# A Generalization of Exponential Class and Its Applications

Authors: Hongya Gao; Chao Liu; Hong Tian
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

A function space,Lθ,∞)(Ω), 0≤θ<∞, is defined. It is proved that Lθ,∞)(Ω) is a Banach space which is a generalization of exponential class. An alternative definition of Lθ,∞)(Ω) space is given. As an application, we obtain weak monotonicity property for very weak solutions of 𝒜-harmonic equation with variable coefficients under some suitable conditions related to Lθ,∞)(Ω), which provides a generalization of a known result due to Moscariello. A weighted space Lwθ,∞)(Ω) is also defined, and the boundedness for the Hardy-Littlewood maximal operator Mw and a Calderón-Zygmund operator T with respect to Lwθ,∞)(Ω) is obtained.


---

## Body

## 1. Introduction

For1<p<∞ and a bounded open subset Ω⊂Rn, the grand Lebesgue space Lp)(Ω) consists of all functions f(x)∈⋂0<ε≤p-1‍Lp-ε(Ω) such that
(1)∥f∥p),Ω=sup0<ε≤p-1(ε⨍Ω|f|p-εdx)1/(p-ε)<∞,
where ⨍Ω=(1/|Ω|)∫Ω‍ stands for the integral mean over Ω. The grand Sobolev space W01,p)(Ω) consists of all functions u∈⋂0<ε≤p-1‍W01,p-ε(Ω) such that
(2)∥u∥W01,p)=sup0<ε≤p-1(ε⨍Ω|∇f|p-εdx)1/(p-ε)<∞.
These two spaces, slightly larger than Lp(Ω) and W01,p(Ω), respectively, were introduced in the paper [1] by Iwaniec and Sbordone in 1992 where they studied the integrability of the Jacobian under minimal hypotheses. For p=n in [2], imbedding theorems of Sobolev type were proved for functions f∈W01,n)(Ω). The small Lebesgue space L(p(Ω) was found by Fiorenza [3] in 2000 as the associate space of the grand Lebesgue space Lp)(Ω). Fiorenza and Karadzhov gave in [4] the following equivalent, explicit expressions for the norms of the small and grand Lebesgue spaces, which depend only on the nondecreasing rearrangement (provided that the underlying measure space has measure 1):
(3)∥f∥L(p≈∫01‍(1-lnt)-1/p(∫0t‍[f*(s)]pds)1/pdtt,1<p<∞,∥f∥Lp)≈sup0<t<1(1-lnt)-1/p(∫t1‍[f*(s)]pds)1/p,1<p<∞.
In [5], Greco et al. gave two more general definitions than (1) and (2) in order to derive existence and uniqueness results for p-harmonic operators. For 1<p<∞ and 0≤θ<∞, the grand Lp space, denoted by Lθ,p)(Ω), consists of functions f∈⋂0<ε≤p-1‍Lp-ε(Ω) such that
(4)∥f∥θ,p)=sup0<ε≤p-1εθ/p∥f∥p-ε<∞,
where
(5)∥f∥p-ε=(⨍Ω|f|p-εdx)1/(p-ε).
The grand Sobolev space Wθ,p)(Ω) consists of all functions f belonging to ⋂0<ε≤p-1‍W1,p-ε(Ω) and such that ∇f∈Lθ,p)(Ω). That is,
(6)Wθ,p)(Ω)={f∈⋂0<ε≤p-1‍W1,p-ε(Ω):∇f∈Lθ,p)(Ω)}.Grand and small Lebesgue spaces are important tools in dealing with regularity properties for very weak solutions of𝒜-harmonic equation as well as weakly quasiregular mappings; see [6, 7].The aim of the present paper is to provide a generalizationLθ,∞)(Ω), 0≤θ<∞, of exponential class EXP(Ω) and prove that it is a Banach space. An alternative definition of Lθ,∞)(Ω) is given in terms of weak Lebesgue spaces. As an application, we obtain weak monotonicity property for very weak solutions of 𝒜-harmonic equation with variable coefficients under some suitable conditions related to Lθ,∞)(Ω). This paper also considers a weighted space Lwθ,∞)(Ω) and some boundedness result for classical operators with respect to this space.In the sequel, the letterC is used for various constants and may change from one occurrence to another.


## 2. A Generalization of Exponential Class

Recall thatEXP(Ω), the exponential class, consists of all measurable functions f such that
(7)∫Ω‍eλ|f|dx<∞
for some λ>0. It is a Banach space under the norm
(8)∥f∥EXP=inf{λ>0:∫Ω‍e|f|/λdx≤2}.In this section, we define a spaceLθ,∞)(Ω), 0≤θ<∞, which is a generalization of EXP(Ω), and prove that it is a Banach space.Definition 1.
Forθ≥0, the space Lθ,∞)(Ω) is defined by
(9)Lθ,∞)(Ω)={f(x)∈⋂1≤p<∞‍Lp(Ω):sup1≤p<∞1pθ(⨍Ω|f(x)|pdx)1/p<∞}.It is not difficult to see that(9)′Lθ,∞)(Ω)={g(x)∈⋂1≤p<∞‍Lp(Ω):limsupp→∞1pθ(⨍Ω|g(x)|pdx)1/p<∞⋂1≤p<∞‍}.There are two special cases ofLθ,∞)(Ω) that are worth mentioning since they coincide with two known spaces.Case 1 (θ=0).
In this case,(10)L0,∞)(Ω)={f(x)∈⋂1≤p<∞‍Lp(Ω):sup1≤p<∞(⨍Ω|f(x)|pdx)1/p<∞⋂1≤p<∞‍}.
From the fact (see [8, page 12])
(11)L∞(Ω)={f∈⋂1≤p<∞‍Lp(Ω):limp→∞∥f∥p<∞},
we get L0,∞)(Ω)=L∞(Ω).Case 2 (θ=1).
The following proposition shows thatLθ,∞(Ω) can be regarded as a generalization of EXP(Ω).Proposition 2.
L
1
,
∞
)
(
Ω
)
=
EXP
(
Ω
).Proof.
In order to realize that a function in theL1,∞)(Ω) space is in EXP(Ω), it is sufficient to read the last lines of [2]. The vice versa is also true; see for example [9, Chap. VI, exercise no. 17].It is clear that for any0≤θ<θ′≤∞ and any q<∞, we have the inclusions
(12)L∞(Ω)⊂Lθ,∞)(Ω)⊂Lθ′,∞)(Ω)⊂Lq(Ω).The following theorem shows that ifθ>0, then Lθ,∞)(Ω) is slightly larger than L∞(Ω).Theorem 3.
Forθ>0, the space L∞(Ω) is a proper subspace of Lθ,∞)(Ω).Proof.
In the proof of Theorem3 we always assume θ>0. Let f(x)∈L∞(Ω), then there exists a constant M<∞, such that |f(x)|≤M, a.e. Ω. Thus,
(13)sup1≤p<∞1pθ(⨍Ω|f(x)|pdx)1/p≤sup1≤p<∞Mpθ=M<∞,
which implies that f(x)∈Lθ,∞)(Ω).
The following example shows thatL∞(Ω)⊂Lθ,∞)(Ω) is a proper subset. Since we have the inclusion (12), then it is no loss of generality to assume that θ≤1. Consider the function f(x)=(-lnx)θ defined in the open interval (0,1). It is obvious that f(x)∉L∞(0,1). We now show that f(x)∈Lθ,∞)(0,1). In fact, for m a positive integer, integration by parts yields
(14)∫01‍(-lnx)mdx=x(-lnx)m|01-∫01‍xd(-lnx)m=-limx→0+x(-lnx)m+m∫01‍(-lnx)m-1dx.
By L'Hospital's Law, one has
(15)limx→0+x(-lnx)m=limx→0+(-lnx)m1/x=limx→0+m(-lnx)m-11/x=⋯=m!limx→0+x=0.
This equality together with (14) yields
(16)∫01‍(-lnx)mdx=m∫01‍(-lnx)m-1dx.
By induction,
(17)∫01‍fm(x)dx=m∫01‍(-lnx)m-1dx=⋯=m!∫01‍dx=m!.
Recall that the function
(18)p⟼(⨍Ω|f(x)|pdx)1/p
is nondecreasing; thus (17) yields
(19)sup1≤p<∞1pθ(⨍01|f(x)|pdx)1/p=sup1≤p<∞[1p(∫01‍(-lnx)pθdx)1/pθ]θ≤sup1≤p<∞[1p(∫01‍(-lnx)[pθ]+1dx)1/([pθ]+1)]θ=sup1≤p<∞[([pθ]+1)!1/([pθ]+1)p]θ≤sup1≤p<∞[[pθ]+1p]θ≤2,
where we have used the assumption θ≤1 and [pθ] is the integer part of pθ. The proof of Theorem 3 has been completed.For functionsf1(x),f2(x)∈Lθ,∞)(Ω), and α∈R, the addition f1(x)+f2(x) and the multiplication αf1(x) are defined as usual.Theorem 4.
L
θ
,
∞
)
(
Ω
) is a linear space on R.Proof.
This theorem is easy to prove, so we omit the details.Forf(x)∈Lθ,∞)(Ω), we define
(20)∥f∥θ,∞),Ω=sup1≤p<∞1pθ(⨍Ω|f(x)|pdx)1/p.
We drop the subscript Ω from ∥·∥θ,∞),Ω when there is no possibility of confusion.Theorem 5.
∥
·
∥
θ
,
∞
) is a norm.Proof.
(1) It is obvious that∥f∥θ,∞)≥0 and ∥f∥θ,∞)=0 if and only if f=0 a.e. Ω.
(2) For anyf1(x),f2(x)∈Lθ,∞)(Ω), Minkowski’s inequality in Lp(Ω) yields
(21)∥f1+f2∥θ,∞)=sup1≤p<∞1pθ(⨍Ω|f1+f2|pdx)1/p≤sup1≤p<∞1pθ[(⨍Ω|f1|pdx)1/p+(⨍Ω|f2|pdx)1/p]≤sup1≤p<∞1pθ(⨍Ω|f1|pdx)1/p+sup1≤p<∞1pθ(⨍Ω|f2|pdx)1/p=∥f1∥θ,∞)+∥f2∥θ,∞).
(3) For allλ∈R and all f(x)∈Lθ,∞)(Ω), it is obvious that ∥λf∥θ,∞)=|λ|∥f∥θ,∞).Theorem 6.
(
L
θ
,
∞
)
(
Ω
)
,
∥
·
∥
θ
,
∞
)
) is a Banach space.Proof.
Suppose that{fn}n=1∞⊂Lθ,∞)(Ω), and for any positive integer p,
(22)∥fn+p-fn∥θ,∞)⟶0,n⟶∞.
Since Ω is σ-finite, then Ω=⋃m=1∞‍Ωm with |Ωm|<∞. It is no loss of generality to assume that the Ωms are disjoint. Equation (17) implies that for any positive integer p,
(23)∫Ωm‍|fn+p(x)-fn(x)|dx⟶0,n⟶∞.
Thus, by the completeness of L1(Ωm), there exists f(m)(x)∈L1(Ωm), such that
(24)fn(x)⟶f(m)(x),n⟶∞,inL1(Ωm).
Hence for any positive integer m, there exists a subsequence {fn(m)(x)} of {fnm-1(x)}, {fn(0)(x)}={fn(x)}, such that
(25)fn(m)(x)⟶f(m)(x),n⟶∞,a.e.x∈Ωm.
If we let
(26)f(x)=f(m)(x),x∈Ωm,m=1,2,…,
then
(27)fn(n)(x)⟶f(x),n⟶∞,a.e.x∈Ω.
It is no loss of generality to assume that the subsequence {fn(n)(x)} of {fn(x)} is itself; thus
(28)fn(x)⟶f(x),n⟶∞,a.e.x∈Ω.
We now prove f(x)∈Lθ,∞)(Ω) and ∥fn-f∥θ,∞)→0, (n→∞). In fact, by (22), for any ε>0, there exists N=N(ε), such that if n>N, then
(29)sup1≤q<∞1qθ(⨍Ω|fn+p(x)-fn(x)|qdx)1/q<ε.
Let p→∞; one has
(30)sup1≤q<∞1qθ(⨍Ω|fn(x)-f(x)|qdx)1/q<ε,n>N.
Hence f(x)∈Lθ,∞)(Ω), and ∥fn(x)-f(x)∥θ,∞)→0, n→∞. This completes the proof of Theorem 6.Definition 7.
The grand Sobolev spaceWθ,∞)(Ω) consists of all functions f belonging to ⋂1≤p<∞‍W1,∞)(Ω) and such that ∇f∈Lθ,∞)(Ω). That is,
(31)Wθ,∞)(Ω)={f∈⋂1≤p<∞‍W1,∞)(Ω):∇f∈Lθ,∞)(Ω)}.This definition will be used in Section4.


## 3. An Alternative Definition ofLθ,∞(Ω)

In this section, we give an alternative definition ofLθ,∞)(Ω) in terms of weak Lebesgue spaces. Let us first recall the definition of weak Lp(0<p<∞) spaces or the Marcinkiewicz spaces, Lweakp(Ω); see [10, Chapter 1, Section 2], [11, Chapter 2, Section 5], or [12, Chapter 2, Section 18].Definition 8.
Let0<p<∞. We say that f∈Lweakp(Ω) if and only if there exists a positive constant k=k(f) such that
(32)f*(t)=|{x∈Ω:|f(x)|>t}|≤ktp
for every t>0, where |E| is the n-dimensional Lebesgue measure of E⊂Rn and f*(t)=|{x∈Ω:|f(x)|>t}| denotes the distribution function of f.Forp>1, we recall that if f∈Lweakp(Ω), then f∈Lq(Ω) for every 1≤q<p, and f∈Lweakp(Ω) if and only if for every measurable set E⊂Ω, the following inequality holds
(33)∫E‍|f(x)|dx≤c|E|(p-1)/p
for some constant c>0.Equation (32) is equivalent to
(34)Mp(f)=[1|Ω|supt>0tpf*(t)]1/p<∞.
Recall also that
(35)∫Ω‍|f(x)|sdx=s∫0∞‍ts-1f*(t)dt<∞.Definition 9.
Forθ≥0, the weak space Lweakθ,∞(Ω) is defined by
(36)Lweakθ,∞(Ω)={f∈⋂1≤p<∞‍Lweakp(Ω):sup1≤p<∞Mp(f)pθ<∞}.
The following theorem shows thatLweakθ,∞(Ω)=Lθ,∞)(Ω); thus, Lweakθ,∞(Ω) can be regarded as an alternative definition of the space Lθ,∞)(Ω).Theorem 10.
(37)
L
weak
θ
,
∞
(
Ω
)
=
L
θ
,
∞
)
(
Ω
)
.Proof.
We divided the proof into two steps.
Step
1
(
L
weak
θ
,
∞
(
Ω
)
⊂
L
θ
,
∞
)
(
Ω
)
). If 1≤s<p, for each a>0, one can split the integral in the right-hand side of (35) to obtain
(38)∫Ω‍|f|sdx=s∫0a‍ts-1f*(t)dt+s∫a∞‍ts-1f*(t)dx≤|Ω|as+sas-pp-s|Ω|Mpp(f).
The second integral has been estimated by the inequality f*(t)≤|Ω|t-pMpp(f), which is a direct consequence of the definition of the constant Mp(f) (see (34)). Setting a=Mp(f), we arrive at
(39)⨍Ω|f|sdx≤Mps(f)+sp-sMps(f)=pp-sMp(f).
This implies that
(40)1sθ(⨍Ω|f|sdx)1/s≤1sθ(pp-s)1/sMp(f).
Therefore
(41)sup1≤s<∞1sθ(⨍Ω|f|sdx)1/s=max{sup1≤s<21sθ(⨍Ω|f|sdx)1/s,sup2≤s<∞1sθ(⨍Ω|f|sdx)1/s}≤max{∥f∥2,sup2≤s=p-1<∞1sθ(⨍Ω|f|sdx)1/s}≤max{∥f∥2,sup2≤s<∞1sθ(s+1)1/sMs+1(f)}≤max{∥f∥2,4sup1≤s<∞Ms(f)sθ}<∞,
here we have used (40) and the definition of Lweak∞(Ω).
Step
2
(
L
θ
,
∞
)
(
Ω
)
⊂
L
weak
∞
(
Ω
)
). Since for any t>0,
(42)tpf*(t)=tp∫{x∈Ω:|f(x)|>t}‍dx≤∫{x∈Ω:|f(x)|>t}‍|f|pdx≤∫Ω‍|f|pdx,
then
(43)supt>0tpf*(t)≤∫Ω‍|f|pdx.
This implies that
(44)Mp(f)=[1|Ω|supt>0tpf*(t)]1/p≤(⨍Ω|f|pdx)1/p.
Thus
(45)sup1≤p<∞Mp(f)pθ≤sup1≤p<∞1pθ(⨍Ω|f|pdx)1/p<∞.
The proof of Theorem 10 has been completed.


## 4. An Application

In this section, we give an application of the spaceLθ,∞)(Ω) to monotonicity property of very weak solutions of the 𝒜-harmonic equation
(46)div𝒜(x,∇u(x))=0,
where 𝒜:Ω×Rn→Rn is a mapping satisfying the following assumptions:(1)
the mappingx↦𝒜(x,ξ) is measurable for all ξ∈Rn,(2)
the mappingξ↦𝒜(x,ξ) is continuous for a.e. x∈Rn, for all ξ∈Rn, and a.e. x∈Rn,(3)
(47)〈𝒜(x,ξ),ξ〉≥γ(x)|ξ|p,(4)
(48)|𝒜(x,ξ)|≤τ(x)|ξ|p-1,where 1<p<∞, 0<γ(x)≤τ(x)<∞, a.e. Ω.Conditions (1) and (2) insure that the composed mappingx↦𝒜(x,g(x)) is measurable whenever g is measurable. The degenerate ellipticity of the equation is described by condition (3). Finally, condition (4) guarantees that for any 0≤θ<∞ and any ε>0, 𝒜(x,∇u) can be integrated for u∈Wθ,p(Ω) against functions in W1,(p-ε)/(1-pε)(Ω) with compact support.Definition 11.
A functionu∈Wloc1,r(Ω), max{1,p-1}<r≤p, is called a very weak solution of (46) if
(49)∫Ω‍〈𝒜(x,∇u(x)),∇φ(x)〉dx=0
for all φ∈W01,r/(r-p+1)(Ω).A fruitful idea in dealing with the continuity properties of Sobolev functions is the notion of monotonicity. In one dimension a functionu:Ω→R is monotone if it satisfies both a maximum and minimum principle on every subinterval. Equivalently, we have the oscillation bounds oscIu≤osc∂Iu for every interval I⊂Ω. The definition of monotonicity in higher dimensions closely follows this observation.A continuous functionu:Ω→Rn defined in a domain Ω⊂Rn is monotone if
(50)oscBu≤osc∂Bu
for every ball B⊂Rn. This definition in fact goes back to Lebesgue [13] in 1973 where he first showed the relevance of the notion of monotonicity in the study of elliptic PDEs in the plane. In order to handle very weak solutions of 𝒜-harmonic equation, we need to extend this concept, dropping the assumption of continuity. The following definition can be found in [14]; see also [6, 7].Definition 12.
A real-valued functionu∈Wloc1,1(Ω) is said to be weakly monotone if for every ball B⊂Ω and all constants m≤M such that
(51)|M-u|-|u-m|+2u-m-M∈W01,1(B),
we have
(52)m≤u(x)≤M
for almost every x∈B.For continuous functions (51) holds if and only if m≤u(x)≤M on ∂B. Then (52) says we want the same condition in B, that is, the maximum and minimum principles.Manfredi's paper [14] should be mentioned as the beginning of the systematic study of weakly monotone functions. Koskela et al. obtained in [15] that 𝒜-harmonic functions are weakly monotone. In [16], the first author obtained a result which states that very weak solutions u∈Wloc1,p-ε(Ω) of the 𝒜-harmonic equation are weakly monotone provided that ε is small enough. The objective of this section is to extend the operator 𝒜 to spaces slightly larger than Lp(Ω).Theorem 13.
Letγ(x)>0, a.e. Ω, τ(x)∈Lθ1,∞)(Ω). If u∈Wθ2,p)(Ω) is a very weak solution to (46), then it is weakly monotone in Ω provided that θ1+θ2<1.Proof.
For any ballB⊂Ω and 0<ε<1, let
(53)ψ=(u-M)+-(m-u)+∈W01,p-ε(B).
It is obvious that
(54)∇ψ={0,form≤u(x)≤M,∇u,otherwise,say,onasetE⊂B.
Consider the Hodge decomposition (see [6]),
(55)|∇ψ|-pε∇ψ=∇φ+h.
The following estimate holds
(56)∥h∥(p-ε)/(1-pε)≤Cε∥∇ψ∥p-ε1-pε.
Definition 11 with φ acting as a test function yields
(57)∫E‍〈𝒜(x,∇u),|∇u|-pε∇u〉dx=∫E‍〈𝒜(x,∇u),h〉dx.
Hölder's inequality together with the conditions (3) and (4) and equations (56) and (57) yields
(58)∫E‍γ(x)|∇u|p(1-ε)dx≤∫E‍〈𝒜(x,∇u),|∇u|-pε∇u〉dx=∫E‍〈𝒜(x,∇u),h〉dx≤∫E‍τ(x)|∇u|p-1|h|dx≤∥τ∥(p-ε)/(p-1)ε∥∇u∥p-εp-1∥h∥(p-ε)/(1-pε)≤Cε∥τ∥(p-ε)/(p-1)ε∥∇u∥p-εp(1-ε)=C|E|ε·ε-θ2(1-ε)[p-ε(p-1)ε]θ1[(p-1)εp-ε]θ1×(⨍E|τ|(p-ε)/(p-1)εdx)(p-1)ε/(p-ε)×εθ2(1-ε)(⨍E|∇u|p-ε)p(1-ε)/(p-ε).
The condition τ∈Lθ1,∞)(Ω) implies that
(59)limε→0+[(p-1)εp-ε]θ1(⨍E|τ|(p-ε)/(p-1)εdx)(p-1)ε/(p-ε)≤∥τ∥θ1,∞)<∞.
Since u∈Wθ2,p)(Ω), then
(60)limε→0+εθ2(1-ε)(⨍E|∇u|p-ε)p(1-ε)/(p-ε)≤∥∇u∥θ2,p)p<∞.
By θ1+θ2<1, we have
(61)limε→0+ε·ε-θ2(1-ε)[p-ε(p-1)ε]θ1=(pp-1)θ1limε→0+ε1-θ2(1-ε)-θ1=0.
Combining (58)–(61) and taking into account the assumption γ(x)>0, a.e. Ω, we arrive at ∇u=0, a.e. E. This implies that (u-M)+-(m-u)+ vanishes a.e. in B, and thus (u-M)+-(m-u)+ must be the zero function in B, completing the proof of Theorem 13.Remark 14.
We remark that the result in Theorem13 is a generalization of a result due to Moscariello; see [17, Corollary 4.1].


## 5. A Weighted Version

A weight is a locally integrable function onRn which takes values in (0,∞) almost everywhere. For a weight w and a measurable set E, we define w(E)=∫E‍w(x)dx and the Lebesgue measure of E by |E|. The weighted Lebesgue spaces with respect to the measure w(x)dx are denoted by Lwp with 0<p<∞. Given a weight w, we say that w satisfies the doubling condition if there exists a constant C>0 such that for any cube Q, we have w(2Q)≤Cw(Q), where 2Q denotes the cube with the same center as Q whose side length is 2 times that of Q. When w satisfies this condition, we denote w∈Δ2, for short.A weight functionw is in the Muckenhoupt class Ap with 1<p<∞ if there exists C>1 such that for any cube Q(62)(⨍Qw(x)dx)(⨍Qw(x)1-p′dx)p-1≤C,
where 1/p+1/p′=1. We define A∞=⋃1<p<∞‍Ap.Letw be a weight. The Hardy-Littlewood maximal operator with respect to the measure w(x)dx is defined by
(63)Mwf(x)=supQ∋x1w(Q)∫Q‍|f(x)|w(x)dx.
We say that T is a Calderón-Zygmund operator if there exists a function K which satisfies the following conditions:
(64)Tf(x)=p.v.∫Rn‍K(x-y)f(y)dy,|K(x)|≤CK|x|n,|∇K(x)|≤CK|x|n+1,x≠0.Forw a weight and 0≤θ<∞, we define the space Lwθ,∞)(Ω) as follows:
(65)Lwθ,∞)(Ω)={f(x)∈⋂1<p<∞‍Lwp(Ω):∥f∥Lwθ,∞)(Ω)<∞},
where
(66)∥f∥Lwθ,∞)(Ω)=sup1<p<∞1pθ(1w(Ω)∫Ω‍|f(x)|pw(x)dx)1/p.
The following lemma comes from [18].Lemma 15.
If1<p<∞ and w∈Δ2, then the operator Mw is bounded on Lwp(Ω).Theorem 16.
The operatorMw is bounded on Lwθ,∞)(Ω) for 0≤θ<∞ and w∈Δ2.Proof.
By Lemma15, since for 1<p<∞ and w∈Δ2, the operator Mw is bounded on Lwp(Ω), then
(67)(∫Ω‍|Mwf(x)|pw(x)dx)1/p≤C(∫Ω‍|f(x)|pw(x)dx)1/p.
This implies that
(68)∥Mwf∥Lwθ,∞)(Ω)=sup1<p<∞1pθ(1w(Ω)∫Ω‍|Mwf(x)|pw(x)dx)1/p≤Csup1<p<∞1pθ(1w(Ω)∫Ω‍|f(x)|pw(x)dx)1/p=∥f∥Lwθ,∞)(Ω),
completing the proof of Theorem 16.The following lemma can be found in [19].Lemma 17.
Ifw∈A∞, then there exists q∈(1,∞) such that w∈Aq.The following lemma can be found in [20, 21].Lemma 18.
If1<p<∞ and w∈Ap, then a Calderón-Zygmund operator T is bounded on Lwp(Ω).Theorem 19.
A Calderón-Zygmund operatorT is bounded on Lwθ,∞)(Ω) for 0≤θ<∞ and w∈A∞.Proof.
Byw∈A∞ and Lemma 17, one has w∈Aq for some q∈(1,∞). For 1<p<q<∞, Hölder's inequality yields
(69)∫Ω|Tf(x)|p‍w(x)dx=∫Ω|Tf(x)|p‍‍w(x)p/qw(x)(q-p)/pdx≤(∫Ω‍|Tf(x)|qw(x)dx)p/q(∫Ω‍w(x)dx)(q-p)/q.
Thus,
(70)1pθ(1w(Ω)∫Ω‍|Tf(x)|pw(x)dx)1/p≤1pθ(1w(Ω)∫Ω‍|Tf(x)|qw(x)dx)p/q×(1w(Ω)∫Ω‍w(x)dx)(q-p)/q=1pθ(1w(Ω)∫Ω‍|Tf(x)|qw(x)dx)p/q.
Lemma 18 yields
(71)∥Tf∥Lwθ,∞)(Ω)=max{sup1<p<q1pθ(1w(Ω)∫Ω‍|Tf(x)|pw(x)dx)1/p,supq≤p<∞1pθ(1w(Ω)∫Ω‍|Tf(x)|pw(x)dx)1/p}=max{sup1<p<q1pθ(1w(Ω)∫Ω‍|Tf(x)|qw(x)dx)p/q,supq≤p<∞1pθ(1w(Ω)∫Ω‍|Tf(x)|pw(x)dx)1/p}≤max{sup1<p<q(qp)θ,1}×supq≤p<∞1pθ(1w(Ω)∫Ω‍|Tf(x)|pw(x)dx)1/p≤Cqθsupq≤p<∞1pθ(1w(Ω)∫Ω‍|Tf(x)|pw(x)dx)1/p≤Cqθ∥f∥Lwθ,∞)(Ω),
as desired.



---

*Source: 476309-2013-06-17.xml*