WEIGHTED VARIABLE HARDY SPACES ASSOCIATED WITH OPERATORS SATISFYING DAVIES-GAFFNEY ESTIMATES Текст научной статьи по специальности «Математика»

66

Probl. Anal. Issues Anal. Vol. 11 (29), No3, 2022, pp. 66-90

DOI: 10.15393/j3.art.2022.11130

UDC 517.98, 512.642

B. Laadjal, K. Saibi, O. Melkemi, Z. Mqkhtari

WEIGHTED VARIABLE HARDY SPACES ASSOCIATED WITH OPERATORS SATISFYING DAVIES-GAFFNEY

ESTIMATES

Abstract. We introduce the weighted variable Hardy space H^Lw associated with the operator L, which has a bounded holomorphic functional calculus and fulfills the Davies-Gaffney estimates. More precisely, we establish the molecular characterization of HPL{W (Rn) and we show that the new weighted variable bounded mean oscillation-type space represents the dual space of

hL'W(Rn), where L* denotes the adjoint operator of L on L2(Rn).

Key words: weighted Hardy spaces, variable exponent, Davies-Gaffney estimates, molecular decomposition, maximal function, dual space

2020 Mathematical Subject Classification: 42B35

1. Introduction. The theory of Hardy spaces in Mn was first introduced by Stein and Weiss [23] and was originally tied closely to the theory of harmonic functions. On the other hand, the real-variable methods were introduced by Fefferman and Stein [7]. It is well-known that the classical Hardy space Hp(Rn) is a suitable substitute of the Lebesgue space Lp(Rn) for any p E (0,1]; for example, when p E (0,1], various well-known operators from harmonic analysis, such as Hilbert and Riesz transforms, are bounded on Hp(Rn), but not on the classical Lebesgue spaces Lp(Rn). As a generalization of the classical Hardy spaces, Nakai and Sawano [20] introduced and studied the atomic characterization of the Hardy space Hp')(Mn) with variable exponent. Independently, Cruz-Uribe and Wang [4] studied the variable Hardy spaces

Hp(--)(En) with p(-) satisfying some conditions slightly weaker than those used in [20]. Recently, Zhuo et al. [27] investigated the intrinsic square function characterizations

© Petrozavodsk State University, 2022

of the variable Hardy spaces, then Saibi [21] extended the results of [27] to the variable Hardy-Lorentz spaces.

The weighted variable Lebesgue space is a natural generalization for the classical weighted Lebesgue space and the variable exponent Lebesgue space. This space has been considered in a series of papers, see for example [3], [17]. Regarding the theory of the Hardy spaces, Ho [12], presented the atomic characterization for the variable weighted Hardy spaces. Additionally, Melkemi et al. [19] explored the weighted Hardy spaces with variable exponents on a proper open subset ^ of E".

In the last decade, the study of function spaces associated with different operators has been a very active area of research in harmonic analysis and has attracted the attention of many researchers. In particular, Yang and Zhuo [26] introduced the variable Hardy space HPL{ (E") associated with the operator L, where £>(•): EE" ^ (0,1] is a measurable function satisfying the globally log-Holder continuous condition and L is a linear operator on L2(E™), which generates an analytic semigroup {e-tL}t>o with kernels having pointwise upper bounds. As a generalization of these results, Yang et al. [24] considered the variable Hardy spaces HPL{ °(Rra) associated with the operator L, which obeys the Davies-Gaffney estimates. More generally, Zuo et al. [28] investigated the variable Hardy-Lorentz spaces associated with operators satisfying Davies-Gaffney estimates. We point out that the notion of the Davies-Gaffney estimates (or the so-called L2 off-diagonal estimates) of the semigroup {e-tL}t>o was first introduced by Gaffney [8] and Davies [5], which is considered as a generalization of the Gaussian upper bound of the associated heat kernel.

The main purpose of this paper is to introduce and study the weighted variable Hardy space tfLw (E") associated with the operator L, which satisfies Davies-Gaffney estimates. We establish its molecular characterization by means of the atomic decomposition of the weighted variable tent space. Furthermore, using this molecular characterization, we formulate the dual space of the variable weighted Hardy space (E"). The rest of this paper is arranged as follows: in Section 2, we describe the Assumption(A) and Assumption(5) imposed on the operator L and we recall some definitions and basic properties of the weighted Lebesgue spaces with variable exponent. In Section 3, we introduce the weighted variable tent space T^ >(E++1), establish its atomic characterization, and give the definition of the weighted Hardy space with variable exponents associated to the operator L in terms of the square function of the heat

semi-group generated by L. Our main results on the molecular characterization of HPL{1 (E") is given in this section (see theorem 3 below). In Section 4, we introduce the weighted BMO space with variable exponent BMOL*>>WM(E"), where M e N and L* denotes the adjoint operator of L on L2(E"), and we establish the duality between H(E") and BMOL^ (E").

We end this introduction by describing the basic notation. We denote by N the set {1, 2,...} and by Z+ the set NU {0}. The square function SL associated with L is defined by setting, for any f e L2(E") and x e E",

slv)(x) :=

t2Le-t2L (f) (y)

dydt t"+1

L 0 B(x,t)

1/2

The symbols A < B and A & B stand for the inequalities A ^ CB and A < B < A, respectively, and C denotes a positive constant independent of the parameters, which can vary from line to line. Finally, for a measurable subset Q C E" we denote by |Q| and the Lebesgue measure of Q and the characteristic function of Q, respectively.

2. Preliminaries. We first give some notions, notation, and useful definitions; we also describe the assumptions required for the operator L considered in this paper.

Let E++1 := E" x (0, ro). For any a e (0, w) and x e E", define

ra(x) := {(y,t) e E++1: |y - x| < at}.

If a = 1, for the sake of simplicity, we write r (x) instead of ra(x). For any ball B := B(xB, rB) C E" with xB e E" and rB e (0, ro), A e (0, ro) and j e N, let \B := B(xB,XrB),

B = {(y,t) e E++1, dist(y,Bc) ^ t}.

Now, we recall some notions of bounded holomorphic calculi which was introduced by McIntosh [18]. Let 0 ^ rq < n. The closed sector in the complex plane C is defined as follows:

Sv = {z e C: |argz\ ^ 'q} U {0}

and its interior denoted by is defined by

S° = {Z e C\{0}: |arg z| <V}.

2

Denote the set of all holomorphic functions on SJ by H(S0) and for any b E H(SJ) we define ||6|U by

||6|U = sup{|6(z)|: z E SJ}.

The set of all b E H(S0) satisfying ||&||^ < ro is denoted by H^(S(J) and define the set ^(5*0) by

S0) = { ^ E H(X>(SJ): 3 u,C> 0: |^(z)| ^ , E .

Let rq E [0,^) and denote the spectrum of L by a(L). Then, we say that the closed operator L on L2(Rn) is of type rq if

1) <j(L) is a subset of Sv,

2) for any v E (rq,^), there exists a positive constant Cv, such that for all X E Sv :

||(L - XI )-1|£(L2(Rn)) ^ Cv ^^

where £(L2(Rra)) denotes the set of all linear continuous operators from L2(Mra) to itself and for any operator T E C(L2(M.n)), its norm is denoted by ||T¡¿^(r™)).

Let rq E [0,^), L be a one-to-one operator of type rq in L2(Era), v E (rq,n) and ^ E ). The operator tp(L) is defined as follows:

^(L)=2ViS%KX)(XI - L)-1dX, (1)

©

where 6 := {re™: r E (0, ro)} U {re-™: r E (0, ro)}, v E (rq,v) is the curve consisting of two rays parameterized anti-clockwise. It is well-known that the integral in (1) is absolutely convergent in L2(Rn) (see [9], [18] for more details) and ^>(L) does not depend on the choice of v (see, for instance, [1, Lecture 2]). By a limiting procedure, we can extend the above holomorphic functional calculus on v(S°) to H^(S0U) (the reader is referred to [18] for more details). Let 0 < v < n; we say that the operator L has a bounded S®)-calculus in L2(Rn) if there exists a positive constant C, such that for all ^ E H^(S<°),

U(L)h(L2(RK)) < CUh-^o).

We assume that L is an operator satisfying the following assumptions: Assumption (A). L is one-to-one operator of type r] in L2(Mra) with r] E [0, |) and has a bounded holomorphic functional calculus. Assumption (B). The semigroup {e-tL}t>o generated by L satisfies the Davies-Gaffney estimates, namely, there exist positive constants c\ and c2, such that, for any function f in L2(Mra) and closed sets E and F of Mra with supp f CE,

lie-tL(f)\\L*{F) ^ Cle|/|l2(e),

where dist(E, F) := inf{|x — y|: x E E,y E F}.

A measurable function p(-): Era ^ (0, w] is called a variable exponent. We set

p- = ess infx^P(x) and p+ = esssup^eRnp(x),

and

V(Rra) := |p(-) variable exponent: 0 < p- ^ p+ < wj.

Let p(-) E V(Era). The variable Lebesgue space consists of all

measurable functions f: Era ^ C, such that / | f(x )lp(x) dx < w, equipped

Rn

with the Luxemburg quasi-norm

/(x)|

f frl f( x)\ ip(x)

LP(0(Rn) := inf j A > 0 : [^^J dx ^ n.

We recall that for any f E L11oc(Era), the Hardy-Littlewood maximal operator M is defined for all x E Rn by setting

M(f)(x) :=sup -BJ | f(y)\dy,

B

where the supremum is taken over all balls B of Rn containing x.

Let w: Rra ^ (0, ro) be a locally integrable function. The weighted Lebesgue space with variable exponent Lw()(Rra) is defined as

Lpw() (R) := {/ : R ^ C: ||/||L,(-)(R„) = ||/w||LP(.)(R.) < w}.

Note that if w = 1, then LW(0(Rra) = IP()(Wn) and if p(-) is a constant, then Lw '^1 (Kra) is the classical weighted Lebesgue space L'' (Era).

We recall in the following lemma the Holder inequality, for the proof see [6, Lemma 3.2.20].

Lemma 1. Letp(-): Rra ^ [1, ro) and w : Rra ^ (0, ro) be two Lebesgue measurable functions. Then

J\f(x)g(x)\dx ^ 2\\/||lp()(R„)IMIlp'(.)(Rn),

Rn

where, p'(-) denotes the conjugate function of p(-), that is: ^ + ^(y = 1. Let p = min{p_, 1}.

Definition 1. Let p(-) E V(Era) and w: Era ^ (0, ro) be a Lebesgue measurable function. We denote by Wp() the set of all Lebesgue measurable functions w, such that

• \\XB\\Lp(p)/p(Rn) < ro and \\Xb\\l(p(-)/p)'(Rn) < ro, for any baJl B c Mra;

w— w —

• there exist k > 1 and s > 1, such that the Hardy-Littlewood maximal operator is bounded on L^1^./] /k(Rra).

Remark 1. It is easy to see that Lsp1'/S (Mra) is the s-convexißcation of

LPW ). ™

For any w E Wp(•), set

Sw = inf{s ^ 1: M is bounded on (Era)}

and

§w = {s ^ 1: M is bounded on /k(Era), for some k> 1}.

For any fixed s E E>w, we define

ksw = sup{fc > 1: M is bounded on L^l'/k(Era)}.

The following theorem is the Fefferman-Stein vector-valued maximal inequalities on LW()(Era). For the proof, we refer to [12, Theorem 3.1].

Theorem 1. Let £>(•): Era ^ (0, oo) be a Lebesgue-measurable function with 0 < p- ^ p+ < o and q E (1, o). Ifw E Wp(^, then, for any r > sw, we have

fe m w)

^ ieN '

1/9

^ C

rrP( • )

J 1/r w1' '

l)

fei')

v ieN 7

1/9

¿rp1(/)r (R")

w1' r

The following lemma plays a key role in the proofs of the main results of this paper; we refer the reader to [12, Lemma 5.4].

Lemma 2. Let p(^) E V(Rra), w E Wp(.), r E (0,1], such that 1 G and q E ( r(kw/r)', <x). Then there exists a positive constant C, such that for any sequence [Bj}jeN of balls in Rra, [Xj}jeN C C and functions [a,j }j€N satisfying the condition that for any j E N, Supp aj C Bj and IIaj IIlï(R") ^ lBj|1/?,

(£ IXjajr) \ j=i /

^ C

Ll{ )(

)

(it ix*^r)

v j=i /

LWW °(R")

3. Weighted variable Hardy spaces. In this section, we introduce the weighted variable Hardy space associated to the operator L, and we establish its molecular characterization via the atomic decomposition of the weighted variable tent spaces given in this section. We begin by recalling some notation.

For all measurable functions f on E++1 and for any x E Era, define the operator A by

A( f)(x)

(//

r(*)

I f(y, t)l

2dy dt \ 1/2

Let £>(•) E (0, o). The tent space Tp (E++1) is the space of all measurable functions f on E++1, such that

H/HlftR'

"+1\

M( /)|Lp(R") <

Let £>(•) E V(Era) and w: Era ^ (0, o). The weighted variable tent space TW()(E++1) is defined to be the space of all measurable functions f on E++1, such that A( f) E LW(0(Era). For any f E TW(0(E++1), define

tW()(r++1)

M(/)IU • )

LW()(R")-

+

Let F be a closed set in Rra and O = Fc; we denote by O the tent over O, which is the set

5 := {(x, t) e E++1: dist(x, F) ^ t}.

Next, we give the definition of (p(-),w, w)-atoms:

Definition 2. Let p(-) e V(Rra), w: Rra ^ (0, w) be a Lebesgue-measur-able function and r e (1, w). A function a on E++1 is called a (p(-),w, w)-atom, if

(i) there exists a ball B C Rra, such that supp a C B;

ii) \\a\\Tr(R++1) ^ \B\1/r\\XB||

-i

The theorem presents the atomic characterization of TW( (K++1):

Theorem 2. Let p(-) eV (Era), w e Wp(). Then, for any f e TW(0(R++1), there exist (p(-),w, w)-atoms {a^en associated with the balls {B,,}^, respectively, and numbers {A^^ C C, such that for almost every (x,t) e R++1,

f (x,t) = ^2 Kai(x,t). (2)

i£N

Moreover, there exists a constant C > 0, such that for all f e TW( )(M++1) :

A({A,},eN, {Bz}teN) ^ C\\f \\

T T1

'P(0 (Rn+1

(3)

where for any sequence of numbers {A^e C and sequence of balls {Bi} ieN

A({Aj}

(y ' \^i\XBi '

v i£N )

(4)

Here and hereafter 9 := sup{s 1: s e Sw}.

Proof. Let f e TW(0(R++1). Let ^ = {x e Era : A(f)(x) > 2i} for

any

i e Z. Since f e TW((R™+1), it is easy to check that Qi is a proper open set and |Qi| < w for each i e Z. By a similar argument used in the

proof of [15, Theorem 3.2], we can show that supp f C

Utez Q* J U E

where E C 1 satisfying J ^^ = 0. Thus, for each i G Z, by applying

E

the Whitney decomposition (see [22, p. 167]) to Q*, we get a sequence Wij'ljeN of disjoint cubes, such that

1) U Qi,j = Qf and {Qjj^eN have disjoint interiors, jeN

2) for all j E N,

c1VnlQiJ ^ dist (Qi, j, (Q* )C) ^ C^^nlQi j ,

where lQij denotes the side-length of the cube Qi,j with dist^^Q*)C) := inf{|x -yl :x E Qhl,y E (Q*)C}.

For each j E N, choose a ball Bij with the same center as Qij and with radius hy/nl (QiJ). Define

Aij = Bi,3 n (Qij x (0, o)) n (Of \ Q+1),

aU = 2 l\\xBi,j ||rP(0(En JXAij and \3 = 2i||xBi,j |U)(

)

Note that {(QiJ- x (0, o)) n (Q* \ Q*+1)} C Bitj. Following the proof used in

[27, Theorem 2.16], we can show that &i,j is a ($(•), w, o)-atom associated

to the ball Bij for any i E Z and j E N. We see that f = Yl Yl Aijai,j

iezjeN

almost everywhere. Then remains to show that A({Ai}ieNh {Bi}ieN) ^ CW/W

Indeed, by the definition of Ai, , we have

A({Aij}iezjeN {Bij}iezjeN) = ^ ( ^ (^b^)

iez

eN

<

<E2 [T,(XB„:

iez ^ jeN

<

<

E 2

iez

|x IIVi?

lXn|||LP0)/e (e„)

Since 1 E §w and 1 > sw. Hence, by Theorem 1 and the fact that Xfi* < [M(XQi ^ , we find

A({ Ai, j}iez,jeN, {Bi, j}iezjeN) <E 2 ||M(x

Qi)W Ll( • )(En) <

iez

<

HAm

L& )(En)

•)(E++1)'

9

Now the proof is completed. □

We denote by Tftf (R++1) and T2(R++1) respectively, the set of all functions in Tw()(R++1) and T2(R++1) which have compact supports.

Proposition 1. Letp(^) e V(Rn),w e Wp(.). Then TP{J(R++1) C T2(R++1) in the meaning of sets.

Proof. By [16, Lemma 3.3(i)], we know that Tf (R++1) C T2(R++1) for any q e (0, w). Then it is sufficient to show that Tftf (r++1) C T/ (R++1), for some q' e (0, w). Indeed, let f e T^J(R++1) be such that supp f C E, where E is a compact set of R++1. Let B be a ball of R++1 such that E C B. Then supp Af C B and by Lemma 1 we have:

J[Af (x)]p- dx = J [Af (x)]p- dx + J [Af (x)]p- dx <

Rn {x£B:Af(x)^1} {x£B:Af(x)>1}

< \B\ + \\(AfY-\\lv,)/v_\\xb\Uo/p)' < \B\ + \\AfIt,).

LwP L -p lw

w w —

□

Next, we establish the molecular characterization of the weighted variable Hardy spaces associated with operators satisfying the Davies-Gaffney estimates. These kind of spaces are denoted by Hp,'^ (Rn). We begin with some definitions.

Definition 3. Let p(^) e V(Rn) and w : Rn ^ (0,w) be a Lebesgue measurable function. Let L be an operator satisfying Assumption (A) and Assumption (B). The weighted variable Hardy space H(Rn) is defined as the completion of the space H(Rn),

St:!(Rn) := if e L2(Rn): \ \ Sp(f) \\ pW(• )(R») < w},

with respect to the quasi-norm

^ (f)

\ 1 f 11 <W (Rn) := 1 1 SL(f) \ \ LW( • )(R") = A > 0: ^-)'w( "X") ^ ,

where Pp{.),w (^) := / [^(f)(Ax)w(x)|}p(x)dx.

Rn

To introduce the molecular weighted variable Hardy spaces HpL('l'M'e (Rn), we give the definition of a (p(^), w, M, e)L-molecule.

Definition 4. Let L be an operator satisfying Assumption (A) and Assumption (B) and £>(•) E V(Rra) and w: Rra ^ (0, w) be a Lebesgue measurable function. Assume that M E N and t E (0, w). A function m E L2(Rra) is called a (p(^),w, M, e)L-molecule, if m E R(LM) (the range of LM) and there exists a ball B := B(xb, rB) C Rra, where xb E Rra and rB > 0 is such that for every k = 0,..., M and j E Z+:

ik'-2L-1m\\L2(Uj(b)) ^ wB\1/2\\xb||-i(.)(

where for j E N:

U (B) := B(xb, 2r-B)\b(xb, 2J-1 tb),

and for = 0:

Uo(B) := B(xb, tb).

Definition 5. Let L be an operator satisfying Assumption (A) and Assumption (B). Let p(^) E V(Rra) and w E Wp{.). Assume that M E N

and t E (0, w). For a measurable function f on Rra, f = Xjmj is called

3 = 1

a molecular (p(^) ,w, M, e) -representation of f, if [mj }jeN is a family of (p(^) ,w, M, t)L-molecules, the sum converges in L2(Rra) and [Xj }jeN C C satisfies

A ([ ^}j&h [bj}jeN) < w,

where A ([Xj}jen, [Bj}jen) is as in (4) and for any j E N,Bj is the ball associated with mj. The space

is defined to be the set of all functions f E L2(Rra), which has a molecular (p(^),w, M, e)-representation. The molecular weighted variable Hardy spaces

is the completion of HpL(iM(Rra) with respect to the quasi-norm

!

Hp(0'M'e(R™) := inM A ([X3}зeN, [Bj}jeN) :

<x

f = ^^ Xjmj is a molecular (p(^),w, M, e) — representation >.

3 = 1 >

To establish the molecular characterization of (Rra), we need

the following technical lemmas. Let L be an operator satisfying Assumptions (A) and (B) and M E N. The next lemma can be proved by using an argument similar to that used in the proof of [24, Proposition 3.10].

Lemma 3. Let e V(Rn),w e Wp(.),M e (n - , w) n N and t > f. There exist a constant C and a e (f, w), such that, for any j e and (p(),w, M,e)L-molecule m, associated with the ball B := B(xB, rB) C Rn, where xB E Rn and rB > 0,

i i SL(m) \1 „(Uj ( b)) ^ C Bll/2\\Xb\\

i-i

Ll{ • (Rn) '

Proposition 2. Let p(^) eV(Rn),w e Wp{.). Let M e N and e e (0, w). Then the set of all finite linear combinations of (p(^),w, M,e)L-molecule denoted by is dense in HpL{')M,e(Wn) with respect to the quasi-

norm |1 • | |

H

P( • ),M,e

(Rn)'

Proof. Let g E HpL()M'e(Rra). Then, by the definition of HpL()M'e(Rra), we

know that for any 5 e (0, w) there exists a function f e HBl such that

1 1 9 - f11 <2'M'e(Rn) ^ 2'

,w (

),

By the definition of HpL(')M,e(Rn) , we conclude that there exist {Aj}j&i C C and a family {m,j}jeN of (p(•), w, M, e)L-molecules, associated with the balls {Bj}-eN of Rn, such that

<x

f = Y, ^mj in L2(Rn) and a({\,} j&h{Bj}j&!) < W. i=i

Let Jn = j=1 mj for any N E N; then we have \ \ f - In \ \ „p(• M

• ),M,e(Rn)

E

j=N +1

m3

<

H

<

A (i^i }T=N+1 AB, }J=N+1

oo

p( • ),M,e L,w

3 .

(R")

E

j=N+1

E

j=N +1

e^ 1/d

- \ \ \ \ Lt • '(Rn) - / Ll( • )(Rn)

e 1/0

\\ \\ • )(R"). Lp(e)/e (Rn)

W

On the other hand, since

A ([Xj}f=N+1,[Bi}i=w+0 =

E

j=N+1

\Xj \XBj

\\xbj\\lp( • ) (en)

1/0

<,

Lp(e)/e (Rn)

it follows that for almost every x E Rn,

\Xi \xb,

lim >

N ^^

j=N+1

WlXBj\\pp(. • )(Rn)_

0.

Combining this and the dominated convergence theorem (see, for example, [6, Lemma 3.2.8]), we obtain

lim

W-s-oo

E

j=N +1

x\x

b.

lxbj\\lp( •)(!»)

1/0

pVt-V« (Rn)

0.

Thus, we conclude that

lim \\ f — fN \\

W-s-oo

H

p( • ),M,ef

)

0.

Hence, we find that for any 8 E (0, w) there exists some N0 E N, such that for any N > N0:

— fN\\

H

p( • ),M,t L.w

5

>) < 2.

Obviously, for any N E N, fN E Hp^(Rn). Then, for any S E (0, w), when N > N0:

WO — fN\\ HP(. • ),M,e

(TOn ) r^J Wu J \ \ H

L.w ^ '

< \\g — f\\Hp(.• ).M.£(fn) + \\ f — In\\hp(.• ).M.c,n„, < 8.

L.w ' L.w

e(R")

Then we conclude that Hp^^^R"") is dense in HpL('l'M'e(Rn) with respect

to the quasi-norm \\ • \\

H

pi. • ).M.t

(Rn)

. □

The following theorem deals with the molecular characterization of

(Rn).

Theorem 3. Let L be an operator satisfying Assumption (A) and Assumption (B). Let p(0 E V(Rn),w E Wp{.). Let M E (n[ J — ±], w) n N

w

w

I,w

) and H^'l (Rn) coincide with the

and let e E (n, ro). Then H^ equivalent quasi-norms.

Let first show that Hf M'\Rn) C H' (Rn) n L2 (Rn)

Proposition 3. Let L be an operator satisfying Assumption (A) and (B). Let pO E V(Rn),w E Wp{.). Let M E (n[1 - 2], ro) n N and let t E (f, ro). Then there exists a positive constant C, such that for any f E Hf M'\Wn),

HP(■ )(Rn) ^ c\\f\\Sv( ),M,c

L.w v ' L.w

(Rn) '

Proof. Let f E Hl£W

P( -), M,€/

l). Then, by Definition 5, we know that there

exist {\jC C and a family {m,jof (p(^),w,M, e)L-molecules associated with the balls {Bj}jeN of Rn, such that f = Y1 Xjmj in L2(Rn)

and

3 = 1

H

p( • ),M,e

(Rn)

A ({Xi}зeN, {Bj}jeN)

Since the operator SL is bounded on L2(Rn), we find that

N

Xî) - Sl( ^Xjmj)

3 = 1

L2 (Rn )

N

0.

Hence, there exists a subsequence |sl^, such that for

almost every x E Rn

X3m^j

yfe^ \/^X3m3 ) (x) = SL(f)(x).

Thus, for almost every x E Rn we have

oo oo

SL(f)(x) < J2Y1 X\SL(mi )(x)XUi(B] )(x). j=1 i=0

Then

\Sl( f ^L* • )(Rn)

[Sl(/)]é

Lp(e)/e (Rn)

rS^

w

^ E E ^iö frfo

i=0 j=l <x r ro

«e ei^r n

i=0 L 3=1

W

i/e e

m-

LPJ • >(R")

(6)

By Lemma 3, we find that for any j G N and i G Z+,

II^Kn^w,)) = II^KW,)L(rb) < 2-WJ№ll/*\\XBi II

-l

ht • )(R");

where a e (f, w). Multiplying (7) by 11 Xb, | |

(7)

H • ' (R")

we obtain

2* 11 Xb3\\

SL(m3)Xu(B]) r_ N < I2%|l/2.

L2(Rn)

We apply Lemma 2 for a3 := 2%° 11 XBj 11 lp(■ )(Rn)5,p(mi)x^i(Bj), to conclude

that

JE l\3 f [SL(m3)XU(B])]'} I 3=i )

s <X ill

1/0

Lvw{ • '(Rn)

2-WJ 11 XB, I I

-l

LPJ • >(R")

1^3 Ix

3 IX2iBj

"3) <

r^J

e i/e

l

Lvw{ • '(Rn)

From the fact that x^bj ^ 2mm(xb , )(^), we deduce that

i/o

f ^ V

E ^ iö frk w,)]'

I ,=i J

<

LPJ • >(R")

<

f <x k 3 = i

2-IIto, 11 -;>)(R„)I^i |2"M (xb, )

^ i/o

Lpw{ • >(R")

We choose r e (0,9) such that a > nr 1. Then, by Remark 1 and Theorem 1, we have

iE I0[SL(m3)XUi(B,)]'} I 3=i )

i/e

<

Lpw{ • >(R")

<

E

3 = i

(2 -^) (2m)r 11 Xb, I I

Ll(• '(R«)K 3

1^3 Ir M (xb, )

e/r>. i/e

i

<

lpj •) (R")

e

< 2-^-7)

< 2-i(*- V)

( r

IgH Ii** ii

^ .7=1

A |

B 11 Lt • )(R")

Xb,-

)] )

r x 1 ö/r>! r/0

Xb

l/

<

LpJr)/r (Rn)

iiXb 11L^ • '(Rn)

l/

<

< 2-*?^ ({A,}i€N, (B,}i€N) - 2-i(°-?)

H

p( • ),m,.

L,w

(R™)"

From this, (5) and (6), we infer that for any f G HL(p'M'e(En),

H^ (Rn)

L^'-'(R") <

s <X y i=0

2-i("-V)

l/

<2'M,£(Rn) -

H^ (Rn)'

is a subset

which is the desired result. □

The following proposition shows that H^p (En) n L2(En) of the space HL[p'M'e(En).

Proposition 4. Let L be an operator satisfying Assumptions (A) and (B). Let pO G V(En), w G Wp{.). Let M G N and let e G (0, w). Then

for any f G HL^(En) n L2(En) there exist [Xj}j€n C C and a family

[m,j}jeN of (p(^),w, M, e)L-molecules, associated with the balls [Bj}jeN of

<x

En, such that f = E Xjmj in L2(En), j=i

A([XjbeN [Bj}j€N) ^ CWfWH^l(Rn).

Proof. Let f G HpL['l (En) n L2(En) and F(x, t) := t2Le-2Lf (x), for all (x, t) G E++1. Then F G Tp0 (En) n L2(En). Hence, by Theorem 2, there exist ($(•), w, w)-atoms [aj}jen, associated with the balls [Bj}jeN, respectively, and numbers [Xj}jeN C C, such that for almost (x, t) G E++1:

F(x, t) = ^ X3a3 (x, t), in Tp{■)(En) nL2(En)

j€N

and

A ({A,}i€N, (B,}i€N) ^CIF||,

Tp(' )(Rn)

<2 (R")'

By the Ho —calculi of L, we know that

oo

f = CM J(t2L)M+1^t2L(t2Le~t2L(f )) J = nM,L(F), in L2(Rn), 0

where CM f t2(M+2)e f2 J- = 1. From the fact that nM,L is a bounded map 0

from T2(R++1) to L2(Rn), it follows that

f = Cm x kml^ E ai) = °m( E X3%m,p(aj)) , in L2(Rn).

jen jen

Following the argument used for [24, Lemma 3.11], we can prove that mj = KM,L(aj) is a multiple of a (p(•), w, M, e)L-molecule adopted to Bj; this implies the desired result. □

Next, we give the proof of Theorem 3. Proof. By Proposition 3, Proposition 4, and the density argument, we

'n [ 1 _ 1 ]

■ 2 U 2]

have for M G (n[ J — 1 ], œ) n N and each e G (n,œ): the spaces H^(En)

and Hp(')M,e(Rn) coincide with equivalent quasi-norms. □

4. Dual space. In this section, we study the duality of H^R). Here and hereafter, we denote by L* the adjoint operator of L in L2(Rn). Let us first recall some basic notions and definitions.

Definition 6. Let p(^) e V(Rn) and w: Rn ^ (0, w) be a Lebesgue measurable function. Let L be an operator satisfying Assumption (A) and Assumption (B). Then for any M e N and e e (0, w) define

MLp(^(Rn) :=j f = LM (g) e L2(Rn): g eV(LM), | | f 11 ^(¿m,^) < w}

where V(LM) denotes the domain of the operator LM and

i M ~k \ Mv( ■ ),M,E(R„) := SU^2J(£- 2 )| | Xb(0,1) | 0(R„^E ^^ (f )| |L2(Ui (B(0,1)) \'

•w K k=o J

is defined

The dual space of M^,M,e(Rn) denoted by ¡Mp)^(Rn) as the set of all bounded linear functionals on Mpp('l((M,e(Rn). Then,

for any f G IMf^'6(En)j * and g G Mf0'M,e(En), the duality between iml^'"(En)l * and MfWM'*(En) is denoted by (f, g)M. Let

mlw

p{n

) = n

e€(0,ro)

MLW

Definition 7. Let p(^) G V(En) andw: En ^ (0, w) be a Lebesgue measurable function. Let M G N and L be an operator satisfying Assumption (A) and Assumption (B). We say that an element f G MPL'WM'* (En) is in

BMOp)'M (En) if

B MOp(t)'M(Rn) • =

|B|1/2

sup

bcrn wxb\\lpj.• )(r„)

|(j- er%L*)M(f)(x)\2dx

B

1/2

< oo,

where the supremum is taken over all balls of En.

The following result can be seen as an extension of [24, Proposition 4.3] to the weighted MP(W'M'e(En).

Proposition 5. Let p(^) G V(En) and w G Wp(.). Let M G N and e G (0, w). If f G MpPP'WWpM'e(En), then f is a harmless positive constant multiple of a (p(^),w,M, e)L-molecule associated with the ball B(0,1). Conversely, if m is a ($(•), w, M, e)L-molecule associated with the ball B C En, then m G MfpM''(En).

The following three estimates play important roles in the proofs of our main results in this section. The proof of the next lemma can be done with similar arguments as in [24, Lemma 9].

Lemma. Letp(^) G V(En), w G Wp(.), M G N. Then f G BMOfWWM(En) is equivalent to

BMOpL(-¿•M-res(M") • =

•- suP h H

SCR" \\XB\\lp(-)(Rn)

IBI1/2

(I- (/ + r2BL)~1M(f)(x)

x

B

1/2

<,

where the supremum is taken over all balls of En. Moreover, there exists a positive constant C, such that for any f G BMOfpM(En) we have

C

1

BMO'

A • ),m

(Rn)

BMOPL,)M-(Rn ) ^ C \\f\\

BMofl*M (Rn) '

■i

2

Similarly to [24, Lemma 4.5], we have the next lemma. The proof is left to the reader.

Lemma 4. Let p(^) E V(Rn) and w E Wp(.). Let t, t E (0, w) and M E N and M>M + ? + f. Suppose that f E M^'M'* (Rn) satisfies

4 '

[I - (I + L*)-1 }M(/)(x)'2

-dx < oo.

J 1 + \x\n+e

Rn

Then, for any (p(^),w, M, t)L-molecule m, it is true that

(f,m)M = Cm [ i (t2L*)Me-t2L* (f )(x)t2Le-2L(m)(x)

dxdt

jn + l

where CM is a positive constant, depending on M, which satisfies

Cm j tM+1 e2i2 j = 1. 0

The proof of the following lemma is similar to that of [14, Lemma 8.3] and [24, Lemma 4.7]:

Lemma 5. Let p(^) E V(Rn) and w E Wp{,) and M E N. Then there exists a positive constant C, such that for any f E BMO^'WM(Rn):

sup

IB |

1/2

\(t2L)M e-t2L(f )(x)|2 dxdt

t

B

1/2

SCR" | | XB | | L^)(Rn)

where the supremum is taken over all balls of Rn.

^ C 1 1 f 11 BMOpL((Rn),

Proposition 6. Let p(^) E V(Rn) and w E Wp(.). For any e E (2n, w), M E N and f E BMOL^M (Rn). Then f satisfies (8).

Proof. We can check that f satisfies (8) by following the argument used for [25, Remark 4.8], with X = Rn and p- replaced by 9. □

In the following result, we prove the duality of the space HL*'W(Rn). Theorem 4. Let p(^) E V(Rn) and w E Wp(.) such that p+ E (0,1]. et M E (n[ 1 - 2], w) n N, M > M + 2n + n and e E (n, w). Then, HPL(W (Rn)l * coincides with BMO^f (Rn) in the following sense:

t

(i) Let g G \hL('W (En)j *. Then g G BMO^i'M (En) and for any

f G HPp(W

M,n(En) the following holds true: g(f) — (g, fu, and there

*

exists a positive constant C, such that for any g G Hf;(En) : WaWsMO^ (Rn) ^ C (R")]* •

(ii) Conversely, let g G BMOf :)wM(En). Then, for any f G Hf-W^n (En), the linear functional lg defined by lg( f) — (g, f )m has a unique bounded extension to Hf-p (En) and there exists a positive C, such

that for any g E BMO^f (Rn):

\\4 W^ (Rn)]* ^ C MBMO

PL (*);M (Rn )'

Proof. First we show (i). Let gG [Hf(W (En)] *. For any fGHf(W (En), we have:

\g( /)| ^ w9w[hi,i(En)]*Wf Wh^i(En). We also know that for any ($(•), w, M, e)L-molecule m \\m\\•)( ) < 1.

L,w( )

Thus,

\g(m)\ <\\g\\[h!})(En)] * • (9)

On the other hand, by Proposition 5, we find that for any h G Mpf('WwM'e(En) with \\h\\. m• ),M,eIV!ns — 1: h is multiple of a (rp(^),w,M, e)L-molecule up

ML,w (E )

to harmless positive constant associated with the ball B (0,1). From (9), we know that for any e G (0, w), g G Mpf'WW(M'" (En) *. Hence, g G MfWM'* (En), and for any h G M^'M'e(En):

(g ,h)M = g(h).

Next, we show that

\ \ 9 \\ BM^(E-) ^ C\\9\\ [h^(E-)] * •

We take a ball B C En, h G L2(B) with \\h\\L2(B) — 1. Following the argument used in [25], we learn that 71—P-/-(l — erBL)M(h) is a harmless

1 J' •) ( En/ ' V '

positive constant multiple of a (p(-), w, M, e)L-molecule. Therefore,

IB |1/2

I I Xb | | lpj• )(Rn) J

B

I- er2L

\ M

) (g)(x)h(x)dx

IBI1/2

II Xb ||

(i

r*L\M rnL I

I - rr2L) (h)\ bIILi )(Rn ) V ' ' M

< I I 911 [H^i (Rn)j*

which implies that for any ball B C M"

IB |

1/2

II Xb II

BIILi()(R") b

(I - er*L*)M(g)(x)

dx

}

1/2

<

II 9 II

[ <i (Rn)j* •

Hence, we get the desired result.

Now we turn to prove (ii). Let g E BMOv^}'^ (Rn). We define

4(f) := / f(x)g(x)dx

for f e m^R). Since f e HLÎM^R) C EvL[l(R"), we have

p( ■),M,eira,n\

M •) /

t2Le-{2Lf e •)(R++1). Then, by Theorem 2, t2Le-{'Lf = E A

•3 aj,

jen

where [aj}jen is a sequence of (p(-),w, œ)-atoms supported by [Bj}jen. By Proposition 6, we know that g satisfies inequality (8) for H> . Thus, it follows from Lemma 4, the Holder inequality, and Lemma 5 that:

I g (f )

CW (t2 LT e (g)(x)t2Le-2L(f )(x)

dxdt

jn + l

<

^^ n n

< E ^ i// \(№)

i=1 R++1

2t*\M- t2 L

e~ (g)(x)\Iaj (x,t)I

dxdt

< r^j

< E ^ I =1

(t2L*)M e-t 2L* (g)(x)\

i dxdt

1/2

\aj(x,t) \

>.dxdt

1/2

<

Bi

Bi

< E IXi I 11 g 11 BM&Vf (Rn ) <

3=1 ,i

9

2

t

t

t

t

<

<

in) ~

a (ixjben, {bjben) bmopj:)'m (e")

L* ,w v '

where in the third inequality uses the fact that

hpl(^ (Rn) wy wbmovl( (Rn),

D = £ IXjI ^ A ({Xj}jeN, {Bj}jeN) ' j=i

Indeed,

ß p(x)

(£ [ Ä ])°w<x>ß - >/g

IXi]IXBi

d\\xbA.

XBi

p(x)

M

D

\\ X^i W J

w(x)ßp(x)dx > p(x)

w(x)ßp(x)dx > 1'

Hence, the proof of Theorem 4 is finished. □

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Received November 27, 2021. In revised form, July 10, 2022. Accepted July 13, 2022. Published online September 7, 2022.

B. Laadjal

Laboratory of applied mathematics University of Biskra, Biskra 07000, Algeria E-mail: baya.laadjal@univ-biskra.dz

K. Saibi

Department of mathematics, Zhejiang Normal University, Jinhua, China E-mail: saibi.khedoudj@yahoo.com E-mail: khedoudjsaibi@zjnu.edu.cn

O. Melkemi

Laboratory of partial differential equations and applications, University of Batna 2, Batna 05000, Algeria E-mail: ou.melkemi@univ-batna2.dz

Z. Mokhtari

Laboratory of partial differential equations and applications, University of Batna 2, Batna 05000, Algeria E-mail: z.mokhtari@univ-batna2.dz