CC BY
11DOI: 10.15393/j3.art.2023.15310
UDC 517.518, 517.982, 517.44
X. Yan
LITTLEWOOD-PALEY ^-FUNCTION CHARACTERIZATIONS OF MUSIELAK-ORLICZ HARDY SPACES ON SPACES OF HOMOGENEOUS TYPE
Abstract. Let (X,d,y) be a space of homogeneous type, in the sense of Coifman and Weiss, and ^: X x [0, to) ^ [0, to) satisfy that, for almost every x p X, <p(x, ■) is an Orlicz function and that <p(-,t) is a Muckenhoupt weight uniformly in t p [0, to). In this article, by using the aperture estimate of Littlewood-Paley auxiliary functions on the Musielak-Orlicz space LV(X), we obtain the Littlewood-Paley ¿^-function characterization of Musielak-Orlicz Hardy space HV(X). Particularly, the range of A coincides with the best-known one.
Key words: space of homogeneous type, Musielak-Orlicz Hardy space, Littlewood-Paley auxiliary function, -function
2020 Mathematical Subject Classification: 46E36, 42B25, 42B30, 30L99
1. Introduction. It is well known that the real-variable theory of Hardy-type spaces on Rra, including various equivalent characterizations and the boundedness of singular integral operators, plays a fundamental role in harmonic analysis and partial differential equations (see, for instance, [26]). Recall that the classical Hardy space Hp(Rn) with p e (0,1] was originally introduced by Stein and Weiss [27]; this initiated the study of the real-variable theory of Hp(Rn). Particularly, Calderon and Torchin-sky [4] established Littlewood-Paley function characterizations of Hp(Rn). Up to now, many new variants of classical Hardy spaces have sprung up and their real-variable theories have been well developed in order to meet the increasing demand from harmonic analysis, partial differential equations, and geometric analysis (see, for instance, [14], [25]).
The bilinear decompositions of the product of Hardy spaces and their dual spaces play key roles in improving the estimates of many nonlinear
© Petrozavodsk State University, 2024
quantities, such as div-curl products (see, for instance, [31]), weak Jaco-bians (see, for instance, [18]), and commutators (see, for instance, [20]). Bonami et al. [3] showed that, for any given f e H 1(Mra), there exist two bounded linear operators Sf: BMO(Rra) ^ L1(Rn) and Tf: BMO(Rra) ^ H*(Rra) such that, for any g e BMO(Rra), fxg = Sfg+Tfg, where H*(Rra) denotes the weighted Orlicz-Hardy space associated to the weight function w(x) := 1/log(e + |x|) for any x e Rra and to the Orlicz function
$(i) := -----, @t e [0, 8).
( ) log(e + i), [, )
This result was essentially improved by Bonami et al. in [2], where they further proved the following bilinear decomposition:
H 1(Rra) x BMO(Rra) c L1(Rra) + Hlog(Rra),
where Hlog(Rra) denotes the Musielak-Orlicz Hardy space related to the Musielak-Orlicz function
V(x, t) —7-—7-¡-pr, @x e Rra, Vie[0, 8).
' ) log(e + i) + log(e + |x|V [
Bonami et al. in [2] also concluded that Hlog (Rra) is the smallest space in the dual sense. Motivated by these, Ky [21] introduced the Musielak-Orlicz Hardy space H^(Rra) with <p being a Musielak-Orlicz function, which generalizes both the Orlicz-Hardy space of Janson [19] and the weighted Hardy space of Stromberg and Torchinsky [28], and established both the atomic and the grand maximal function characterizations of H^(Rra). Since then, the real-variable theory of Musielak-Orlicz Hardy spaces has rapidly been developed. Precisely, Hou et al. [15] characterized Hv(Rn) by the Lusin-area function and the molecule; Liang et al. [24] further established several other real-variable characterizations, respectively, in terms of various maximal functions and Littlewood-Paley -function and <7*-function.
On the other hand, there has been an increasing interest in extending the above results of Musielak-Orlicz Hardy spaces from the Euclidean space to more general underlying spaces, such as the anisotropic Euclidean space; see, for instance, [22], [23]. In particular, Coifman and Weiss [5] originally introduced the concept of the space X of homogeneous type and the atomic Hardy space Hcw(X) with p e (0,1] and q e (p, 8] x [1, 8], and proved that H™(X) is independent of the choice of q. In the same
article, they also posed a question: to what extent the geometrical condition of X is necessary for the validity of the radial maximal function characterization of H^w(X). Since then, lots of efforts have been made to establish various real-variable characterizations of the atomic Hardy spaces on X with few geometrical assumptions. However, due to the lack of Calderon reproducing formulae on X, many existing results of both function spaces and boundedness of operators require some additional geometrical assumptions on X, such as the reverse doubling condition of ^ (see, for instance, [6]).
Recently, He et al. [12] first introduced a kind of approximations of the identity with exponential decay and then obtained new Calderoon reproducing formulae on X. Later, He et al. completely answered the aforementioned question of Coifman and Weiss by developing a quite complete real-variable theory of the Hardy space and its localized version on X, respectively, in [11] and [13]. Fu et al. [7] further generalized the corresponding results in [11] to Musielak-Orlicz Hardy spaces H^(X). Indeed, let rq e (0,1), u be the upper dimension of X, and p a growth function, with uniformly lower type p e (0,1], satisfying that
P i u
q(^) + 'q
where q(^) is the critical weight index of Fu et al. in [7, Theorem 6.16] characterized HV(X) via the Littlewood-Paley g*-function with
A e (.[2<M + 1],»).
In this article, we first establish an aperture estimate of Littlewood-Paley auxiliary functions on the Musielak-Orlicz space LV(X), and then obtain the Littlewood-Paley g*-function characterizations of HV(X) with A e (2uq(p)/p, 8), which improves the corresponding results in [7, Theorem 6.16] via widening the range of A into the best-known one.
The organization of the remainder of this article is as follows.
In Section 2, we recall some notation and concepts that are used throughout this article. More precisely, in Subsection 2.1, we recall the definition of a space X of homogeneous type and state some basic properties of X. In Subsection 2.2, we introduce the concepts of the uniformly Muckenhoupt condition, the Musielak-Orlicz space LV(X), the spaces of test functions and distributions, the system of dyadic cubes, and approximations of the identity with exponential decay on X. Then, via the
Lusin-area function Sa with a e (0,8), we introduce the Musielak-Orlicz Hardy space HV(X).
In Section 3, we first recall the concepts of Littlewood-Paley g*-function and auxiliary function SÍ"q. Then, by an argument similar to that used in the proof of [11, Lemma 5.11], we establish an aperture estimate of SÍ"q on LV(X) (see Lemma 5 below). Finally, from Lemma 5 and the fact that g\ and SÍ"q are pointwisely comparable, we further obtain the Littlewood-Paley ^-function characterizations of Hv (X) with the best known range A e (2uq(ip)/p,8).
At the end of this section, we make some conventions on notation. Let N := {1, 2,...} and Z+ := N y {0}. We denote by C a positive constant which is independent of the main parameters, but may vary from line to line. We use Cpa,... q to denote a positive constant depending on the indicated parameters a,.... The symbol f < g means f ^ Cg and, if f < g < f, then we write f „ g. If f ^ Cg and g = h or g ^ h, we then write f < g „ h or f < g < h, rather than f < g = h or f < g ^ h. If E is a subset of X, we denote by 1® its characteristic function and by EA the set X\E. For any a e R, we denote by [aj the biggest integer not greater than a. For any index q e [1,8], we denote by q' its conjugate index, namely, 1/q + 1/q' = 1. For any x,x0, e X and r,$ e (0, 8), let Vr(x) := y(B(x, r)),
V(x,y) :=
{^(B(x, d(x, y))), if x ^ y,
o, if x = y,
and
Pl9(xo,x; r) : =
1
V (xo) + V (xo,x)
r + d(x0,x)
2. Preliminaries. In this section, we recall some basic concepts about the space X of homogeneous type and Musielak-Orlicz Hardy spaces.
2.1. Spaces of Homogeneous Type. In this subsection, we recall the concept of spaces of homogeneous type and some related basic estimates.
Definition 1. A quasi-metric space (X, d) is a non-empty set X equipped with a quasi-metric d, namely, a non-negative function defined on X x X such that for any x,y,z e X:
(i) d(x, y) = 0 if and only if x = y;
§
(ii) d(x,y) = d(y,x);
(iii) there exists a constant A0 e [1, 8), independent of x, y, and z, such that
d(x,z) ^ Ao[d(x,y) + d(y,z)]. (2)
The ball B of X, centered at x0 e X with radius r e (0, 8), is defined by setting
B := B(x0, r) := {x e X: d(x, x0) < r} .
For any ball B and any r e (0,8), we denote B(x0,rr) by tB if B := B(x0, r) for some x0 e X and r e (0, 8).
Definition 2. Let (X,d) be a quasi-metric space and ^ a non-negative measure on X. The triple (X, d, ^) is called a space of homogeneous type if ^ satisfies the following doubling condition: there exists a constant C(M) e [1, 8), such that for any ball B c X:
p(2B) ^ Cp,)^(B).
The above doubling condition implies that for any ball BcX and any A e [1, 8),
f,(\B) ^ C^X*p(B), (3)
where u := log2 C(^,) is called the upper dimension of X.
Throughout this article, according to [5, pp. 587-588], we always make the following assumptions on (X,d,y):
(i) for any x e X, the balls {B(x,r)}re(o,c8) form a basis of open neighborhoods of x;
(ii) ^ is Borel regular, which means that all open sets are ^-measurable and every set AcX is contained in a Borel set E, such that ^(A)= »(E);
(iii) for any x e X and r e (0, 8), ^(B(x, r)) e (0, 8);
(iv) diam X := sup d(x,y) = 8, and (X,d,y) is non-atomic, which
x, yeX
means ^({x}) = 0 for any x e X.
Notice that diam X = 8 implies that ^(X) = 8 (see, for instance, [1, p. 284]). From this, it follows that, under the above assumptions, ^(X) = 8 if and only if diam X = 8.
The following basic estimates are from [9, Lemma 2.1], which can be proved by using (3).
Lemma 1. Let x,y e X and r e (0, 8). Then V(x, y) „ V(y,x) and
Vr(x) + V(y) + V(x, y) „ V(x) + V(x, y) „
„ Vr(y) + V(x, y) „ y(B(x, r + d(x, y))).
Moreover, if d(x, y) ^ r, then Vr (x) „ Vr (y). Here the positive equivalence constants are independent of x, y, and r.
2.2. Musielak—Orlicz Hardy Spaces. Throughout this article, we always let (X, d, y) be a space of homogeneous type. In this subsection, we recall the concept of Musielak-Orlicz Hardy spaces and state some known results.
A function $: [0, 8) ^ [0, 8) is called an Orlicz function if it is non-decreasing, $(0) = 0, $(t) > 0 for any t e (0, 8), and limt^8 $(t) = 8.
Now, we recall the concept of uniformly upper and lower types, which was introduced in [16, p. 1924].
Definition 3. For a given function <p: X x [0, 8) ^ [0, 8), such that, for almost every x e X, p(x, •) is an Orlicz function, p is said to be of uniformly upper (resp., lower) type p for some p e (0, 8) if there exists a positive constant Cppq, depending on p, such that, for almost every x e X, s e [1, 8) (resp., s e [0,1]), and t e [0, 8),
<p(x,S t) ^ Cp SPp(x, t).
Next, we recall the concept of the uniformly Muckenhoupt condition from [16, Definition 2.6].
Definition 4. A function <p: X x [0, 8) ^ [0, 8) is said to satisfy the uniformly Muckenhoupt condition for some q e [1, 8), denoted by p e Aq(X), if, when q e (1, 8),
Ma, (*) := sup sup pTm^ t)dll(x){\[^(y, t)] ¿M} <8
ic
B B
or
Mai(*) := suP su^ ^T^^ I t) dy{x) (esssup B[<fi(y, t)] 1) < 8
ie(0,œ) BcX Mm) J B
where the first suprema are taken over all t e (0, 8) and the second ones over all balls B cX.
Throughout this article, let
Ac(X) := |J Ag(X).
qe\\,rn)
For any ( e A8(X), ^-measurable set EcX, and t e [0, 8), let
(p(E, t) := ip(x, t)dfi(x).
For any given p e (0, 8), a function f is said to be locally p-integrable if, for any x e X, there exists an re (0, 8), such that
\ f(y)\P d»(y)<8.
B(x,r)
Denote by L\oc(X) the set of all the locally p-integrable functions on X. In what follows, we always let M denote the Hardy-Littlewood maximal operator defined by setting, for any f e Lloc(X) and x e X,
M(f)(x) := sup -1- i\f(y)\d^(y), (4)
B3X ^(-D ) J
B
where the supremum is taken over all balls B of X containing x.
Now, we state some basic properties of Aq(X) with q e [1, 8), which are just parts of [7, Lemma 2.6] (see also [30, Lemma 1.1.3] for the corresponding Euclidean case).
Lemma 2. The following conclusions hold true:
(i) Ai(X)cAp(X)cAq(X) for any p, q satisfying 1 ^ p ^ q <8.
(ii) If( e Aq(X) with q e [1, 8), then there exists a positive constant C, such that, for any ball BcX, ^-measurable set EcB, and t e (0, 8),
((B, t) <r (p(E, t) <
p(E)
(iii) If q e (1,8) and p e Aq(X), then there exists a positive constant C, such that, for any f e L11oc (X) and t e [0, 8),
j[M(f)(x)]\(x, t) dii(x) ^ C^\f(x)\qp(x, t)d^(x),
A A"
where M is the same as in (4). The critical weight index q(p) of p e A8(X) is defined by setting
q(p) := inf {qe[1, 8) :p e Aq(X)} . (5)
The following concept of growth functions was first introduced in [16, Definition 2.7].
Definition 5. A function p: X x [0, 8) ^ [0,8) is called a growth function if the following conditions are satisfied:
(i) p is a Musielak-Orlicz function, namely,
(i)i the function p(x, ■): [0, 8) ^ [0, 8) is an Orlicz function for almost every x e X;
(i)2 the function p(■, t) is ^-measurable for any t e [0, 8).
(ii) p e A»(*).
(iii) p is of uniformly lower type for some e (0, 1] and of uniformly upper type 1 .
Next, we recall the definition of Musielak-Orlicz spaces, which was first introduced in [16, Definition 2.8].
Definition 6. Let p be a growth function in Definition 5. The Musielak-Orlicz space LV(X) is defined to be the set of all the ^-measurable functions f, such that
J P(x, \f(x)\) dfi(x) < 8,
equipped with the Luxemburg (also called the Luxemburg-Nakano) (quasi-)norm
Wfhnx) := inf {A e (0, 8): ^(x, ^M) dfi(x) ^ l}.
A"
Now, we recall some basic properties of LV(X), which were first given in [7, Lemma 2.8] (see also [30, Lemmas 1.1.6 and 1.1.10] for the corresponding Euclidean case).
Lemma 3. Let p be a growth function in Definition 5. Then the following conclusions hold true.
(i) p is uniformly a-quasi-subadditive on X x[0, 8), namely, there exists a positive constant C, such that, for any (x, tj) e X x [0, 8) with
3 6 N,
v{x, 2 ^ C 2p(x, fj).
jeN jeN
(ii) For any f e Lip(X)\{0}
I x -
\f \\lv{X )
dy(x) = 1.
V t\\T,viy\ /
A"
(iii) For any (x, t) e X x [0, 8),
t
r(x, t) := ds
is a growth function and equivalent to p, namely, there exists a positive constant C, such that, for any (x, t) e X x [0, 8),
C¡^p(x, t) ^ p(x, t) ^ Cp(x, t).
Moreover, for almost every x e X, ip(x, •) is continuous and strictly increasing.
Next, we introduce the Musielak-Orlicz Hardy spaces via the Lusin-area functions. To this end, we first recall the concept of spaces of test functions on X, which was originally introduced by Han et al. [9, Definition 2.2] (see also [10, Definition 2.8]).
Definition 7. Let x0 e X, re (0, 8), q e (0,1], and e (0, 8). A function f on X is called a test function of type (x0,r, q,"&), denoted by f e Q(x0, r, q, •&), if there exists a positive constant C, such that
(T1) for any x e X,
\f(x)\ ^CP^(xo,x; r), (6)
here and thereafter, P$ is the same as in (1); (T2) for any x,y e X satisfying d(x, y) ^ [r + d(x0,x)]/(2A0) with Ao the same as in (2),
\f(x)-f(y)\ ^C" fx V) ^ Y P-&(xo,x; r). (7)
Lr + d(x0, x) J
Moreover, for any f e Q(x0, r, q, '), define
Wf\\g(xo,r,e,v) := inf {C: C satisfies (6) and (7)} . The subspace Q(x0, r, q, ') is defined by setting
G(xo,r, q,') := |/ e Q(x0,r, q,') : J f(x) d/i(x) = 0|
A"
equipped with the norm \\ ■ \\g{xQ:= \\ ■ \\g(x0,r,e,v).
Fix an xo e X. Denote Q(x0, 1,Q,') simply by Q(q,'&). Obviously, Q(q,') is a Banach space. Note that, for any fixed x e X and r e (0, 8), Q(x,r, q,') = Q(q,') with equivalent norms, but the positive equivalence constants may depend on x and .
Fix e e (0,1] and g,' e (0,e]. Let G0(q,') be the completion of the the set G(e, e) in G(q,'). Furthermore, the norm of £?0(8,') is defined by setting \\ ■ \\:= \\ ■ \\s(e,tf). The space q,') is called the
space of test functions. The dual space (G0(8,'))' is defined to be the set of all continuous linear functionals from Q0(8,') to C, equipped with the weak-* topology. The space (@0(8,'))' is called the space of distributions.
The following system of dyadic cubes of (X,d,y) was established by Hytonen and Kairema in [17, Theorem 2.2].
Lemma 4. Suppose that constants 0 < c0 ^ Co <8 and 8 e (0,1) satisfy 12A0Co# ^ c0 with Ao the same as in (2). Assume that a set of points {zk: k e Z,a e Ak} c X with Ak, for any k e Z, being a set of indices, has the following properties: for any k e Z,
d (zk, Zp) ^ c08k if a ^ 0, and min d (x, zk) < Co8k for any x e X. Then there exists a family of sets {Qk : k e Z, a e Ak}, satisfying
(i) for any k e Z, [Jke^fc Qk = ^ and {Qk: a e Ak} is disjoint;
(ii) if k,l e Z and k ^ I then, for any a e Ak and ft e Ai, either Ql/3 c Qk or Qli X Qka = H;
(iii) for any ke Z and a e Ak, B( zk, (3A$)-1co8k) c Qk cB( ^, 2A)Co£k). Throughout this article, for any k e Z, define
£k := A+i\A and := {z^1^ =: {ykaUfc ,
and, for any x e X, define
d(x, yk) := inf d(x, y).
yey k
Now, recall the concept of approximations of the identity with exponential decay introduced in [12, Definition 2.7].
Definition 8. Let 8 be the same as in Lemma 4. A sequence {Qk}keZ of bounded linear integral operators on L2(X) is called an approximation of the identity with exponential decay (for short, exp-ATI) if there exist constants C,u e (0, 8), a e (0,1], and y e (0,1), such that, for any k e Z, the kernel of the operator Qk, a function on X x X, which is still denoted by Qk, has the following properties:
(i) (the identity condition) XjkeZ Qk " ^ in L2(X), where I denotes the identity operator on L2(X);
(ii) (the size condition) for any x,y e X,
\Qk(x, y)\ ^ CEk(x, y),
here and thereafter,
„ i ^ 1 ( [d(x,y)ia) Ek(x, y) := , -exM ~v \l ^
k( , (x)Vs*(y) L ^ J /
max{d(x, yk), d(y, ykH"1»
x exp ^ — v
k
(iii) (the regularity condition) for any x,x',y e X with d(x,x') ^ 8'
\Qk(x,yq — Qk^^yq\ + \Qk(y— Qk(y,x>)\ ^
~d(x,x')iv
^ C
k
Ek (x, yq;
(iv) (the second difference regularity condition) for any x,x',y, y' e X with d(x,x') ^ 5k and d(y, y') ^ 5k,
\[ Qk(x, y)-Qk(x', y)] - [Qk(x, y')-Qk(x', y')]\ ^
<C [ ^ f [ -^ ^ Ek (x,y);
(v) (the cancellation condition) for any x,y e X, Qk(x, y') dy(y') = 0 = J Qk(x', y) dy(x').
X A"
Next, we recall the concept of the Lusin-area function (see, for instance, [11, Section 5]).
Definition 9. Let 5 and y be the same, respectively, as in Lemma 4 and Definition 8, and let q,$ e (0, y). Assume that f e (q,i9))' and {Qk}keZ is an exp-ATI. For any a e (0, 8), the Lusin-area function Sk(f) of f with aperture a is defined by setting, for any x e X,
SM)(x) :=| 2 J \Qkf (y)f
kpZ B(x,kS k)
here and thereafter, for any y e X,
d^(yq
VaS k (x)
! ' •
Qkf (y) Qk(y,x)f(x)dy(x). x
When a := 1, we simply write S := S1.
Now, we recall the concept of Musielak-Orlicz Hardy spaces, which was first introduced in [7, Definition 6.2].
Definition 10. Let y be the same as in Definition 8 and <p a growth function in Definition 5 with uniformly lower type p e (0,1] satisfying
u
>
and let
Q,ê e
q( ip) u + y Q( ,
)
where q(p) and u are the same, respectively, as in (5) and (3). The Musielak-Orlicz Hardy space HV(X) is defined by setting
Hip(X) := {f e (<30 (e'j)': \\S (f)\\mX) < 8) and, moreover, for any f e Hv (X), let
\f\Hv(X) : = \\5(f)\\Lf(X).
Remark 1.
(i) As was proved in [7, Theorem 6.3], the space HV(X) in Definition 10 is independent of the choices of exp-ATIs in S(f).
(ii) Combining [29, Remark 3.17(iii)], [7, Theorems 5.4 and 6.15], and [7, Proposition 6.12], we conclude that the space HV(X) in Definition 10 is independent of the choices of (GO(Q,'))' whenever
ee (u[Q(p){p- 1] v).
3. Littlewood—Paley -Function Characterizations of HV(X). In this section, we establish Littlewood-Paley -function characterizations of HV(X), which improves the corresponding results in [7, Theorem 6.16] by widening the range of the parameter A into the best-known one. To this end, we first recall the concept of Littlewood-Paley -function (see, for instance, [11, Section 5]).
Definition 11. Let 5 and r] be the same, respectively, as in Lemma 4 and Definition 8, and let q,' e (0, rj). Assume that f e (Q0(q,'))' and {Qk}kei is an exp-ATI. The Littlewood-Paley -function (f) of f, with any given A e (0, 8), is defined by setting, for any x e X,
fi m»-{g Of [ ^ r }1.
The following theorem is the main result of this section.
Theorem 1. Let r] be the same as in Definition 8 and p a growth function in Definition 5 with uniformly lower type p e (0,1] satisfying
p u
q( p) u +
and let
( < )
g,ê e ^u
where q(<<) and u are the same, respectively, as in (5) and (3). Further, assume that
X e ( , 8
Then f e H*(X) if and only iff e (Q (e,ti))' and g*(f) e L*(X). Moreover, there exists a constant C e [1, 8), such that, for any f e HV(X),
C_1 }9*(f)}Lvpx) ^ WfWHvpxq ^ C IIg*x(f)}Lv(x).
To prove Theorem 1, we need more preparations. Let 6 and y be the same as in Theorem 1. For any q,i9 e (0, y), a e (0, 8), and f e (Qg(q, "&))', recall that the Littlewood-Paley auxiliary function Sa ) of f with aperture a is defined by setting, for any x e X,
Sia)(f)(x) := J \Qkf (^ H)}^. (8)
keZB(x,aS k)
Particularly, when a = 1, by Lemma 1, we conclude that, for any fe (Q(e,ti))' and x e X,
SPa)(/)(x)„S(/)(x), (9)
where the implicit positive constant is independent of both and x.
The following conclusion, which shows an aperture estimate of Sja\f) on LV(X), plays a key role in the proof of Theorem 1.
Lemma 5. Let q e (1, 8) and < e Aq(X) with uniformly lower type p e (0,1]. If a e [1, 8), then there exists a positive constant C, such that
< (x,Sia)(f)(x)) dy(x) ^ Ca"q (x,Sia)(/)(x)) dy(x), (10)
A A"
where u is the same as in (3).
Proof. For any non-negative function g and any x e X, define M(g)(x) := sup sup 1 , I g(z) dy(z),
keZ d{x,y)<aSk V&k (V) J
B(y, ¿k)
1
where 6 is the same as in Lemma 4. Moreover, for any t e (0, 8) and f e (@0j(Q,'))' with r] being the same as in Definition 8 and g,' e (0, rj), define
Et := {x e X: S[a)(f)(x) > t} and Et := |x e X: M(1Et )(x) > 1 j .
On the one hand, by [11, p. 2252], we conclude that, for any nonnegative function g and any x e X,
M(g)(x) < a"M(g)(x)
with M as in (4), which, combined with Lemma 2(iii), further implies that, for any t e (0, 8),
p(Et,?) =^|x e *: M(1Et)(x)> 1| ^
e X :Ca" M(1e )(x) > 1} ^ (U)
^ Ca"q ^[M(1Et)(x)fp(x, t) dn(x) ^ Ccfqp (Et, t), x
where C is a positive constant independent of both a and . On the other hand, fix t e (0, 8) and, for any yeX, let
P(y) := inf d(x, y).
xeeA
Obviously, for any k e Z and x,y e X, x e Et x B(y,a8k) implies that p(y) < a5k. Moreover, by an argument similar to that used in [11, p. 2253], we conclude that, for any k e Z and any ye X satisfying p(y) < a5k,
H {Et nB(y, 6k)) ^ (B(y, )) .
From these, (8), Tonelli theorem, (3), and Lemma 2(ii), we deduce that, for any t e (0, 8),
[ Sia)( f)(x)]2p(x, t) d/i(x) =
E
IS
\Qk f (y)!2 ê^fix, t) dy(x) ^
ÈA keZB(x,aSk)
Vôk (y)
< S [ ! qk f (y)\2f (j xB (y, aôk ), t)
keZ ,
y)<a&k
d^(y)
VSk (y)
<
< S f ! Qk f (v)\2<P (B(y, aôk ), t)
keZ , J
dy(y)
<
p( y)<a&k
À awq S f \Qkf (y)\2<fi {B(y, ôk), t)
kpZ p{ y)<aè k
À awq S (\Qkf (y)\2f (J x B(y, ôk), t)
VSk (y)
kW\ d^(y)
keZ
A
Vôk (y)
dn(y)
V&k (y)
<
a
Sf f \Qk f (?/)\2 PÈf(x,t)dn(x)
keZ
Ètc B(x,ôk)
V&k (y)
a
wq
M 12
si(/)(x) f(x, t)dy(x).
This, together with Tonelli theorem and the fact that < is of uniformly upper type 1, further implies that, for any t e (0, 8),
(JSC X {x e X : Sa\f)(x)> t} ,t) = f(x,t) dy(x) ^
Èztx{xeX : S^\f)(x)>t}
<
ÈA x{xeX : s£a) (f)(x)>t}
saa)u)(x) t
f(x, t) dy(x) ^
^ t-2\ [saa)(/)(x)]2f(x,t)dy(x)
<
< awqt~
p a) 2
si U)(x) f(x, t) dMx)
2
2
aTq r
< aTqt~
S[a\f)(x)1
1 2
{yeX: S[a\f)(y)<t}
(x) p(x, t) dji(x)
A"
sP^fX^l (a) (x)
1 ()( ) {yeX: s[a)(f)(y)^t}( )
X
8
sp(x, t) ds dfi(x) „
sp(x, t) dji(x) ds
0 {xeX: S(a)(f )(x)1 (a) (x)>s}
{ 1 (J)( ) {yeX: s\a) (f) (y)^t}( ) }
sp(x, t) dji(x) ds <
{xeX :Sla)(f )(x)1 ,a) (x)>s}
{ 1 (J)( ) {yeX: s1a)( f)(y)<,t}( ) }
s • - • p(x, s) dfi(x) ds
0 {xeX :S\a)(f)(x)1 (a) (x)>s}
{yeX : S^ ( f ) ( y)<,t} ( a)
oTt-1 pNx e *:S(a)(f)(x)1{yeX
1°)(f)(.y)<t}(x) > s}'s ) ds <
<
aTqt-1 (p({x e X: S[a)(f)(x) >s} , s) ds
(12)
Combining (11) and (12), we find that, for any t e (0, 8),
p ({x eX :S(:)(f)(x)> t} , t) ^
^ p (Et, tS + p (Ect x {x e X: S(aa)(f)(x) > t} , ^ <
t
p (Et, t)+ t-1 fp ({x e X: Sla)(f)(x) > s J , s) ds
< aTq
(13)
Moreover, from Lemma 3(iii) and Tonelli theorem, we deduce that, for any a e (0, 8),
Sia)(f)(x)
p (x,Sja)(f)(x))dfi(x)
p( x, )
dt dji(x)
x
0
2
2
2
2
2
f(x, t)dji(x) dt
{yeX : Sia\f ){y)>t}
rV ({x e *: Sia)(f)(x) > t} , t) dt,
which, together with (13) and Tonelli theorem, further implies that < (x,Sia)(f)(x)) dy(x)„
X
t-lf ({x e X : S^mx) > t} , t)dt
<
< awq
co t 2
0
c
a
wq
0
c
a
wq
rV (Et, t)dt + j t-2 J e X : sla\f)(x) > s },s)dsdt
00
co co
Î-V (Et, t) dt^^ J"i-2^{x e *: s[a)(f)(x) > s},s)dids
0 s
c
rV (Et, t) dt + s-lf (Es, s) ds
0
c
0
p a)
awq | rl^{x e X : s[a\f)(x) >t} ,tj dt
f (x,s!a\f)(x)) dMx).
x
This finishes the proof of (10) and, hence, of Lemma 5. □
Now, we prove Theorem 1.
Proof of Theorem 1. We first prove the sufficiency. Let f e (Q0j(g,i)))' and g*(f) e LV(X). By Definition 11, Lemma 1, and Definition 9, we conclude that, for any x e X,
9l(f)(x) H S \Qkf(y)\
keZ
B(x,Sk )
k
ôk + d(x, y)
dy(y)
l/2
V&k (x) + V&k (y)
l
A
2
„{2 f \Qkf(?/)\2 M}1/2 „S(f)(x), ^pZbJ, k) (x)J
which, combined with Definition 10, further implies that
Wf}H'P{X) = }S(/)}L^(X) S WS,*(/)WL^(A) .
This shows f e HV(X) and, hence, finishes the proof of the sufficiency.
Next, we prove the necessity. By Definition 11 and (8), we find that, for any f e Hip(X) and x e X,
0/*( /)(x)]2 = [2
8
+2 2
keZB{x,5k) J ° keZB(x,2i+i<5k)\B(x,2iSk)
\Qkf (y)\2 X
k
1 A
dy(y)
. 5k + d(x,y)J V&k (x) + Vik (y) ,2
<
s 2 f \Qkf(?/)\s
B( x, k) 8
Vsk (y)
+
8
+ 2 2"°+1)A 2 f
\Qkf (y)i
B(x,2i+1S k)
dy(y)
V&k (y)
2
Sr(f)(x)\ + £ 2-^A[S2a)(/)(x) i-1
2
2 2-
-jA
S2a )(/)(x)
12
which further implies that
/- 8
0a(/)(x)S 2 2- A №\f)(x)
i2 ^v 2 8
1 s 2 2-^(/xx).
J j=°
'14)
Moreover, from X > , (5), and Lemma 2(i), it follows that there
exists a q > q(<), such that X > and << e Aq(X). By this, (14), the facts that <(x, •) is non-decreasing for almost every x e X and < is of uniformly lower type p, Lemma 3(i), and Lemma 5, we conclude that, for
x
x x j"0
any f e H*>(X),
~ 8
vUgtmx)) d^x) < p(x, 2 2-^S$\f)(x)) d/i(x) <
3=0
<p (x, 2-£s$\f )(x)) d^x)<
2 2-^ • 2^ \<p (x,S[a\f)(x)) d^x) ' 0 x
<P (x,sia\f)px)) d^x),
<
x
which, together with (9), Lemma 3(ii), the positive homogeneity of both g* and Sf^, and the fact that p is of uniformly upper type 1, further implies that
d»{x)= f ^(x,g*Jf-^-)(x)) <
v \\j \\hv{Xy J V y\\j \\HV{X)> >
A" A"
X
< I jjf-)(x)) dvi(x)
pU „^){x) ) d»{x)„ 1.
V \\b (T )\\Lvpx ) '
X
Thus, there exists a positive constant C, such that, for any f e HV{X),
h\u )\\lv(X) ^ C\\f \\hv{X).
This finishes the proof of the necessity and, hence, of Theorem 1. □
Remark 2. Let u and rq be the same, respectively, as in (3) and Definition 8.
(i) Assume that p e + rq), 1] and
<p{x,t) := tp, @x e X, @ t e [0, 8).
Then HV{X) is just the classical Hardy space HP(X). In this case, Theorem 1 shows the Littlewood-Paley g*-function characterization of HP{X) with the best known range X e (2u/p, 8), which coincides with [11, Theorem 5.12].
(ii) Recall that Fu et al. established the Littlewood-Paley g*-function characterization of H^(X) with X e (u[^^ + 1], 8) in [7, Theorem 6.16]. Thus, Theorem 1 improves the conclusion of [7, Theorem 6.16] by widening the range of X into X e (2cvgJip), 8).
Acknowledgment. The author would like to thank Xing Fu for some helpful discussions on the subject of this article. The author also would like to thank both referees for their careful reading and helpful comments which indeed improved the presentation of this article. This project is partially supported by China Postdoctoral Science Foundation (Grant No. 2022M721024), the National Natural Science Foundation of China (Grant No. 12301112), and the Open Project Program of Key Laboratory of Mathematics and Complex System of Beijing Normal University (Grant No. K202304).
References
[1] Auscher P., Hytonen T. Orthonormal bases of regular wavelets in spaces of homogeneous type. Appl. Comput. Harmon. Anal., 2013, vol. 34, no. 2, pp. 266-296. DOI: http://doi.org/10.1016/j.acha.2012.05.002
[2] Bonami A., Grellier S., Ky, L. D. Paraproducts and products of functions in BMO(Rra) and H 1(Rra) through wavelets. J. Math. Pures Appl. (9), 2012, vol. 97, no. 3, pp. 230-241.
DOI: https://doi.org/10.1016/j.matpur.2011.06.002
[3] Bonami A., Iwaniec T., Jones P., Zinsmeister M. On the product of functions in BMO and H1. Ann. Inst. Fourier (Grenoble), 2007, vol. 57, no. 5, pp. 1405-1439. DOI: https://doi.org/10.5802/aif.2299
[4] Calderon A.-P., Torchinsky A. Parabolic maximal functions associated with a distribution. II. Adv. Math., 1977, vol. 24, no. 1, pp. 101-171.
DOI: https://doi.org/10.1016/S0001-8708(77)80016-9
[5] Coifman R. R., Weiss G. Extensions of Hardy spaces and their use in analysis. Bull. Amer. Math. Soc., 1977, vol. 83, no. 4, pp. 569-645. DOI: https://doi.org/10.1090/S0002-9904-1977-14325-5
[6] Duong X.-T., Yan L. Hardy spaces of spaces of homogeneous type. Proc. Amer. Math. Soc., 2003, vol. 131, no. 10, pp. 3181-3189.
DOI: https://doi.org/10.1090/S0002-9939-03-06868-0
[7] Fu X., Ma T., Yang D. Real-variable characterizations of Musielak-Orlicz Hardy spaces on spaces of homogeneous type. Ann. Acad. Sci. Fenn. Math., 2020, vol. 45, no. 1, pp. 343-410.
DOI: https://doi .org/10.5186/aasfm.2020.4519
[8] Grafakos L. Classical Fourier Analysis. Third edition, Graduate Texts in Mathematics 249, Springer, New York, 2014.
DOI: https://doi.org/10.1007/978-1-4939-1194-3
[9] Han Y., Müller D., Yang D. Littlewood-Paley characterizations for Hardy spaces on spaces of homogeneous type. Math. Nachr., 2006, vol. 279, no. 1314, pp. 1505-1537. DOI: https://doi.org/10.1002/mana.200610435
[10] Han Y., Müller D., Yang D. A theory of Besov and Triebel-Lizorkin spaces on metric measure spaces modeled on Carnot-Caratheodory spaces. Abstr. Appl. Anal., 2008, Art. ID 893409, 250 pp.
DOI: https://doi.org/10.1155/2008/893409
[11] He Z., Han Y., Li J., Liu L., Yang D., Yuan W. A complete real-variable theory of Hardy spaces on spaces of homogeneous type. J. Fourier Anal. Appl., 2019, vol. 25, no. 5, pp. 2197-2267.
DOI: https://doi.org/10.1007/s00041-018-09652-y
[12] He Z., Liu L., Yang D., Yuan W. New Calderon reproducing formulae with exponential decay on spaces of homogeneous type. Sci. China Math., 2019, vol. 62, no. 2, pp. 283-350.
DOI: https://doi.org/10.1007/s11425-018-9346-4
[13] He Z., Yang D., Yuan W. Real-variable characterizations of local Hardy spaces on spaces of homogeneous type. Math. Nachr., 2021, vol. 294, no. 5, pp. 900-955. DOI: https://doi.org/10.1002/mana.201900320
[14] Ho K.-P. Sublinear operators on weighted Hardy spaces with variable exponents. Forum Math., 2019, vol. 31, no. 3, pp. 607-617.
DOI: https://doi .org/10.1515/forum-2018-0142
[15] Hou S., Yang D., Yang S. Lusin area function and molecular characterizations of Musielak-Orlicz Hardy spaces and their applications. Commun. Contemp. Math., 2013, vol. 15, no. 6, 1350029, 37 pp.
DOI: https://doi .org/10.1142/S0219199713500296
[16] Hou S., Yang D., Yang S. Musielak-Orlicz BMO-type spaces associated with generalized approximations to the identity. Acta Math. Sin. (Engl. Ser.), 2014, vol. 30, no. 11, pp. 1917-1962.
DOI: https://doi.org/10.1007/s10114-014-3181-9
[17] Hytonen T., Kairema A. Systems of dyadic cubes in a doubling metric space. Colloq. Math., 2012, vol. 126, no. 1, pp. 1-33.
DOI: https://doi .org/10.4064/cm126-1-1
[18] Iwaniec T., Onninen J. %l-estimates of Jacobians by subdeterminants. Math. Ann., 2002, vol. 324, no. 2, pp. 341-358.
DOI: https://doi.org/10.1007/s00208-002-0341-5
[19] Janson S. Generalizations of Lipschitz spaces and an application to Hardy spaces and bounded mean oscillation. Duke Math. J., 1980, vol. 47, no. 4, pp. 959-982. DOI: https://doi.org/10.1215/S0012-7094-80-04755-9
[20] Ky L. D. Bilinear decompositions and commutators of singular integral operators. Trans. Amer. Math. Soc., 2013, vol. 365, no. 6, pp. 2931-2958. DOI: https://doi.org/10.1090/S0002-9947-2012-05727-8
[21] Ky L. D. New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators. Integral Equations Operator Theory, 2014, vol. 78, no. 1, pp. 115-150.
DOI: https://doi.org/10.1007/s00020-013-2111-z
[22] Li B., Fan X., Yang, D. Littlewood-Paley characterizations of anisotropic Hardy spaces of Musielak-Orlicz type. Taiwanese J. Math., 2015, vol. 19, no. 1, pp. 279-314. DOI: https://doi.org/10.11650/tjm.19.2015.4692
[23] Li B., Yang D., Yuan W. Anisotropic Hardy spaces of Musielak-Orlicz type with applications to boundedness of sublinear operators. The Scientific World Journal, 2014, Article ID 306214, 19 pp.
DOI: https://doi.org/10.1155/2014/306214
[24] Liang Y., Huang J., Yang D. New real-variable characterizations of Musielak-Orlicz Hardy spaces. J. Math. Anal. Appl., 2012, vol. 395, no. 1, pp. 413-428. DOI: https://doi.org/10.1016Zj.jmaa.2012.05.049
[25] Nakai E., Sawano Y. Hardy spaces with variable exponents and generalized Campanato spaces. J. Funct. Anal., 2012, vol. 262, no. 9, pp. 3665-3748. DOI: https://doi.org/10.1016/jofa.2012.01.004
[26] Stein E. M. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series 43, Princeton University Press, Princeton, NJ, 1993.
DOI: https://doi.org/10.1515/9781400883929
[27] Stein E. M., Weiss G. On the theory of harmonic functions of several variables. I. The theory of Hp-spaces. Acta Math., 1960, vol. 103, no. 1-2, pp. 25-62. DOI: https://doi.org/10.1007/BF02546524
[28] Strömberg J.-O., Torchinsky A. Weighted Hardy Spaces. Lecture Notes in Mathematics 1381, Springer-Verlag, Berlin, 1989.
DOI: https://doi .org/10.1007/BFb0091154
[29] Yan X., He Z., Yang D., Yuan W. Hardy spaces associated with ball quasi-Banach function spaces on spaces of homogeneous type: Characterizations of maximal functions, decompositions, and dual spaces. Math. Nachr., 2023, vol. 296, no. 7, pp. 3056-3116.
DOI: https://doi.org/10.1002/mana.202100432
[30] Yang D., Liang Y., Ky L. D. Real-Variable Theory of Musielak-Orlicz Hardy Spaces. Lecture Notes in Mathematics 2182, Springer-Verlag, Cham, 2017. DOI: https://doi.org/10.1007/978-3-319-54361-1
[31] Yang D., Yuan W., Zhang Y. Bilinear decomposition and divergence-curl estimates on products related to local Hardy spaces and their dual spaces. J. Funct. Anal., 2021, vol. 280, no. 2, Paper No. 108796, 74 pp.
DOI: https://doi.org/10.1016Zj.jfa.2020.108796
Received April 24, 2023. In revised form, October 10, 2023. Accepted November 03, 2023. Published online December 02, 2023.
Institute of Contemporary Mathematics, School of Mathematics and Statistics, Henan University
Kaifeng 475004, The People's Republic of China E-mail: xianjieyan@henu.edu.cn
