R E S E A R C H
Open Access
Global Poincaré inequalities for the
composition of the sharp maximal operator
and Green’s operator with Orlicz norms
Bao Gejun
1and Yi Ling
2**Correspondence: [email protected] 2Department of Mathematical
Science, Delaware State University, Dover, 19901, USA
Full list of author information is available at the end of the article
Abstract
In this paper, we establish the global Poincaré-type inequalities for the composition of the sharp maximal operator and Green’s operator with Orlicz norm.
Keywords: Poincaré-type inequalities; Orlicz norm; sharp maximal operator; Green’s operator
1 Introduction
TheLp-theory of solutions of the homogeneousA-harmonic equationdA(x,du) = for
differential formsuhas been very well developed in recent years. ManyLp-norm
esti-mates and inequalities, including the Poincaré inequalities, for solution of the homoge-neousA-harmonic equation have been established; see [, ]. The Poincaré inequalities for differential forms is an important tool in analysis and related fields, including partial differential equations and potential theory. However, the study of the nonhomogeneous A-harmonic equationsdA(x,du) =B(x,du) has just begun [–]. In this paper, we focus
on a class of differential forms satisfying the well-known nonhomogeneousA-harmonic equationdA(x,du) =B(x,du).
Let us first introduce some necessary notation and terminology. will refer to a bounded, convex domain inRnunless otherwise stated andBis a ball inRn,n≥. We
useσBto denote the ball with the same center asBand withdiam(σB) =σdiam(B),σ> . We do not distinguish the balls from cubes in this paper. We use |E|to denote then -dimensional Lebesgue measure of the setE⊆Rn. We saywis a weight ifw∈L
loc(Rn) and
w> a.e. For a functionu, we denote the average ofuoverBby
uB=
|B|
B
u dx,
where|B|is the volume ofBand theμ-average ofuoverBby
uB,μ=
μ(B)
B
u dμ.
Let∧l=∧l(Rn) be the set of alll-forms inRn, letD(,∧l) be the space of all differ-entiall-forms on, and letLp(,∧l) be thel-formsu(x) =
IuI(x)dxI onsatisfying
|uI|pdx<∞ for all orderedl-tuples I, l= , , . . . ,n. We denote the exterior
deriva-tive byd:D(,∧l)→D(,∧l+) forl= , , . . . ,n– , and define the Hodge star
opera-tor:∧k→ ∧n–k as follows. Ifu=u
IdxI,i<i<· · ·<ik, is a differentialk-form, then
u= (–)(I)u
IdxJ, whereI= (i,i, . . . ,ik),J={, , . . . ,n}–I, and
(I) =k(k+)+kj=ij.
The Hodge codifferential operator
d:D,∧l+→D,∧l
is given byd= (–)nl+donD(,∧l+),l= , , . . . ,n– . We write
u s,=
|u|sdx /s
.
The well-known nonhomogeneousA-harmonic equation is
dA(x,du) =B(x,du), ()
whereA:× ∧l(Rn)→ ∧l(Rn) andB:× ∧l(Rn)→ ∧l–(Rn) satisfy the conditions:
A(x,ξ) ≤a|ξ|p–, A(x,ξ)·ξ≥ |ξ|p, B(x,ξ) ≤b|ξ|p– ()
for almost every x∈ and allξ ∈ ∧l(Rn). Here,a,b> are constants and <p<∞
is a fixed exponent associated with (). If the operator B= , equation () becomes dA(x,du) = , which is called the (homogeneous)A-harmonic equation. A solution to ()
is an element of the Sobolev spaceWloc,p(,∧l–) such that
A(x,du)·dϕ+B(x,du)·ϕ=
for allϕ∈Wloc,p(,∧l–) with compact support. LetA:× ∧l(Rn)→ ∧l(Rn) be defined
byA(x,ξ) =ξ|ξ|p–withp> . ThenAsatisfies the required conditions anddA(x,du) =
becomes thep-harmonic equation
ddu|du|p–= ()
for differential forms. If uis a function (-form), equation () reduces to the usual p -harmonic equation div(∇u|∇u|p–) = for functions. A remarkable progress has been
made recently in the study of different versions of the harmonic equations, see [] for more details.
LetC∞(,∧l) be the space of smoothl-forms onand
W,∧l=u∈Lloc,∧l:uhas generalized gradient.
The harmonicl-fields are defined by
H,∧l=u∈W,∧l:du=du= ,u∈Lpfor some <p<∞.
The orthogonal complement ofHinLis defined by
Then Green’s operatorGis defined as
G:C∞,∧l→H⊥∩C∞,∧l
by assigningG(u) to be the unique element ofH⊥∩C∞(,∧l) satisfying Poisson’s equation
G(u) =u–H(u), whereHis the harmonic projection operator that mapsC∞(,∧l) onto
Hso thatH(u) is the harmonic part ofu. See [] for more properties of these operators. In harmonic analysis, a fundamental operator is the Hardy-Littlewood maximal opera-tor. The maximal function is a classical tool in harmonic analysis but recently it has been successfully used in studying Sobolev functions and partial differential equations. For any locallyLs-integrable formu(y), we define the Hardy-Littlewood maximal operatorM
sby
Ms(u) =Ms(u)(x) =sup r>
|B(x,r)|
B(x,r)
u(y) sdy
s
, ()
whereB(x,r) is the ball of radiusr, centered atx, ≤s<∞. We writeM(u) =M(u) if
s= . Similarly, for a locallyLs-integrable formu(y), we define the sharp maximal operator
M#
s by
M#
s(u) =M
#
s(u)(x) =sup r>
|B(x,r)|
B(x,r)
u(y) –uB(x,r)
s
dy
s
. ()
Some interesting results about these operators have been established, see [, ] and [] for more details.
The purpose of this paper is to estimate the global Poincaré-type inequalities for the composition of the sharp maximal operator and Green’s operator with Orlicz norm.
2 Definitions and lemmas
We now introduce the following definition and lemmas that will be used in this paper.
Definition We say the weightw(x) satisfies theAr() condition,r> , writew∈Ar()
ifw(x) > a.e., and
sup
B
|B|
B
w dx
|B|
B
w
r–
dx r–
<∞ ()
for any ballB⊂.
Definition A proper subdomain⊂Rnis called aδ-John domain,δ> , if there exists a pointx∈ which can be joined with any other pointx∈ by a continuous curve γ ⊂so that
d(ξ,∂)≥δ|x–ξ|
Lemma [] Eachhas a modified Whitney cover of cubesV={Qi}such that
i
Qi=,
Qi∈V χ
Qi≤N χ
and some N > ,and if Qi∩Qj=∅,then there exists a cube R(this cube need not be a
member ofV)in Qi∩Qj such that Qi∪Qj⊂NR.Moreover,ifisδ-John,then there is
a distinguished cube Q∈V which can be connected with every cube Q∈V by a chain of
cubes Q=Qj,Qj, . . . ,Qjk =Q fromV and such that Q⊂ρQji,i= , , , . . . ,k,for some
ρ=ρ(n,δ).
3 Poincaré inequalities
In this section, we prove the global Poincaré inequalities for the composition of the sharp maximal operator and Green’s operator withLpnorm.
To get our result, we rewrite our Theorem in [] as follows.
Lemma Let u be a smooth differential form satisfying A-harmonic equation ()in a
bounded domain ,let G be Green’s operator,and let Ms be the sharp maximal
oper-ator defined in()with <s≤p,q<∞.Then there exists a constant C,independent of u,
such that
B
M
s
G(u)–MsG(u)B qdμ
/q
≤C(δ,)|B|+n–p+q
σB
|u|pdμ
/p
for all balls B withσB⊂,and a constantσ> ,where the measureμis defined by dμ=
w(x)dx and w(x)∈Ar()with w≥δ> for some r> and a constantδ.
Theorem Let u∈Lt
loc(,∧l),l= , , . . . ,n,be a smooth differential form satisfying
A-harmonic equation(),let G be Green’s operator,and letMsbe the sharp maximal operator
defined in()with <s<t<∞.Then there exists a constant C(n,t,δ,N,),independent
of u,such that
M
s
G(u)–M
s
G(u)Q
t
dμ
/t
≤C(n,t,δ,N,)
|u|tdμ
/t
()
for any bounded and convexδ-John domain⊂Rn,where the fixed cube Q
⊂,the
constant N> appeared in Lemma,and the measureμis defined by dμ=w(x)dx and
w(x)∈Ar()with w≥δ> for some r> and a constantδ.
Proof First, we use Lemma for the bounded and convexδ-John domain. There is a
modified Whitney cover of cubesV={Qi}forsuch that=
Qi, and
Qi∈Vχ
Qi≤
Nχfor someN> . Moreover, there is a distinguished cubeQ∈V which can be
such thatQ⊂ρQji,i= , , , . . . ,k, for someρ=ρ(n,δ). Then, by the elementary
inequal-ity (a+b)t≤t(|a|t+|b|t),t≥, we have
M
s
G(u)–MsG(u)Q
t
dμ
/t
≤
Qi∈V
t Qi M s
G(u)–M
s
G(u)Q
i
t
dμ
+ t
Qi
M
s
G(u)Q
i–
M
s
G(u)Q
t
dμ
/t
≤C(t)
Qi∈V
Qi
M
s
G(u)–MsG(u)Q
i
t
dμ
/t
+
Qi∈V
Qi
M
s
G(u)Q
i–
M
s
G(u)Q
t
dμ
/t
. ()
The first sum in () can be estimated by using Lemma .
Qi∈V
Qi
M
s
G(u)–M
s
G(u)Q
i
t
dμ
≤C(n,t,δ,)
Qi∈V
ρiQi
|u|tdμ
≤C(n,t,δ,)
Qi∈V
|u|tdμ
≤C(n,t,N,δ,)
|u|tdμ, ()
where the measureμis defined bydμ=w(x)dxandw(x)∈Ar() withw≥δ> for some
r > and a constantδ.
To estimate the second sum in (), we need to use the property ofδ-John domain. Fix a cubeQi∈Vand letQ=Qj,Qj, . . . ,Qjk=Qibe the chain in Lemma . Then we have
M
s
G(u)Q
i–
M
s
G(u)Q
≤ k– i= M s
G(u)Q
ji–
M
s
G(u)Q
ji+
. ()
The chain{Qji}also has the property that for eachi,i= , , . . . ,k– ,Qji∩Qji+=∅. Thus,
there exists a cubeDisuch thatDi⊂Qji∩Qji+andQji∪Qji+⊂NDi,N> . So,
max{|Qji|,|Qji+|}
|Qji∩Qji+|
≤max{|Qji|,|Qji+|}
|Di| ≤
N. ()
Note that
μ(Q) =
Q
dμ=
Q
w(x)dx≥
Q
By (), () and Lemma , we have
M
s
G(u)Q
ji–
M
s
G(u)Q
ji+
t
=
μ(Qji∩Qji+)
Qji∩Qji+
M
s
G(u)Q
ji–
M
s
G(u)Q
ji+
t
dμ
≤
δ|Qji∩Qji+|
Qji∩Qji+
M
s
G(u)Q
ji–
M
s
G(u)Q
ji+
t
dμ
≤ N
δmax{|Qji|,|Qji+|}
Qji∩Qji+
M
s
G(u)Q
ji–
M
s
G(u)Q
ji+
t
dμ
≤C(n,t,δ,N,)
i+
k=i
|Qjk|
Qjk
M
s
G(u)–M
s
G(u)Q
jk
t
dμ
≤C(n,t,δ,N,)
i+
k=i
|Qjk| +n
|Qjk|
σjkQjk |
u|tdμ
=C(n,t,δ,N,)
i+
k=i
|Qjk| n
σjkQjk|
u|tdμ
≤C(n,t,δ,N,)
i+
k=i
||n
|u|tdμ
≤C(n,t,δ,N,)
Qi∈V
|u|tdμ
≤C(n,t,δ,N,)
|u|tdμ. ()
Then, by (), () and the elementary inequality|Mi=ti|s≤Ms–
M
i=|ti|s, we finally
ob-tain
Qi∈V
Qi
M
s
G(u)Q
i–
M
s
G(u)Q
t
dμ
≤C(n,t,δ,N,)
Qi∈V
Qi
|u|tdμ
dμ
=C(n,t,δ,N,)
Qi∈V
Qi
dμ
|u|tdμ
≤C(n,t,δ,N,)
dμ
|u|tdμ
=C(n,t,δ,N,)μ()
|u|tdμ
=C(n,t,δ,N,)
|u|tdμ. ()
4 Poincaré inequality with Orlicz norm
In this section, we give a global Poincaré inequality with Orlicz norm for the composition of the sharp maximal operator and Green’s operator.
Definition Letϕbe a continuously increasing convex function on [,∞) withϕ() = , and letbe a domain withμ() <∞. Ifuis a measurable function in, then we define the Orlicz norm ofuby
u L(ϕ,,μ)=inf
k> :
μ()
ϕ
| u(x)|
k
dμ≤
. ()
A continuously increasing functionψ: [,∞)→[,∞) withϕ() = is called an Orlicz function. A convex Orlicz functionϕis often called a Young function.
In [], Buckley and Koskela gave the following class of functions.
Definition We say a Young functionϕlies in the classG(p,q,C), ≤p<q<∞,C≥, if (i) /C≤ϕ(t/p)/g(t)≤Cand (ii) /C≤ϕ(t/q)/h(t)≤Cfor allt> , wheregis a convex
increasing function andhis a concave increasing function on [,∞).
From [] and [], we know that the classG(p,q,C) contains some very interesting func-tions, such asϕ(t) =tpandϕ(t) =tplogα
+(t),p≥,α∈R, and each ofϕ,gandhis doubling
in the sense that its values attand tare uniformly comparable for allt> , and the con-sequent fact that
Ctq≤h–
ϕ(t)≤Ctq, Ctp≤g–
ϕ(t)≤Ctp, ()
whereCandCare constants.
Now, we are ready to give our another global Poincaré inequality with Orlicz norm.
Theorem Letϕbe a Young function in the class G(p,q,C), ≤p<q<∞,C≥,let u∈
Ltloc(,∧l),l= , , . . . ,n,be a smooth differential form satisfying A-harmonic equation()
in,let G be Green’s operator,and letMsbe the sharp maximal operator defined in()
with <s≤t<∞.Then there exists a constant C,independent of u,such that
M
s
G(u)–MsG(u)Q
L(ϕ,,μ)≤C u L(ϕ,,μ)
for any bounded and convexδ-John domain⊂Rnwithμ() <∞,where the fixed cube
Q⊂appeared in Lemma,and the measureμis defined by dμ=w(x)dx and w(x)∈
Ar()with w≥δ> for some r> and a constantδ.
Proof Letg,hbe the functions in theG(p,q,C) condition. Note thatϕis an increasing
function. Using Theorem , (i) in Definition , and Jensen’s inequality, we obtain
ϕ
k
M
s
G(u)–MsG(u)Q
t
dμ
/t
≤ϕ
kC
|u|tdμ
=ϕ
ktC
t
|u|tdμ
/t
≤Cg
ktC
t
|u|tdμ
=Cg
ktC
t
|u|tdμ
≤C
g
ktC
t
|u|t
dμ. ()
Again, from (i) in Definition , we have
g(x)≤Cϕ
xt.
Thus, we obtain
g
ktC
t
|u|t
dμ≤C
ϕ
kC|u|
dμ. ()
Combining () and () yields
ϕ k M s
G(u)–M
s
G(u)Q
t
dμ
/t
≤C
ϕ
kC|u|
dμ
=C
ϕ
kC|u|
dμ. ()
Now, using Jensen’s inequality forh–, () and (ii) in Definition , and noticing thatϕis doubling, we see
ϕ
|M
s(G(u)) –Ms(G(u))Q|
k dμ =h h– ϕ |M
s(G(u)) –Ms(G(u))Q|
k dμ ≤h h– ϕ |M
s(G(u)) –Ms(G(u))Q|
k dμ ≤h C |M
s(G(u)) –M
s(G(u))Q|
k
t
dμ
≤Cϕ
C
|M
s(G(u)) –Ms(G(u))Q|
k
t
dμ
t
=Cϕ
k C M s
G(u)–M
s
G(u)Q
t
dμ
t
≤Cϕ
k M s
G(u)–MsG(u)Q
Substituting () into () and using the fact thatϕis doubling, we get
ϕ
|M
s(G(u)) –Ms(G(u))Q|
k
dμ ()
≤C
ϕ
kC|u|
dμ
≤C
ϕ
k|u|
dμ. ()
Therefore, from Definition , we have
M
s
G(u)–MsG(u)Q
L(ϕ,,μ)≤C u L(ϕ,,μ).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors completed the paper together. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, P.R. China.2Department of Mathematical
Science, Delaware State University, Dover, 19901, USA.
Acknowledgements
The first author was supported by NSF of P.R. China (No. 11071048).
Received: 23 April 2013 Accepted: 8 October 2013 Published:11 Nov 2013
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