Weighted Hardy spaces associated with elliptic operators. Part II: Characterizations of $H^1_L(w)$

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by  José María Martell

WEIGHTED HARDY SPACES ASSOCIATED WITH ELLIPTIC OPERATORS.
PART II: CHARACTERIZATIONS OF H1(w)
L
JOSÉMARÍAMARTELLANDCRUZPRISUELOS-ARRIBAS
Abstract. Given a Muckenhoupt weight w and a second order divergence form elliptic operator
7 L, we consider different versions of the weighted Hardy space H1(w) defined by conical square
L
1 functions and non-tangential maximal functions associated with the heat and Poisson semigroups
0 generatedby L. Weshow thatallofthemareisomorphicandalsothat H1(w)admitsamolecular
L
2 characterization. One of the advantages of our methods is that our assumptions extend naturally

b theunweightedtheorydevelopedbyS.HofmannandS.Mayborodain[19]andwecanimmediately
e recovertheunweightedcase.Someofourtoolsconsistinestablishingweightednorminequalitiesfor
F thenon-tangentialmaximalfunctions,aswellascomparingthemwithsomeconicalsquarefunctions
inweightedLebesguespaces.
6

]
A
C
. Contents
h
t
a
1. Introduction 2
m
2. Preliminaries 3
[

2.1. Muckenhoupt weights 3
3
v 2.2. Ellipticoperators 5
0
2.3. Off-diagonalestimates 5
2
9 2.4. Conicalsquarefunctions andnon-tangential maximalfunctions 6
0
0 3. Definitionsandmainresults 7
.
1 3.1. MolecularweightedHardyspaces 7
0
3.2. WeightedHardyspacesassociated withoperators 8
7
1 3.3. Mainresults 8
:
v 4. Auxiliaryresults 9
i
X 5. Characterization ofthe weighted Hardy spaces defined bysquare functions associated
r withtheheatsemigroup 11
a
5.1. ProofofProposition 5.2 13
5.2. ProofofProposition 5.1,part(a) 16
5.3. ProofofProposition 5.1,part(b) 22
Date: January10,2017.Revised:February1,2017.
2010MathematicsSubjectClassification. 42B30,35J15,42B37,42B25,47D06,47G10.
Key words and phrases. Hardy spaces, second order divergence form elliptic operators, heat and Poisson semi-
groups, conical squarefunctions, non-tangential maximal functions, moleculardecomposition, Muckenhoupt weights,
off-diagonalestimates.
TheresearchleadingtotheseresultshasreceivedfundingfromtheEuropeanResearchCouncilundertheEuropean
Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC agreement no. 615112 HAPDEGMT. Both authors
acknowledgefinancialsupportfromtheSpanishMinistryofEconomyandCompetitiveness,throughthe“SeveroOchoa
ProgrammeforCentresofExcellenceinR&D”(SEV-2015-0554). BothauthorswouldliketothankP.Auscherforhis
usefulcommentsandsuggestions.
1
2 JOSÉMARÍAMARTELLANDCRUZPRISUELOS-ARRIBAS
6. Characterization ofthe weighted Hardy spaces defined bysquare functions associated
withthePoissonsemigroup 23
6.1. ProofofProposition 6.1,part(a) 24
6.2. ProofofProposition 6.1,part(b). 26
6.3. ProofofProposition 6.1,part(c). 26
7. Non-tangential maximalfunctions 27
7.1. ProofofProposition 7.1,part(a) 27
7.2. ProofofProposition 7.1,part(b) 27
7.3. ProofofProposition 7.2 30
7.4. Characterization oftheweightedHardyspacesassociated with and 37
H P
N N
References 42
1. Introduction
This is the second of a series of three papers whose aim is to study and develop a theory for
weightedHardyspacesassociatedwithdifferentoperators. ThestudyofHardyspacesbeganinthe
early 1900s in the context of Fourier series and complex analysis in one variable. It was not until
1960whenthetheoryinRnstarteddevelopingbyE.M.SteinandG.Weiss([25]). Afewyearslater
R.R. Coifman in [12] and R.H. Latter in [22] gave an atomic decomposition of the Hardy spaces
Hp, 0 < p 1. This atomic decomposition turns out to be a very important tool when studying
≤
theboundedness ofsomesingularintegraloperators, sinceinmostcaseschecking theactionofthe
operator inquestion onthese simpler elements (atoms) suffices toconclude its boundedness inthe
corresponding Hardyspace.
AnotherstageintheprogressofthetheoryofHardyspaceswasdonebyJ.García-Cuervain[15]
(see also [26]) when he considered Rn with the measure given by a Muckenhoupt weight. These
spaces were called weighted Hardy spaces, and among other contributions, he also characterized
themusinganatomicdecomposition.
In general, the development of the theory of Hardy spaces has contributed to give us a better
understanding of some other topics as in the theory of singular integrals operators, maximal func-
tions, multiplier operators, etc. However, there are some operators that escape from the theory
of these classical Hardy spaces. These are, for example, the operators associated with a second
order divergence form elliptic operator L, such as the conical square functions and non-tangential
maximal functions defined by the heat and Poisson semigroups generated by the operator L, (see
(2.14)-(2.19)and(2.20)–(2.21)fortheprecisedefinitions oftheseoperators).
Thetheory of Hardy spaces associated with elliptic operators L was initiated in an unpublished
workbyP.Auscher,X.T.DuongandA.McIntosh[3]. P.AuscherandE.Russin[9]consideredthe
caseonwhichtheheatkernelassociatedwithLissmoothandsatisfiespointwiseGaussianbounds,
this occurs for instance for real symmetric operators. There, among other things, it was shown
thatthecorresponding Hardyspaceassociated with Lagreeswiththeclassical Hardyspace. Inthe
settingofRiemannianmanifoldssatisfyingthedoublingvolumeproperty, Hardyspacesassociated
with the Laplace-Beltrami operator are introduced in [8] by P. Auscher, A. McIntosh and E. Russ
and it is shown that they admit several characterizations. Simultaneously, in the Euclidean set-
ting, thestudy ofHardyspaces related totheconical squarefunctions andnon-tangential maximal
functions associated with the heat and Poisson semigroups generated by divergence form elliptic
operators L was taken by S. Hofmann and S. Mayboroda in [19], for p = 1. The new point was
WEIGHTEDHARDYSPACESASSOCIATEDWITHELLIPTICOPERATORS 3
that only a form of decay weaker than pointwise bounds and satisfied in many occurrences was
enoughtodevelopatheory. ThiswasfollowedlateronbyasecondarticleofS.Hofmann,S.May-
boroda, and A.McIntosh [20], for ageneral pand simultaneously byan article ofR. Jiang and D.
Yang[21]. AnaturallineofstudyinthecontextofthesenewHardyspacesisthedevelopmentofa
weightedtheoryforthem,asJ.García-Cuervadidintheclassicalsetting. Someinterestingprogress
has been done in this regard by T.A. Bui, J. Cao, L.D.Ky, D. Yang, and S. Yang in [10, 11]. The
results obtained in [11] in the particular case ϕ(x,t) := tw(x), where w is a Muckenhoupt weight,
give characterizations of the weighted Hardy spaces that, however, only recover part of the results
obtained intheunweighted casebysimplytakingw = 1.
In this paper we take a further step, and present a different approach to the theory of weighted
Hardy spaces H1(w) (the general case Hp(w) will be treated in the forthcoming paper [24]) as-
L L
sociated with a second order divergence form elliptic operator, which naturally generalizes the
unweighted setting developed in [19]. Wedefine weighted Hardy spaces associated with the coni-
calsquarefunctions considered in(2.14)–(2.19)whicharewrittenintermsoftheheatandPoisson
semigroups generated by the elliptic operator. Also, we use non-tangential maximal functions as
defined in (2.20)–(2.21). We show that the corresponding spaces are all isomorphic and admit a
molecular characterization. This isparticularly useful to prove different properties ofthese spaces
ashappens intheclassical settingandinthecontext ofsecond orderdivergence formelliptic oper-
atorsconsidered in[19].
Some of the ingredients that are crucial in the present work are taken from the first part of this
seriesofpapers[23],wherewealreadyobtained optimalrangesfortheweightednorminequalities
satisfied by the heat and Poisson conical square functions associated with the elliptic operator.
Here, we need to obtain analogous results for the non-tangential maximal functions associated
with the heat and Poisson semigroups (see Section 7). All these weighted norm inequalities for
the conical square functions and the non-tangential maximal functions, along with the important
fact that our molecules belong naturally to weighted Lebesgue spaces, allow us to impose natural
conditions that in particular lead to fully recover the results obtained in [19] by simply taking the
weight identically one. It is relevant to note that in [10, 11] their molecules belong to unweighted
Lebesgue spaces and also their ranges of boundedness of the conical square functions are smaller.
Thismakestheirhypothesissomehowstronger(althoughsometimestheycannotbecomparedwith
ours) and, despite making a very big effort to present a very general theory, the unweighted case
doesnotfollowimmediatelyfromtheirwork.
The plan of this paper is as follows. In the next section we present some preliminaries con-
cerning Muckenhoupt weights, elliptic operators and introduce the conical square functions and
non-tangential maximal functions. In Section 3 we define the different versions of the weighted
Hardyspaces andstate our mainresults. Section 4contains someauxiliary results. Sections 5and
6dealwiththecharacterization oftheweighted Hardy spaces definedintermsofsquare functions
associated with the heat and Poisson semigroups, respectively. Finally, in Section 7 we study the
non-tangential maximalfunctions andtheweightedHardyspacesassociated withthem.
2. Preliminaries
2.1. Muckenhouptweights. Wewillwork with Muckenhoupt weights w, which are locally inte-
grablepositivefunctions. Wesaythatw A if,foreveryball B Rn,thereholds
1
∈ ⊂
w(x)dx Cw(y), fora.e. y B,
− ≤ ∈
B
Z
4 JOSÉMARÍAMARTELLANDCRUZPRISUELOS-ARRIBAS
or, equivalently, w Cw a.e. where denotes the uncentered Hardy-Littlewood maximal
u u
M ≤ M
operator overballsinRn. Foreach1 < p< ,wesaythatw A ifitsatisfies
p
∞ ∈
p 1
w(x)dx w(x)1−p′dx − C, B Rn.
− − ≤ ∀ ⊂
(cid:18)ZB (cid:19)(cid:18)ZB (cid:19)
The reverse Hölder classes are defined as follows: for each 1 < s < , w RH if, for every ball
s
∞ ∈
B Rn,wehave
⊂ 1
s
w(x)sdx C w(x)dx.
− ≤ −
(cid:18)ZB (cid:19) ZB
For s= ,w RH providedthatthereexistsaconstantC suchthatforeveryball B Rn
∞ ∈ ∞ ⊂
w(y) C w(x)dx, fora.e. y B.
≤ − ∈
B
Z
Notice that we have excluded the case q = 1 since the class RH consists of all the weights, and
1
thatisthewayRH isunderstood inwhatfollows.
1
Wesum upsome ofthe properties ofthese classes inthe following result, see forinstance [16],
[14],or[17].
Proposition 2.1.
(i) A A A for1 p q< .
1 p q
⊂ ⊂ ≤ ≤ ∞
(ii) RH RH RH for1< p q .
q p
∞ ⊂ ⊂ ≤ ≤ ∞
(iii) Ifw A ,1 < p < ,thenthereexists1 < q < psuchthatw A .
p q
∈ ∞ ∈
(iv) Ifw RH ,1 < s< ,thenthereexists s < r < suchthatw RH .
s r
∈ ∞ ∞ ∈
(v) A = A = RH .
p s
∞
1 p< 1<s
≤[∞ [≤∞
(vi) If1 < p< ,w Ap ifandonlyifw1−p′ Ap .
∞ ∈ ∈ ′
(vii) Forevery1 < p< ,w A ifandonlyif isboundedonLp(w). Also,w A ifandonly
p 1
∞ ∈ M ∈
if isboundedfrom L1(w)intoL1, (w),where denotesthecenteredHardy-Littlewood
∞
M M
maximaloperator.
Foraweightw A ,define
∈ ∞
(2.2) r := inf 1 r < :w A , s := inf 1 s< : w RH .
w r w s
{ ≤ ∞ ∈ } { ≤ ∞ ∈ ′}
Notice that according to our definition s is the conjugated exponent of the one defined in [5,
w
Lemma4.1]. Given0 p < q ,w A ,andaccording to[5,Lemma4.1]wehave
0 0
≤ ≤ ∞ ∈ ∞
q
(2.3) Ww(p0,q0):= (cid:26)p: p0 < p< q0,w ∈ App0 ∩RH(cid:16)qp0(cid:17)′(cid:27)= (cid:18)p0rw, sw0(cid:19).
If p = 0 and q < it is understood that the only condition that stays is w RH . Analo-
0 0 ∞ ∈ q0 ′
(cid:16) p(cid:17)
gously, if0 < p0 andq0 = theonlyassumption isw A p . Finally w(0, )= (0, ).
∞ ∈ p0 W ∞ ∞
We recall some properties of Muckenhoupt weights. Let w be a weight in A , if w A ,
r
∞ ∈
1 r < ,foreveryball Bandeverymeasurable setE B,
≤ ∞ ⊂
r
w(E) E
(2.4) [w] 1 | | .
w(B) ≥ −Ar B
(cid:18)| |(cid:19)
Thisimpliesinparticular thatwisadoubling measure:
(2.5) w(λB) [w] λnrw(B), B, λ > 1.
≤ Ar ∀ ∀
WEIGHTEDHARDYSPACESASSOCIATEDWITHELLIPTICOPERATORS 5
Besides,ifw RH ,1 s< ,
s
∈ ′ ≤ ∞
1
w(E) E s
(2.6) [w] | | .
RH
w(B) ≤ s′ B
(cid:18)| |(cid:19)
2.2. Elliptic operators. Let A be an n n matrix of complex and L -valued coefficients defined
∞
×
on Rn. We assume that this matrix satisfies the following ellipticity (or “accretivity”) condition:
thereexist0< λ Λ < suchthat
≤ ∞
(2.7) λ ξ2 ReA(x)ξ ξ¯, and A(x)ξ ζ¯ Λ ξ ζ,
| | ≤ · | · |≤ | || |
forall ξ,ζ Cn andalmost every x Rn. Wehaveused the notation ξ ζ = ξ ζ + +ξ ζ and
1 1 n n
therefore ξ∈ζ¯ is the usual inner pro∈duct in Cn. Associated with this m·atrix we defi·n·e· the second
·
orderdivergence formelliptic operator
(2.8) Lf = div(A f),
− ∇
which is understood in the standard weak sense as a maximal-accretive operator on L2(Rn) with
domain (L)bymeansofasesquilinear form.
D
As in [1] and [6], we denote respectively by (p (L),p (L)) and (q (L),q (L)) the maximal
+ +
open intervals on which the heat semigroup e tL −and its gradient √−t e tL are uniformly
− t>0 y − t>0
{ } { ∇ }
bounded onLp(Rn):
(2.9) p (L) := inf p (1, ) :sup e t2L < ,
− Lp(Rn) Lp(Rn)
− (cid:26) ∈ ∞ t>0 k k → ∞(cid:27)
(2.10) p (L) := sup p (1, ): sup e t2L < ,
+ − Lp(Rn) Lp(Rn)
(cid:26) ∈ ∞ t>0 k k → ∞(cid:27)
(2.11) q (L):= inf p (1, ) :sup t e t2L < ,
y − Lp(Rn) Lp(Rn)
− (cid:26) ∈ ∞ t>0 k ∇ k → ∞(cid:27)
(2.12) q (L):= sup p (1, ) :sup t e t2L < .
+ y − Lp(Rn) Lp(Rn)
(cid:26) ∈ ∞ t>0 k ∇ k → ∞(cid:27)
From [1] (see also [6]) we know that p (L) = 1 and p (L) = if n = 1,2; and if n 3 then
+
p (L) < 2n and p (L) > 2n. Moreov−er, q (L) = p (L), q (L∞) p (L) (where q (L≥) is the
So−bolev enx+p2onent o+f q (L) an−s2defined below)−, and we−always+have∗ q≤(L+) > 2, with q+(L)∗= if
+ + +
∞
n = 1.
Notethatinplaceofthesemigroup e tL weareusingitsrescaling e t2L . Wedososince
− t>0 − t>0
{ } { }
all the “heat” square functions are written using the latter and also because in the context of the
off-diagonal estimatesdiscussed belowitwillsimplifysomecomputations.
Besides,forevery K N and0< q < letusset
0
∈ ∞
qn
, if (2K +1)q < n,
qK, := n (2K +1)q
∗  −
, if (2K +1)q n.

∞ ≥
Corresponding tothecase K = 0,wewriteq := q0, .
 ∗ ∗
2.3. Off-diagonalestimates. Webrieflyrecall thenotion ofoff-diagonal estimates. Let T be
t t>0
{ }
a family of linear operators and let 1 p q . We say that T satisfies Lp(Rn) Lq(Rn)
t t>0
≤ ≤ ≤ ∞ { } −
off-diagonalestimatesofexponential type,denotedby T (Lp Lq),ifforallclosedsets
t t>0
{ } ∈ F∞ →
E, F,all f,andallt > 0wehave
Tt(f 1E)1F Lq(Rn) Ct−n(cid:16)1p−1q(cid:17)e−cd(Et,2F)2 f 1E Lp(Rn).
k k ≤ k k
6 JOSÉMARÍAMARTELLANDCRUZPRISUELOS-ARRIBAS
Analogously, givenβ > 0,wesaythat T satisfiesLp Lqoff-diagonalestimatesofpolynomial
t t>0
{ } −
type with order β > 0, denoted by T (Lp Lq) if for all closed sets E, F, all f, and all
t t>0 β
{ } ∈ F →
t > 0wehave
n 1 1 d(E,F)2 −(cid:16)β+2n(cid:16)1p−1q(cid:17)(cid:17)
kTt(f 1E)1FkLq(Rn) ≤Ct− (cid:16)p−q(cid:17) 1+ t2 kf 1EkLp(Rn).
(cid:18) (cid:19)
Theheat and Poisson semigroups satisfy respectively off-diagonal estimates ofexponential and
polynomialtype. Beforemakingthisprecise,letusrecallthedefinitionof p (L), p (L),q (L),and
+
− −
q (L)in(2.9)–(2.10)andin(2.11)–(2.12). Theimportanceoftheseparametersstemsfromthefact
+
that, besides giving the maximal intervals on which either the heat semigroup or its gradient are
uniformly bounded, they characterize the maximal open intervals on which off-diagonal estimates
ofexponential typehold(see[1]and[6]). Moreprecisely, foreverym N ,therehold
0
∈
(t2L)me t2L (Lp Lq) forall p (L) < p q < p (L)
− t>0 +
{ } ∈ F∞ − − ≤
and
t e t2L (Lp Lq) forall q (L) < p q < q (L).
y − t>0 +
{ ∇ } ∈ F∞ − − ≤
Fromtheseoff-diagonal estimateswehave,foreverym N ,
0
∈
(t√L)2me t√L , (Lp Lq),
{ − }t>0 ∈ Fm+12 →
forall p (L) < p q< p (L),and
+
− ≤
t (t2L)me t2L , t (t2L)me t2L (Lp Lq),
y − t>0 y,t − t>0
{ ∇ } { ∇ } ∈ F∞ →
t (t√L)2me t√L (Lp Lq), t (t√L)2me t√L (Lp Lq),
{ ∇y − }t>0 ∈ Fm+1 → { ∇y,t − }t>0 ∈ Fm+21 →
forallq (L) < p q < q (L),(see[23,Section2]).
+
− ≤
2.4. Conical square functionsand non-tangential maximal functions. The operator L gener-
−
atesaC0-semigroup e tL ofcontractions on L2(Rn)whichiscalledtheheat semigroup. Using
− t>0
{ }
thissemigroup andthecorresponding Poissonsemigroup e t√L ,onecandefinedifferentconi-
− t>0
{ }
calsquarefunctions whichallhaveanexpression oftheform
1
dydt 2
(2.13) Qαf(x) = T f(y)2 , x Rn,
| t | tn+1 ∈
(cid:18)ZZΓα(x) (cid:19)
whereα > 0andΓα(x) := (y,t) Rn+1 : x y < αt denotes thecone(ofaperture α)withvertex
+
{ ∈ | − | }
at x Rn (see (4.1)). When α = 1 we just write Qf(x) and Γ(x). More precisely, we introduce
∈
the following conical square functions written in terms of the heat semigroup e tL (hence the
− t>0
{ }
subscript H): foreverym N,
∈
1
(2.14) f(x) = (t2L)me t2Lf(y)2dydt 2 ,
Sm,H | − | tn+1
(cid:18)ZZΓ(x) (cid:19)
and,foreverym N := N 0 ,
0
∈ ∪{ }
1
(2.15) G f(x) = t (t2L)me t2Lf(y)2dydt 2 ,
m,H | ∇y − | tn+1
(cid:18)ZZΓ(x) (cid:19)
1
(2.16) f(x) = t (t2L)me t2Lf(y)2dydt 2 .
Gm,H | ∇y,t − | tn+1
(cid:18)ZZΓ(x) (cid:19)
WEIGHTEDHARDYSPACESASSOCIATEDWITHELLIPTICOPERATORS 7
In the same manner, let us consider conical square functions associated with the Poisson semi-
group e t√L (hencethesubscript P): given K N,
− t>0
{ } ∈
1
dydt 2
(2.17) f(x) = (t√L)2Ke t√Lf(y)2 ,
SK,P | − | tn+1
(cid:18)ZZΓ(x) (cid:19)
andforevery K N ,
0
∈
1
dydt 2
(2.18) G f(x) = t (t√L)2Ke t√Lf(y)2 ,
K,P | ∇y − | tn+1
(cid:18)ZZΓ(x) (cid:19)
1
dydt 2
(2.19) f(x) = t (t√L)2Ke t√Lf(y)2 .
GK,P | ∇y,t − | tn+1
(cid:18)ZZΓ(x) (cid:19)
Corresponding to the cases m = 0 or K = 0 we simply write G f := G f, f := f,
H 0,H H 0,H
G G
G f := G f,and f := f. Besides,weset f := f, f := f.
P 0,P P 0,P H 1,H P 1,P
G G S S S S
Wealsointroducethenon-tangentialmaximalfunctions and associatedrespectivelywith
H P
N N
theheatandPoissonsemigroups:
1
(2.20) f(x) = sup e t2Lf(z)2dz 2
NH | − | tn
(y,t)∈Γ(x)(cid:18)ZB(y,t) (cid:19)
and
1
dz 2
(2.21) f(x) = sup e t√Lf(z)2 .
NP | − | tn
(y,t)∈Γ(x)(cid:18)ZB(y,t) (cid:19)
3. Definitionsandmainresults
As in the classical setting our weighted Hardy spaces will admit several characterizations us-
ing molecules, conical square functions, or non-tangential maximal functions. They will extend
the definitions and results obtained in the unweighted case in [19], to weights w A such that
(p (L),p (L)) , . ∈ ∞
w +
W − ∅
3.1. Molecular weighted Hardy spaces. To set the stage, we take a molecular version of the
weightedHardyspaceastheoriginal definition, andweshallshowthatalltheotherdefinitions are
isomorphictothatoneandoneanother. Inordertoformalizethenotionofmoleculesandmolecular
decomposition weintroduce thefollowingnotation: givenacube Q Rn weset
⊂
(3.1) C (Q) := 4Q, C(Q) := 2i+1Q 2iQ, for i 2, and Q = 2i+1Q, for i 1.
1 i i
\ ≥ ≥
Definition 3.2 (Molecules and molecular representation). Let w A , p (p (L),p (L)),
w +
∈ ∞ ∈ W −
ε > 0,and M Nsuchthat M > n r 1 .
∈ 2 w− p (L)
−
(a) Molecules: We say that a fu(cid:16)nction m (cid:17)Lp(w), (belonging to the range of Lk in Lp(w)), is a
∈
(w,p,ε,M) moleculeif,forsomecube Q Rn,msatisfies
− ⊂
M
kmkmol,w := 2iεw(2i+1Q)1−1p (ℓ(Q)2L)−km 1Ci(Q) Lp(w) < 1.
i 1 k=0
X≥ X(cid:13)(cid:0) (cid:1) (cid:13)
Henceforth,werefertothepreviousexpressi(cid:13)onasthemolecularw-(cid:13)normofm. Additionally,
anycube Qsatisfying thatexpression, iscalled acubeassociated withm. Besides, notethat
ifmisa(w,p,ε,M) molecule,inparticular wehave
−
(3.3) (ℓ(Q)2L)−km 1Ci(Q) Lp(w) ≤ 2−iεw(2i+1Q)1p−1, i = 1,2,...; k = 0,1,...,M.
(cid:13)(cid:0) (cid:1) (cid:13)
(cid:13) (cid:13)
8 JOSÉMARÍAMARTELLANDCRUZPRISUELOS-ARRIBAS
(b) Molecularrepresentation: Foranyfunction f Lp(w),wesaythatthesum λm isa
∈ i N i i
(w,p,ε,M) representation of f,ifthefollowingconditions aresatisfied: ∈
− P
(i) λ ℓ1.
i i N
{ }∈ ∈
(ii) Foreveryi N,m isa(w,p,ε,M) molecule.
i
∈ −
(iii) f = λm inLp(w).
i N i i
∈
Theseobjects aPreaweightedversionoftheonesdefinedin[19]intheunweighted case.
WefinallydefinethemolecularweightedHardyspaces.
Definition3.4(MolecularweightedHardyspaces). Forw A , p (p (L),p (L)),ε > 0,and
w +
∈ ∞ ∈ W −
M N such that M > n r 1 , we define the molecular weighted Hardy space H1 (w)
∈ 2 w− p (L) L,p,ε,M
asthecompletion ofthes(cid:16)et − (cid:17)
∞ ∞
H1 (w) := f = λm : λm isa(w,p,ε,M) representation of f ,
L,p,ε,M i i i i −
( )
i=1 i=1
X X
withrespect tothenorm,
∞ ∞
f := inf λ : λm isa(w,p,ε,M) representation of f .
k kH1L,p,ε,M(w) ( | i| i i − )
i=1 i=1
X X
We shall show below that the Hardy spaces H1 (w) do not depend on the choice of the
L,p,ε,M
allowable parameters p, ε, and M. Hence, atthis point, itis convenient for us to makea choice of
theseparametersanddefinetheweightedHardyspaceastheoneassociated withthischoice:
Notation3.5. Fromnowon,wefixw A , p (p (L),p (L)),ε > 0,and M Nsuchthat
0 w + 0 0
∈ ∞ ∈ W − ∈
M > n r 1 andsetH1(w) := H1 (w).
0 2 w− p (L) L L,p0,ε0,M0
−
(cid:16) (cid:17)
3.2. Weighted Hardy spaces associated with operators. We next define other versions of the
molecularweightedHardyspacesdefinedaboveusingdifferentoperators.
Definition 3.6 (Weighted Hardy spaces associated with an operator). Let w A and take q
(p (L),p (L)). Given a sublinear operator acting on functions of L∈q(w)∞we define th∈e
w +
Wweight−edHardyspace H1 (w)asthecompletion oTftheset
,q
T
(3.7) H1 (w):= f Lq(w) : f L1(w) ,
T,q ∈ T ∈
withrespect tothenorm (cid:8) (cid:9)
(3.8) f := f .
k kH1,q(w) kT kL1(w)
T
In our results will be any of the square functions presented in (2.14)–(2.19), or the non-
T
tangential maximalfunctions definedin(2.20)–(2.21).
3.3. Mainresults.
Theorem 3.9. Given w A , let H1(w) be the fixed molecular Hardy space as in Notation 3.5.
∈ ∞ L
For every p (p (L),p (L)), ε > 0, and M Nsuch that M > n r 1 , the following
∈ Ww − + ∈ 2 w− p (L)
spacesareisomorphicto H1(w)(andthereforeoneanother) withequiva(cid:16)lent no−rms(cid:17)
L
H1 (w); H1 (w), m N; H1 (w), m N ; and H1 (w), m N .
L,p,ε,M Sm,H,p ∈ Gm,H,p ∈ 0 Gm,H,p ∈ 0
In particular, none of these spaces depend (modulo isomorphisms) on the choice of the allowable
parameters p,ε, M,andm.
WEIGHTEDHARDYSPACESASSOCIATEDWITHELLIPTICOPERATORS 9
Theorem 3.10. Given w A , let H1(w) be the fixed molecular Hardy space as in Notation 3.5.
Forevery p (p (L),∈p (∞L)),thefLollowingspacesareisomorphictoH1(w)(andthereforeone
∈ Ww − + L
another) withequivalent norms
H1 (w), K N; H1 (w), K N ; and H1 (w), K N .
SK,P,p ∈ GK,P,p ∈ 0 GK,P,p ∈ 0
Inparticular, noneofthesespacesdepend(moduloisomorphisms) onthechoiceof p,and K.
Theorem 3.11. Given w A , let H1(w) be the fixed molecular Hardy space as in Notation 3.5.
Forevery p (p (L),∈p (∞L)),thefLollowingspacesareisomorphictoH1(w)(andthereforeone
∈ Ww − + L
another) withequivalent norms
H1 (w) and H1 (w).
H,p P,p
N N
Inparticular, noneofthesespacesdepend(moduloisomorphisms) onthechoiceof p.
4. Auxiliaryresults
Inthissection weintroduce somenotation andestablish someauxiliary results thatwillbevery
usefulinordertosimplifytheproofsofTheorems3.9,3.10,and3.11.
LetRn+1 betheupper-half space,thatis,thesetofpoints(y,t) Rn Rwitht > 0. Givenα > 0
+
∈ ×
and x Rn wedefinetheconeofaperture αwithvertexat xby
∈
(4.1) Γα(x) := (y,t) Rn+1 : x y < αt .
+
{ ∈ | − | }
Whenα = 1wesimplywriteΓ(x). ForaclosedsetE inRn,set
(4.2) α(E) := Γα(x).
R
x E
[∈
Whenα = 1wesimplifythenotationbywriting (E)insteadof 1(E).
R R
Besides,forafunction F definedinRn+1 andforevery x Rn,letusconsider
+
∈
1
dydt 2
(4.3) F := F(y,t)2 .
|k k|Γ(x) | | tn+1
(cid:18)ZZΓ(x) (cid:19)
Usingideasfrom[19,Lemma5.4],weobtainthefollowingresult:
Lemma4.4. Forallw A and f L2(Rn). Therehold
∈ ∞ ∈
(a) f . G f ,forallm Nand0 < p < ,
m,H Lp(w) m 1,H Lp(w)
kS k k − k ∈ ∞
(b) f . G f ,forall K Nand0 < p < .
K,P Lp(w) K 1,P Lp(w)
kS k k − k ∈ ∞
Furthermore, one can see that (a) and (b) hold for all functions f Lq(w) with w A and
∈ ∈ ∞
q (p (L),p (L)).
w +
∈ W −
Proof. Westartbyproving part(a). Fix x Rn andt > 0,andconsider
∈
B:= B(x,t), f(y) := (t2L)m−1e−t22Lf(y), and H(y) := f(y) (f)4B,
−
where (f) = f(y)dy. Then, applying the fact that t2Le t2L (L2 L2) and that
t2Le t2L1=4Bt2L1−4=B0(see[1]e),weobtainthat { − }t>0 ∈eF∞ e→
− R
e e
1 1
t2Le−t22Lf(y)2dy 2 = t2Le−t22LH(y)2dy 2
| | | |
(cid:18)ZB (cid:19) (cid:18)ZB (cid:19)
1 1
. |t2eLe−t22L(H14B)(y)|2dy 2 + |t2Le−t22L(H1Cj(B))(y)|2dy 2
(cid:18)ZB (cid:19) j 2 (cid:18)ZB (cid:19)
X≥
10 JOSÉMARÍAMARTELLANDCRUZPRISUELOS-ARRIBAS
1 1
. H(y)2dy 2 + e c4j H(y)2dy 2 =: I + e c4jI .
− − j
| | | |
(cid:18)Z4B (cid:19) j 2 (cid:18)Z2j+1B (cid:19) j 2
X≥ X≥
ByPoincaréinequality, weconcludethat
1
2
I . t f(y)2dy ,
y
|∇ |
(cid:18)Z8B (cid:19)
andthat e
1 j
2
I . f(y) (f) 2dy + 2j+1B1/2 (f) (f)
j 2j+1B 2kB 2k+1B
| − | | | | − |
(cid:18)Z2j+1B (cid:19) k=2
X
e j e 1 e e
2
. 2j+1B1/2 f(y) (f) 2dy
2k+1B
| | − | − |
k=2 (cid:18)Z2k+1B (cid:19)
X
j e e 1
2
. 2(j k)n/22kt f(y)2dy .
− y
|∇ |
k=2 (cid:18)Z2k+2B (cid:19)
X
Then, e
1 1 j 1
t2Le−t22Lf(y)2dy 2 . t yf(y)2dy 2 + e−c4j 2n(j2−k)+kt yf(y)2dy 2
| | |∇ | |∇ |
(cid:18)ZB (cid:19) (cid:18)Z8B (cid:19) j 2 k=2 (cid:18)Z2k+2B (cid:19)
X≥ X
e e 1 e
. e c4j t f(y)2dy 2 ,
− y
| ∇ |
j 1 (cid:18)Z2j+2B (cid:19)
X≥
e
andtherefore
f(x) . e c4jG2j+3 f(x),
Sm,H − m−1,H
j 1
X≥
recallthedefinitionofG2j+3 in(2.13)and(2.15). Then,forevery0 < p < andw A ,taking
the Lp(w) norm in bothmsi−d1e,Hs of the previous inequality and applying chan∞ge of ang∈les∞(see [23,
Proposition 3.2]),weconclude that
f . e c4j G2j+3 f . G f e c4j . G f .
kSm,H kLp(w) − m−1,H Lp(w) k m−1,H kLp(w) − k m−1,H kLp(w)
Xj≥1 (cid:13)(cid:13) (cid:13)(cid:13) Xj≥1
(cid:13) (cid:13)
As for part (b), fix w A , f L2(Rn), and 0 < p < , and note that following the same
argument of[19,Lemma∈5.4]1∞,ther∈e exist adimensional cons∞tant k NandC > 0such that for
0 1
∈
all K Nandk N .
0
∈ ∈
1 1
2k f(x) C G2k+k0 f(x) 2 2k+k0 f(x) 2 ,
SK,P ≤ 1 K−1,P SK,P
(cid:16) (cid:17) (cid:16) (cid:17)
whererecall thedefinitions of 2k andG2k+k0 in(2.13),(2.17),and(2.18). Now,forsomeR > 0,
SK,P K 1,P
tobedeterminate later,consider −
f(x) := ∞ R k 2k f(x) and G f(x) := ∞ R kG2k f(x).
S∗ − SK,P ∗ − K−1,P
k=0 k=0
X X
1WewanttothankSteveHofmannforsharingwithusthisargumentthatwasomittedin[19,Lemma5.4].