

p−ε

, ε ∈ (0, p − 1).

Again by the proof of Theorem 2.1, it is clear that the interesting functions δ to consider in the Banach function norm arethose ones in the class Bp. This motivates the following

Definition 2.6. Let Ω ⊂ Rn, n ≥ 1, be a measurable set of Lebesgue measure |Ω| < +∞, let 1 < p < ∞ and let δ ∈ Bp.The grand Lp space over Ω with respect to δ is the Banach Function Space defined by

Lp),δ(Ω) =

f ∈ Mo : ∥f ∥p),δ = ρp),δ(|f |) = sup

0<ε<p−1δ(ε)

1p−ε ∥f ∥p−ε < +∞

.

It is immediate from the definition that the spaces Lp),δ(Ω) are rearrangement-invariant, include Lp(Ω) and are includedin each Lp−ε(Ω), 0 < ε < p − 1.

We conclude this Section by showing a sufficient condition, and a necessary condition, for the embedding between grandLp spaces built from two functions δ1, δ2 ∈ Bp. Wewill need the following simple lemma, which extends the useful propertyof the grand Lebesgue spaces mentioned at the end of the Introduction.

Lemma 2.7. Let 1 < p < ∞ and 0 < σ < p − 1. If δ ∈ Bp, there exists a constant c = c(p, δ, σ ) such that

sup0<ε<σ

δ(ε)1

p−ε

?Ωf p−εdx

1p−ε

≤ ρp),δ(f ) ≤ c sup0<ε<σ

δ(ε)1

p−ε

?Ωf p−εdx

1p−ε

.

Proof. The left wing inequality is trivial, therefore we need to prove only the right wing one. Fix ε ≥ σ and 0 < µ < σ . ByHölder’s inequality, ∥ · ∥r is increasing in r , therefore we have

supσ≤ε<p−1

δ(ε)1

p−ε ∥f ∥p−ε ≤ ∥f ∥p−µ = δ(µ)−

1p−µ δ(µ)

1p−µ ∥f ∥p−µ

from which

supσ≤ε<p−1

δ(ε)1

p−ε ∥f ∥p−ε ≤ δ(µ)−

1p−µ sup

0<ε<σ

δ(ε)1

p−ε ∥f ∥p−ε, µ ∈]0, σ [.

Passing to the infimum over µ on the right hand side, and recalling that 0 < δ ≤ 1, we get the desired inequality withc = δ(σ )

−1

p−σ .

Proposition 2.8. Let Ω ⊂ Rn, n ≥ 1, be ameasurable set of Lebesguemeasure |Ω| < +∞, let 1 < p < ∞, and let δ1, δ2 ∈ Bp.Then

limε→0

sup0<σ<ε

δ1(ε)

δ2(ε)< ∞ ⇒ Lp),δ2(Ω) ⊂ Lp),δ1(Ω) ⇒ lim

ε→0inf

0<σ<ε

δ1(ε)

δ2(ε)< ∞.

Proof. If

limε→0

sup0<σ<ε

δ1(ε)

δ2(ε)= M < ∞,

then for small ε0 > 0 it is

δ1(ε) < (M + 1)δ2(ε), ε ∈ (0, ε0)

130 C. Capone et al. / Nonlinear Analysis 85 (2013) 125–131

and this immediately implies that Lp),δ2(Ω) ⊂ Lp),δ1(Ω). On the other hand, if this inclusion holds, assume, on the contrary,that

limε→0

inf0<σ<ε

δ1(ε)

δ2(ε)= ∞.

For any fixedM > 1, there exists a small ε0 > 0 such that

δ1(ε) > Mδ2(ε), ε ∈ (0, ε0).

Raising both sides to the power 1p−ε

, multiplying by ∥f ∥p−ε and taking the supremum over (0, ε0), by Lemma 2.7 we get

∥f ∥p),δ1 ≥ cM1p ∥f ∥p),δ2

which is in contradiction with the assumed embedding.

Corollary 2.9. Let Ω ⊂ Rn, n ≥ 1, be a measurable set of Lebesgue measure |Ω| < +∞, let 1 < p < ∞, and let δ1, δ2 ∈ Bp

be equivalent in a neighborhood of the origin. Then Lp),δ1(Ω) = Lp),δ2(Ω).

3. The Hardy inequality for Lp),δ spaces

The classical Hardy inequality states that

Theorem 3.1. Let p > 1 and f be a measurable, nonnegative function in (0, 1). Then 1

0

? x

0fdt

p

dx

1/p

≤p

p − 1

1

0f pdx

1/p

. (3.1)

In this section we extend the Hardy inequality in the context of Lp),δ(0, 1) spaces. We will follow closely the proofs givenin [11].

Theorem 3.2. Let 1 < p < ∞ and δ ∈ Bp. There exists a constant c(p, δ) > 1 such that? x

0fdtp),δ

≤ c(p, δ)∥f ∥p),δ (3.2)

for all nonnegative measurable functions f in (0, 1).

Proof. Let 0 < σ < p − 1. We have

? x

0fdtp),δ

= max

sup

0<ϵ<σ

δ(ϵ)

1

0

? x

0fdt

p−ϵ

dx

1p−ϵ

, supσ≤ϵ<p−1

δ(ϵ)

1

0

? x

0fdt

p−ϵ

dx

1p−ϵ

= max

sup

0<ϵ<σ

δ(ϵ)

1

0

? x

0fdt

p−ϵ

dx

1p−ϵ

, supσ≤ϵ<p−1

δ(ϵ)1

p−ϵ

1

0

? x

0fdt

p−ϵ

dx

1p−ϵ

≤ max

sup

0<ϵ<σ

δ(ϵ)

1

0

? x

0fdt

p−ϵ

dx

1p−ϵ

,

× supσ≤ϵ<p−1

δ(ϵ)1

p−ϵ δ(σ )−

1p−σ δ(σ )

1p−σ

1

0

? x

0fdt

p−σ

dx

1p−σ

≤ max

sup

0<ϵ<σ

δ(ϵ)

1

0

? x

0fdt

p−ϵ

dx

1p−ϵ

,

× supσ≤ϵ<p−1

δ(ϵ)1

p−ϵ δ(σ )−

1p−σ sup

0<ϵ<σ

δ(ϵ)

1

0

? x

0fdt

p−ϵ

dx

1p−ϵ

.

C. Capone et al. / Nonlinear Analysis 85 (2013) 125–131 131

Therefore,

? x

0fdtp),δ

≤ maxσ≤ϵ<p−1

δ(ϵ)1

p−ϵ δ(σ )−

1p−σ sup

0<ϵ<σ

δ(ϵ)

1

0

? x

0fdt

p−ϵ

dx

1p−ϵ

since maxσ≤ϵ<p−1 δ(ϵ)1

p−ϵ δ(σ )−

1p−σ ≥ 1.

Now take 0 < ϵ ≤ σ , so that p − ϵ > 1. Applying the Hardy inequality (3.1) with the exponent p replaced by p − ϵ, andmultiplying both sides by δ(ϵ)

1p−ϵ , we get

δ(ϵ)

1

0

? x

0fdt

p−ϵ

dx

1p−ϵ

≤p − ϵ

p − ϵ − 1

δ(ϵ)

1

0f p−ϵdx

1p−ϵ

.

If we pass to the sup over 0 < ϵ < σ on both sides, the previous inequality implies

sup0<ϵ<σ

δ(ϵ)

1

0

? x

0fdt

p−ϵ

dx

1p−ϵ

≤p − σ

p − σ − 1sup

0<ϵ<σ

δ(ϵ)

1

0f p−ϵdx

1p−ϵ

and therefore? x

0fdtp),δ

≤ maxσ≤ϵ<p−1

δ(ϵ)1

p−ϵ δ(σ )−

1p−σ

p − σ

p − σ − 1sup

0<ϵ<σ

δ(ϵ)

1

0f p−ϵdx

1p−ϵ

≤ maxσ≤ϵ<p−1

δ(ϵ)1

p−ϵ δ(σ )−

1p−σ

p − σ

p − σ − 1sup

0<ϵ<p−1

δ(ϵ)

1

0f p−ϵdx

1p−ϵ

.

Setting

c(p, δ) := inf0<σ<p−1

maxσ≤ϵ<p−1

δ(ϵ)1

p−ϵ δ(σ )−

1p−σ

p − σ

p − σ − 1≥ 1,

we get the inequality (3.2).

References

[1] T. Iwaniec, C. Sbordone, On the integrability of the Jacobian under minimal hypothesis, Arch. Ration. Mech. Anal. 119 (1992) 129–143.[2] L. Boccardo, Quelques problemes de Dirichlet avec donnees dans de grands espaces de Sobolev, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997) 1269–1272.[3] A. Fiorenza, A. Mercaldo, J.M. Rakotoson, Regularity and comparison results in grand Sobolev spaces for parabolic equations with measure data, Appl.

Math. Lett. 14 (2001) 979–981.[4] A. Fiorenza, A.Mercaldo, J.M. Rakotoson, Regularity anduniqueness results in grand Sobolev spaces for parabolic equationswithmeasure data, Discrete

Contin. Dyn. Syst. 8 (4) (2002) 893–906.[5] A. Fiorenza, C. Sbordone, Existence and uniqueness results for solutions of nonlinear equations with right hand side in L1 , Studia Math. 127 (3) (1998)

223–231.[6] L. Greco, T. Iwaniec, C. Sbordone, Inverting the p-harmonic operator, Manuscripta Math. 92 (1997) 249–258.[7] C. Sbordone, Nonlinear elliptic equations with right hand side in nonstandard spaces, Atti Sem. Mat. Fis. Univ. Modena 46 (Suppl.) (1998) 361–368.[8] C. Sbordone, Grand Sobolev spaces and their applications to variational problems, Matematiche 51 (2) (1996) 335–347.[9] C. Capone, A. Fiorenza, G.E. Karadzhov, Grand Orlicz spaces and global integrability of the Jacobian, Math. Scand. 102 (2008) 131–148.

[10] G. Di Fratta, A. Fiorenza, A direct approach to the duality of grand and small Lebesgue spaces, Nonlinear Anal. TMA 70 (2009) 2582–2592.[11] A. Fiorenza, B. Gupta, P. Jain, The maximal theorem for weighted grand Lebesgue spaces, Studia Math. 188 (2) (2008) 123–133.[12] P. Koskela, X. Zhong, Minimal assumptions for the integrability of the Jacobian, Ric. Mat. 51 (2) (2002) 297–311.[13] C. Bennett, R. Sharpley, Interpolation of Operators, Academic Press, 1988.[14] C. Capone, A. Fiorenza, On small Lebesgue spaces, J. Funct. Spaces Appl. 3 (1) (2005) 73–89.[15] A. Fiorenza, G.E. Karadzhov, Grand and small Lebesgue spaces and their analogs, Z. Anal. Anwend. 23 (4) (2004) 657–681.[16] A. Fiorenza, J.M. Rakotoson, Compactness, interpolation inequalities for small Lebesgue–Sobolev spaces and applications, Calc. Var. Partial Differential

Equations 25 (2) (2005) 187–203.[17] H. Rafeiro, A note on boundedness of operators in grand grand Morrey spaces (in press).[18] E. Hewitt, K. Stromberg, Real and Abstract Analysis, Springer-Verlag, 1975.