Musielak-Orlicz-Hardy Spaces Associated with Operators Satisfying Reinforced Off-Diagonal Estimates

Let $\mathcal{X}$ be a metric space with doubling measure and $L$ a one-to-one operator of type $\omega$ having a bounded $H_\infty$-functional calculus in $L^2(\mathcal{X})$ satisfying the reinforced $(p_L, q_L)$ off-diagonal estimates on balls, where $p_L\in[1,2)$ and $q_L\in(2,\infty]$. Let $\varphi:\,\mathcal{X}\times[0,\infty)\to[0,\infty)$ be a function such that $\varphi(x,\cdot)$ is an Orlicz function, $\varphi(\cdot,t)\in {\mathbb A}_{\infty}(\mathcal{X})$ (the class of uniformly Muckenhoupt weights), its uniformly critical upper type index $I(\varphi)\in(0,1]$ and $\varphi(\cdot,t)$ satisfies the uniformly reverse H\"older inequality of order $(q_L/I(\varphi))'$. In this paper, the authors introduce a Musielak-Orlicz-Hardy space $H_{\varphi,\,L}(\mathcal{X})$, via the Lusin-area function associated with $L$, and establish its molecular characterization. In particular, when $L$ is nonnegative self-adjoint and satisfies the Davies-Gaffney estimates, the atomic characterization of $H_{\varphi,\,L}(\mathcal{X})$ is also obtained. Furthermore, a sufficient condition for the equivalence between $H_{\varphi,\,L}(\mathbb{R}^n)$ and the classical Musielak-Orlicz-Hardy space $H_{\varphi}(\mathbb{R}^n)$ is given. Moreover, for the Musielak-Orlicz-Hardy space $H_{\varphi,\,L}(\mathbb{R}^n)$ associated with the second order elliptic operator in divergence form on $\rn$ or the Schr\"odinger operator $L:=-\Delta+V$ with $0\le V\in L^1_{\mathrm{loc}}(\mathbb{R}^n)$, the authors further obtain its several equivalent characterizations in terms of various non-tangential and radial maximal functions; finally, the authors discuss the boundedness of the Riesz transform $\nabla L^{-1/2}$.


Introduction
The real-variable theory of Hardy spaces H (R ) with ∈ (0 1], introduced by Stein and Weiss [70] and systematically developed in the seminal paper of Fefferman and Stein [31], plays a center role in various fields of harmonic analysis and partial differential equations (see, for example, [21,66] and the references therein). One of the main features of the Hardy space H (R ) with ∈ (0 1] is their atomic decomposition characterizations (see [20] for = 1 and [54] for > 1). Later, the theory of weighted Hardy spaces H (R ) with Muckenhoupt weights has been studied by García-Cuerva [33], and Strömberg and Torchinsky [69]. Furthermore, Strömberg [68] and Janson [44] introduced the Orlicz-Hardy space which plays an important role in studying the theory of nonlinear PDEs (see, for example, [13,14,16,36,43]). Recently, in [49], the last two authors of the present paper studied Hardy spaces of Musielak-Orlicz type which generalize the Orlicz-Hardy space in [44,68] and the weighted Hardy spaces in [33,69]. Furthermore, several real-variable characterizations of the Hardy spaces of Musielak-Orlicz type were established in [42,56]. Moreover, the local Hardy space of Musielak-Orlicz type was studied in [73]. It is worth pointing out that Musielak-Orlicz functions are the natural generalization of Orlicz functions (see, for example, [28,29,49,59]) and the motivation to study function spaces of Musielak-Orlicz type is attributed to their extensive applications to many fields of mathematics (see, for example, [13-16, 28, 29, 49, 50, 55] for more details). However, it is now understood that there are many settings in which the theory of the spaces of Hardy type can not be applicable; for example, the Riesz transform ∇L −1/2 may not be bounded from H 1 (R ) to L 1 (R ) when L := −div(A∇) is a second order divergence elliptic operator with complex bounded measurable coefficients (see, for example, [40]). Recently, there has been a lot of studies which pay attention to the theory of function spaces associated with operators. In many applications, the very dependence on the function spaces associated with the operators provides many advantages in studying the boundedness of singular integrals which may not fall within the scope of the classical Calderón-Zygmund theory. Here, we would like to give a brief overview of this research direction. Let L be an infinitesimal generator of an analytic semigroup { − L } >0 on L 2 (R ) whose kernels satisfy the Gaussian upper bound estimates. The theory on Hardy spaces associated with such operators L was investigated in [5,30]. Later, Hardy spaces associated with operators which satisfy the weaker conditions, so-called Davies-Gaffney estimate conditions, were treated in the works of Auscher et al. [8], Hofmann and Mayboroda [40] and Hofmann et al. [39,41]. In [45-47, 57, 71-74], the authors studied the Orlicz-Hardy spaces associated with operators and, in some sense, these results are extensions to Hardy spaces associated with operators. Then, the weighted Hardy spaces associated with operators were also considered in [67] and [17]. Recently, in [74], the last two authors of this paper studied the Musielak-Orlicz-Hardy spaces associated with nonnegative self-adjoint operators satisfying Davies-Gaffney estimates. Furthermore, some special Musielak-Orlicz-Hardy spaces associated with the Schrödinger operator L := −∆ + V on R , where the nonnegative potential V satisfies the reverse Hölder inequality of order /2, were studied by the third author of this paper [51][52][53] and further applied to the study of commutators of singular integral operators associated with the operator L. Very recently, the authors of this paper [12] studied the weighted Hardy space associated with nonnegative self-adjoint operators satisfying the reinforced off-diagonal estimates on R (see Assumption (B) for their definitions in the present setting), which improves those results in [17,67,74] in some sense by essentially extending the range of the considered weights. We would like to describe partly the results in [74] which may be closely related to this paper. Let L be a nonnegative self-adjoint operator on for all ∈ and ∈ [0 ∞), with some α ∈ (0 1], β ∈ [0 ) and γ ∈ [0 2α(1 + ln 2)] (see Section 2.2 for more details). Then, the last two authors of the present paper [74] introduced a Musielak-Orlicz-Hardy space H L ( ), via the Lusin-area function associated with L, and obtained two equivalent characterizations of H L ( ) in terms of the atom and the molecule. Hence, it is natural to raise the question when the condition ( ·) ∈ RH 2/[2−I( )] ( ) can be relaxed. One of the main aims of this paper is to give an affirmative answer to this question. Moreover, motivated by [7,12,18,74], in this paper, we consider more general operators by assuming that the considered operator satisfies Assumptions (A) and (B) in Subsection 2.3 of this paper. Indeed, Assumption (A) is weaker than "the nonnegative and self-adjoint" condition imposing on the operator L in [74]. Meanwhile, in Assumption (B), we first introduce the notion of the reinforced ( L L ) off-diagonal estimates on balls in spaces of homogeneous type (see Definition 2.7 below), which is quite wide so that it can provide a framework to treat almost the results in previous works (see, for example, [5,12,30,39,40,45,46,74]). Under Assumptions (A) and (B), we first introduce the Musielak-Orlicz-Hardy spaces H L ( ) (see Definition 4.1 below), via the Lusin-area function associated with L, and then characterize the spaces H L ( ) in terms of the molecule with ( ·) ∈ RH ( L /I( )) ( ) (see Theorem 4.8 below), where ( L /I( )) denotes the conjugate exponent of L /I( ). In particular, when L is nonnegative self-adjoint and satisfies the Davies-Gaffney estimates, the atomic characterization of H L ( ) is also obtained (see Theorem 5.4 below). It is important to notice that RH 2/[2−I( )] ( ) ⊂ RH ( L /I( )) ( ) whenever L > 2 and hence the results in this paper improve significantly those in [74], by enlarging the range of the weights. In the particular case that the heat kernels associated with { − L } >0 satisfy the Gaussian upper bound estimate, then L = 1 and L = ∞ and hence the class of can be extended to ( ·) ∈ A ∞ ( ). Moreover, we also give a sufficient conditions on L so that our Musielak-Orlicz-Hardy space H L ( ) coincides with the Musielak-Orlicz-Hardy space H ( ) introduced by the third author of this paper in [49] when := R (see Theorem 6.7 below). As applications, we consider Musielak-Orlicz-Hardy spaces H L ( ) in some particular cases, for example, L being the second order elliptic operator in divergence form or the Schrödinger operator. More precisely, for the Musielak-Orlicz-Hardy space H L (R ) associated with the second order elliptic operator in divergence form on R with bounded measurable complex coefficients or the Schrödinger operator L := −∆ + V , where 0 ≤ V ∈ L 1 loc (R ), we further obtain its several equivalent characterizations in terms of the non-tangential and the radial maximal functions (see Theorems 7.5 and 8.3 below); finally, we show that the Riesz transform ∇L −1/2 is bounded from H L (R ) to the Musielak-Orlicz space L (R ) when ( ) ∈ (0 1], from H L (R ) to H (R ) when ( ) ∈ ( +1 1], and from H L (R ) to the weak Musielak-Orlicz-Hardy space W H (R ) when ( ) = +1 is attainable and (· ) ∈ A 1 ( ) (see Theorems 7.8, 7.11, 8.5 and 8.6 below), where ( ) denotes the uniformly critical lower type index of . One of the new ingredients appeared in this paper is the introduction of the notion of the reinforced ( L L ) offdiagonal estimates on balls in spaces of homogeneous type with ∈ N := {1 2 }. We remark that, to study the weighted Hardy space H ω (R ) on the Euclidean space R and to relax the range of the weight ω as wide as possible, the authors introduced a notion of the reinforced ( L L ) off-diagonal estimates in [12], which is particular useful for studying the weighted Hardy space associated with various differential operators of second order in the setting of Euclidean spaces. However, if we consider the differential operators on some more general spaces (for example, the Laplace-Beltrami operator on the Riemannian manifold with a doubling measure), the reinforced ( L L ) off-diagonal estimates in [12] seem no longer suitable (see Remark 2.9(a)). To overcome this difficulty, we introduce the reinforced ( L L ) off-diagonal estimates on balls by combining the ideas of the reinforced ( L L ) off-diagonal estimates from [12] and the off-diagonal estimates on balls from [7]. Also, the order ∈ N makes many differential operators of higher order fall into our scope (see Remark 2.9(c)). We also point out that, in [39,45], the authors introduced a Hardy space associated with operators L in the space of homogenous type by assuming that L satisfies the so-called Davies-Gaffney estimates. However, due to the fact that Davies-Gaffney estimates are equivalent to L 2 -L 2 off-diagonal estimates on balls (see Remark 5.1(ii)), their Hardy spaces can be viewed as a special case of ours. Another interesting ingredient appeared in this paper is the discussion of the role of the L 2 ( ) norm in the definitions of the Hardy space H L ( ) and the atomic or the molecular Hardy space. This discussion has two aspects, the first one is from [12], where the authors asked the question that what happen if we replace L 2 (R ) by L (R ) with = 2 in the definition of the Hardy space. For this question, in the present setting, we prove that the space H L ( ) is invariant when we do this replacement for all ∈ ( L L ) (see Theorem 4.9), which coincides with the result obtained in [12] when ( ) := ( ), with ∈ (0 1], for all ∈ R and ∈ [0 ∞). The second aspect of this discussion can be reduced to the following question: "what happen if we replace the L 2 (R )-convergence of the atomic (resp. molecular) representation by the L (R )-convergence with = 2 in the definition of the atomic and the molecular Hardy space? " This question arises naturally when we study the boundedness of the fractional integral between two different Hardy spaces. For this question, we prove that the atomic and the molecular Hardy spaces are invariant when we do this replacement for all ∈ ( L L ) (see Theorems 5.9 and 4.8).
The organization of this paper is as follows. In Section 2, we discuss the settings which are considered in this papers. This includes the assumptions for the function and the operator L. Then, we establish the results on the L ( )boundedness of two square functions which is useful in what follows. Section 3 is dedicated to studying the Musielak-Orlicz tent spaces. Like the classical result for the tent spaces, we also give out the atomic decomposition for the Musielak-Orlicz tent spaces. In Section 4, we first introduce the Musielak-Orlicz-Hardy space H L ( ) via the Lusin-area function and prove that the operator π L M (see (4.2) below for its definition) maps the Musielak-Orlicz tent space T ( + ) continuously into the space H L ( ) (see Proposition 4.5 below), here and in what follows, + := × (0 ∞). By this and the atomic decomposition of the space T ( + ), we establish the molecular characterization of H L ( ) (see Theorem 4.8 below). Moreover, similar to [12,Theorem 3.4], we show that H L ( ) is invariant if we replace L 2 ( ) by L ( ) with ∈ ( L L ) in the definition of H L ( ) (see Theorem 4.9 below). As a consequence, we see that L ( ) ∩ H L ( ) is dense in H L ( ) whenever ∈ ( L L ) (see Corollary 4.10 below). If L is a nonnegative self-adjoint operator in L 2 ( ) satisfying the reinforced ( L L 1) off-diagonal estimates on balls with L ∈ [1 2), in Section 5, we establish the atomic characterization of the space H L ( ) (see Theorem 5.4 below) by using the finite propagation speed for the wave equation and a similar method used in Section 4. The aim of Section 6 is to give an affirmative answer to the question "when do the Musielak-Orlicz-Hardy spaces H L (R ) and H (R ) coincide? ". More precisely, if the distribution kernel of the heat semigroup { − L } >0 satisfies the Gaussian upper bound estimate, some Hölder regularity and the conservation (see Assumption (C) below for details), then the spaces H L (R ) and H (R ) coincide with equivalent quasi-norms (see Theorem 6.7 below). In Section 7, as a special case, we further study the Musielak-Orlicz-Hardy space H L (R ) associated with the second order elliptic operator in divergence on R with complex bounded measurable coefficients. By making full use of the special structure of the divergence form elliptic operator and establishing a good-λ inequality concerning the nontangential maximal function and the truncated Lusin-area function, we obtain the radial and the non-tangential maximal function characterizations of H L (R ) (see Theorem 7.5 below). We remark that the proof of Theorem 7.5 is similar to that of [74,Theorem 7.4 for all ∈ [0 ∞). In Section 8, we consider the Musielak-Orlicz-Hardy spaces H L (R ) associated with the Schrödinger operator L := −∆ + V , where 0 ≤ V ∈ L 1 loc (R ). Similar to Section 7, we establish several equivalent characterizations of H L (R ) in terms of the radial and the non-tangential maximal functions associated with the heat and the Poisson semigroups of L (see Theorem 8.3 below). Moreover, we also study the boundedness of ∇L −1/2 on the space H L (R ) (see Theorems 8.5 and 8.6 below). It is worth pointing out that Theorems 8.3 and 8.5, respectively, improve [74,Theorem 7.4] and [74,Theorems 7.11 and 7.15] by extending the range of weights (see Remarks 8.4 and 8.7 below for details). Finally we make some conventions on notation. Throughout the whole paper, we denote by C a positive constant which is independent of the main parameters, but it may vary from line to line. We also use C (γ β ) to denote a positive constant depending on the indicated parameters γ, β, . The symbol A B means that A ≤ C B. If A B and B A, for ∈ R denotes the maximal integer not more than . For any given normed spaces and with the corresponding norms || · || and || · || , the symbol ⊂ means that for all ∈ , then ∈ and || || || || . Also given λ > 0, we write λB for the λ-dilated ball, which is the ball with the same center as B and with radius λB = λ B . We also set N :

Preliminaries
In Subsection 2.1, we first recall some notions on metric measure spaces and then, in Subsection 2.2, we state some notions and assumptions concerning growth functions considered in this paper and give some examples which satisfy these assumptions; finally, we recall some properties of growth functions established in [49]. In Subsection 2.3, we describe some basic assumptions on the operator L studied in this paper and then study the L ( )-boundedness of two square functions associated with L.

Metric measure spaces
Throughout the whole paper, we let be a set, a metric on and µ a nonnegative Borel regular measure on . For all ∈ and ∈ (0 ∞), let B( ) := { ∈ : ( ) < } and V ( ) := µ(B( )). Moreover, we assume that there exists a constant C ∈ [1 ∞) such that, for all ∈ and Observe that ( µ) is a space of homogeneous type in the sense of Coifman and Weiss [23]. Recall that in the definition of spaces of homogeneous type in [23,Chapter 3], is assumed to be a quasi-metric. However, for simplicity, we always assume that is a metric. Notice that the doubling property (2.1) implies the following strong homogeneity property that, for some positive constants C and , uniformly for all λ ∈ [1 ∞), ∈ and ∈ (0 ∞). There also exist constants C ∈ (0 ∞) and N ∈ [0 ] such that, for all ∈ and ∈ (0 ∞), Indeed, the property (2.3) with N = is a simple corollary of the triangle inequality for the metric and the strong homogeneity property (2.2). In the cases of Euclidean spaces and Lie groups of polynomial growth, N can be chosen to be 0. Furthermore, for ∈ (0 ∞], the space of -integrable functions on is denoted by L ( ) and the (quasi-)norm of ∈ L ( ) by || || L ( ) .

Growth functions
Recall that a function Φ : [59,62,63]). The function Φ is said to be of upper type (resp. lower type ) for some ∈ [0 ∞), if there exists a positive constant C such that for all ∈ [1 ∞) (resp. ∈ [0 1]) and For a given function : Following [49], (· ) is called uniformly locally integrable if, for all bounded sets K in , The function (· ) is said to satisfy the uniformly Muckenhoupt condition for some ∈ [1 ∞), denoted by ∈ A ( ), if, when ∈ (1 ∞), Here the first supremums are taken over all ∈ (0 ∞) and the second ones over all balls B ⊂ . The function (· ) is said to satisfy the uniformly reverse Hölder condition for some ∈ (1 ∞], denoted by ∈ RH ( ), if, when ∈ (1 ∞), Here the first supremums are taken over all ∈ (0 ∞) and the second ones over all balls B ⊂ .
Let A ∞ ( ) := ∪ ∈[1 ∞) A ( ) and define the critical indices of ∈ A ∞ ( ) as follows: Now we introduce the notion of growth functions.
The function is of positive uniformly upper type 1 and of uniformly lower type 2 for some 2 ∈ (0 1].

Remark 2.3.
From the definitions of the uniformly upper type and the uniformly lower type, we deduce that, if the growth function is of positive uniformly upper type 1 with 1 ∈ (0 1], and of positive uniformly lower type 2 with 2 ∈ (0 1], then 1 ≥ 2 . Clearly, ( ) := ω( )Φ( ) is a growth function if ω ∈ A ∞ ( ) and Φ is an Orlicz function of lower type for some  ( 1]. Recall that if an Orlicz function is of upper type ∈ (0 1), then it is also of upper type 1.
Another typical and useful example of the growth function is as in (1.2). It is easy to show that ∈ A 1 ( ), is of uniformly upper type α, I( ) = ( ) = α, ( ) is not attainable, but I( ) is attainable. Moreover, it worths to point out that such function naturally appears in the study of the pointwise multiplier characterization for the BMO-type space on the metric space with doubling measure (see [60,61]); see also [50][51][52][53] for some other applications of such functions. Throughout the whole paper, we always assume that is a growth function as in Definition 2.2. Let us now introduce the Musielak-Orlicz space. The Musielak-Orlicz space L ( ) is defined to be the set of all measurable functions such that In what follows, for any measurable subset E of and ∈ [0 ∞), we let  . Then is a growth function, which is equivalent to ; moreover, ( ·) is continuous and strictly increasing.

(iii) Let be a growth function as in Definition 2.2. Then
We have the following properties for A ∞ ( ), whose proofs are similar to those in [34,35].

Remark 2.8.
(i) It is easy to see that, for ≤ 1 ≤ 1 ≤ , (ii) Similar to [7, Proposition 2.2], we see that A ∈ (L 1 −L ∞ ) if and only if the associated kernel of A satisfies the Gaussian upper bound, namely, there exist positive constants and C such that, for all ∈ and ∈ (0 ∞), if and only if its dual, A * , belongs to (L − L ). Now, we make the following two assumptions on operators L, which are used through the whole paper. Assumption (A). Assume that the operator L is a one-to-one operator of type ω in L 2 ( ) with ω ∈ [0 π/2), has dense range in L 2 ( ) and a bounded H ∞ -functional calculus in L 2 ( ).

Remark 2.9.
(a) We first point out that in Assumptions (A) and (B), if L is non-negative self-adjoint, is the Euclidean space R and = 1, from [7, Proposition 3.2], it follows that the notion of the reinforced ( L L ) off-diagonal estimates on balls is the same as the reinforced ( L L ) off-diagonal estimates introduced in [12] (see [11,27,32] and their references for the history of the off-diagonal estimates). Here, we use the off-diagonal estimates on balls, because they coincide with the off-diagonal estimates when = R , and the off-diagonal estimates on balls seem more suitable in a general space of homogeneous type. For example, the heat semigroup − ∆ on functions for general Riemannian manifolds with a doubling measure is not L − L bounded when < unless the measure of any ball is bounded below by a power of its radius. However, if we assume the L − L off-diagonal estimates, it then implies the L − L boundedness (see also the discussions above [7, Proposition 3.2]).  (i) the second order divergence form elliptic operators with complex bounded coefficients as in [40] (see also (7.1) below for its precise definition); (ii) the 2 -order homogeneous divergence form elliptic operators interpreted in the usual weak sense via a sesquilinear form, with complex bounded measurable coefficients α β for all multi-indices α and β (see, for example, [10,18]); (iii) the Schrödinger operator −∆ + V on R with the nonnegative potential V ∈ L 1 loc (R ) (see, for example, [39,45] and related references); (iv) the Schrödinger operator −∆ + V on R with the suitable real potential V as in [3]; (v) the nonnegative self-adjoint operators satisfying Gaussian upper bounds, namely, there exist positive constants C and such that, for all ∈ and ∈ (0 ∞), where is the associated kernel of − L and ∈ N; (d) We point out that the condition that L is one-to-one is necessary for the bounded H ∞ functional calculus on L 2 ( ) (see [25,58]). Moreover, from [25,Theorem 2.3], it follows that if T is a one-to-one operator of type ω in L 2 ( ), then T has dense domain and dense range; (e) If L is nonnegative self-adjoint on L 2 ( ) satisfying the reinforced ( L L ) off-diagonal estimates on balls, then the condition that L is one-to-one can be removed and we can introduce another kind of functional calculus by using the spectral theorem. More precisely, in this case, for every bounded Borel function F : [0 ∞) → C, we define the operator F (L) : L 2 ( ) → L 2 ( ) by the formula where E L (λ) is the spectral resolution of L (see [39] for more details). Observe also that a one-to-one nonnegative self-adjoint operator is of type 0.
Assume that the operator L satisfies Assumptions (A) and (B). For all ∈ N, the vertical square function G L is defined by setting, for all ∈ L 2 ( ) and ∈ , which is bounded on L 2 ( ) (see, for example, [58]). When = 1, we write G L instead of G L 1 .

Theorem 2.10.
Let L satisfy Assumptions (A) and (B), ∈ N and, L and L be as in Assumption (B). Then G L is bounded on L ( ) for all ∈ ( L L ).
To prove Theorem 2.10, we need the following two criteria, which are due to [4] (see also [6]).

Lemma 2.11.
where denotes the Hardy-Littlewood maximal function (see Lemma 2.5(vi)). Then T is of strong type ( 0 0 ). Now we prove Theorem 2.10 by using Lemmas 2.11 and 2.12.

Proof of Theorem 2.10.
For the sake of simplicity, we only give the proof for = 1. Since G L is bounded on L 2 ( ), we can assume that L < 2 < L . We now consider the following two cases. Case 1). ∈ ( L 2).
In this case, we apply Lemma 2.11 with . Thus, (2.12) holds. It remains to show that for all ∈ N with ≥ 3, balls B and ∈ L ( ) supported in B, it holds that We first estimate I. Write By the fact that (( 2 which, together with Minkowski's integral inequality and (2.16), implies that Likewise, we also see that This, combined with (2.15) and (2.17), shows that (2.14) holds as long as M > ( +θ 1 )/2 . Thus, as a direct consequence of Lemma 2.12 and interpolation, we conclude that G L is bounded on L ( ) for all ∈ ( L 2). Case 2). ∈ (2 L ).
In this case, we first prove (2.13) for T := G L and A B : . Then, by an argument similar to that used in the proof of Case 1), we conclude that (2.13) holds true for T := G L and A B : To see this, by Minkowski's inequality, we conclude that Thus, (2.19) holds and hence (2.18) holds true. By this and Lemma 2.12, we see that G L is bounded on L ( ) for all ∈ (2 L ), which completes the proof of Theorem 2.10.
For all ∈ N, the non-tangential square functions S L is defined by setting, for all ∈ L 2 ( ) and ∈ , In particular case = 1, we omit the subscript to write S L . It is easy to show that, for all ∈ L 2 ( ), ||S L ( )|| L 2 ( ) ||G L ( )|| L 2 ( ) and hence S L is bounded on L 2 ( ). Moreover, we have the following boundedness of S L on L ( ).

Theorem 2.13.
Let L satisfy Assumptions (A) and (B), ∈ N and L and L be as in Assumption (B). Then S L is bounded on L ( ) for all ∈ ( L L ).
Proof. Without the loss of generality, we may assume that = 1. Similar to the proof of Theorem 2.10, we also consider the following two cases for . Case 1). ∈ ( L 2).
In this case, we apply Theorem 2.11 to this situation for T := S L and A B : To show this, we first write · · · =: I 1 + I 2 Let us first estimate I 1 . Let . This, together with the fact that At this stage, by an argument used in Case 1) of the proof of Theorem 2.10, we conclude that Likewise, for I 2 , we write Notice that in this situation, B ⊂ B and hence = χ B . By an argument similar to that used in the proof of Theorem 2.10 and the fact that ( L) − L ∈ (L − L 2 ) for all ∈ Z + , we see that where is the dimension of appearing in (2.2). From these estimates of K and H , we deduce that (2.21) holds and hence S L is bounded on L ( ) for all ∈ ( L 2). Case 2). ∈ (2 L ).
In this case, for any ∈ L ( /2) ( ), from Fubini's theorem and Hölder's inequality, we infer that At this stage, using Theorem 2.10 and the fact that which implies that S L is bounded on L ( ) for all ∈ (2 L ) and hence completes the proof of Theorem 2.13.

Musielak-Orlicz tent spaces
In this section, we study the Musielak-Orlicz tent space associated with the growth function. We first recall some notions as follows.
Coifman, Meyer and Stein [22] introduced the tent space T 2 (R +1 + ) for ∈ (0 ∞), here and in what follows, R +1 + := R ×(0 ∞). The tent space T 2 ( + ) on spaces of homogenous type was introduced by Russ [65]. Recall that a measurable function is said to belong to the tent space T 2 ( Moreover, Harboure, Salinas and Viviani [37], and Jiang and Yang [45], respectively, introduced the Orlicz tent spaces T Φ (R +1 + ) and T Φ ( + ). Let be as in Definition 2.2. In what follows, we denote by T ( + ) the space of all measurable functions on + such that ( ) ∈ L ( ) and, for any ∈ T ( + ), its quasi-norm is defined by Furthermore, if A is a (T )-atom for all ∈ (1 ∞), we then call A a ( ∞)-atom. For functions in T ( + ), we have the following atomic decomposition.

Let be as in Definition 2.2. Then for any ∈ T ( + ), there exist {λ } ⊂ C and a sequence {A } of (T ∞)-atoms associated with {B } such that, for almost every
Moreover, there exists a positive constant C such that, for all ∈ T ( + ), The proof of Theorem 3.1 is similar to that of [74, Theorem 3.1]. We omit the details here.
The proof of Corollary 3.2 is similar to that of [74, Corollary 3.5] and hence we omit the details here.
Here and in what follows, a function on + is said to have bounded support means that there exist a ball B ⊂ and 0 < 1 < 2 < ∞ such that supp ⊂ B × ( 1 2 ).
The proof of Proposition 3.3 is an application of the uniformly lower type 2 property of for some 2 ∈ (0 1], which is similar to that of [42,Proposition 3.5]. We omit the details.

The Musielak-Orlicz-Hardy space H L ( ) and its molecular characterization
In this section, we first introduce the Musielak-Orlicz-Hardy space H L ( ) associated with the operator L via the Lusin-area function. Then we establish an equivalent characterization of H L ( ) in terms of the molecule. We begin with some notions and notation. Let L satisfy Assumptions (A) and (B), and ∈ N be as in (2.9). For all ∈ L 2 ( ), the Lusin-area function S L is defined as in (2.20). By Theorem 2.13, we know that, for any ∈ ( L L ), where L and L are as in Assumption (B), there exists a positive constant C ( ) , depending on , such that, for all ∈ L ( ), Now we introduce the Musielak-Orlicz-Hardy H L ( ) via the Lusin-area function S L .

Definition 4.1.
Let be as in Definition 2.2 and L satisfy Assumptions (A) and (B). Assume that L and L are as in Assumption (B). A function ∈ L ( ) with ∈ ( L L ) is said to be in H L ( ) if S L ( ) ∈ L ( ) and, moreover, define In what follows, for the simplicity of the notation, we write H L ( ) := H L 2 ( ).
For the operator π L M , we have the following boundedness.
4}, by (i), we know that When ∈ N with ≥ 5, take ∈ L ( ) satisfying || || L ( ) ≤ 1 and supp ⊂ S (B). Then from Hölder's inequality and ∈ ( L 2], we infer that Moreover, by Assumption (B), we see that which, together with (4.5), implies that From this and the choice of , we deduce that, for each ∈ N with ≥ 5, 4}, take ∈ L ( ) satisfying || || L ( ) ≤ 1 and supp ⊂ S (B). Then it follows, from Hölder's inequality and the L ( )-boundedness of S L * M+1− , that When ∈ N with ≥ 5, similar to the proof of (4.5), we know that, for each ∈ {1 M}, which, together with (4.4), (4.6) and (4.7), implies that α is a ( . We now claim that, for any λ ∈ C and ( M ) L -molecule α associated with the ball B ⊂ , If (4.8) holds, from this, the facts that, for all λ ∈ (0 ∞), and hence completes the proof of (ii). Now we prove (4.8). By the definition of α, we see that 4}, by the uniformly upper type 1 and lower type 2 properties of , we see that Now we estimate G . From Hölder's inequality, Theorem 2.13, ∈ RH ( / 1 ) ( ) and Lemma 2.5(vi), we deduce that For H , similarly, we have which, together with (4.10) and (4.11), implies that, for each ∈ Z + and ∈ {0 1 4}, For all ∈ Z + and ∈ , let It is easy to see that when . Then by Hölder's inequality, Fubini's theorem and Assumption (B), we conclude that By using (4.13), similar to the proof of (4.12), we know that for any ∈ Z + and ∈ N with ≥ 5, Now we deal with F . Let 4}, similar to the proof of (4.12), we conclude that For each ∈ Z + and all ∈ , let We first see that, for all ∈ , For G 1 , similar to (4.13), we conclude that, when ∈ N with ≥ 5, For G 2 , by Theorem 2.13, we find that which, together with (4.16) and (4.17), implies that By using this estimate, similar to the proof of (4.14), we see that, for all ∈ Z + and ∈ N with ≥ 5, which, together with (4.9) through (4.15) and > [ 0 2 + 1 2 + θ 1 + θ 2 ] − 1, implies that (4.8) holds true, and hence completes the proof of Proposition 4.5.    with some 0 ∈ . Then from Proposition 3.3, we infer that 2 L − 2 L χ O N ∈ T ( + ) ∩ T 2 2 ( + ), which implies that N ∈ H L 2 ( ). Moreover, by ∈ L ( ) and the L ( )-boundedness of S L , we conclude that S L ( ) ∈ L ( ), which implies that 2 L − 2 L ∈ T 2 ( + ). From this and the definition of T 2 ( + ), it follows that 2 L − 2 L χ O N ∈ T 2 ( + ), which, together with Proposition 4.5(i), implies that N ∈ L ( ). Thus, N ∈ H L 2 ( ) ∩ L ( ). Moreover, which completes the proof of Theorem 4.9.
As a corollary of Theorem 4.9, we have the following conclusion. We omit the details.   [39], where the dense subspace H 2 ( ) of the Hardy space is defined to be the completion of the range of L in L 2 ( ), (L) (see [39] for more details). Recall that L 2 ( ) = (L) (L), where (L) denotes the kernel of L. We know that these Hardy spaces are different from a kernel space (L), which is not essential for our purpose. We make this change in the definition of the Hardy space, because it brings us some conveniences; for example, when = 2, we obtain H L ( ) = L 2 ( ). (ii) The following definition of the L off-diagonal estimates is from [4]. For all ∈ (1 ∞), a family {T } >0 of operators is said to satisfy the L off-diagonal estimates, if there exist two positive constants C and such that holds true for every closed sets E F ⊂ , ∈ (0 ∞) and ∈ L (E). From [7], we deduce that {T } >0 ∈ 1 (L − L ) if and only if {T } >0 satisfies the L off-diagonal estimates. Thus, Assumption (H 2 ) implies that {T } >0 satisfies the L off-diagonal estimates.
To establish the atomic characterization of H L ( ), we first introduce the notion of the following atoms. To prove Theorem 5.4, we need to introduce some operator π Φ L , which can be viewed as a retraction operator from the Musielak-Orlicz-tent space T ( + ), introduced in Section 3, to H L ( ). To this end, we first give some notation. In what follows, for any operator T , we let K T be its integral kernel. Let ( √ L) with ∈ (0 ∞) be the cosine function operator generated by L. By [24,Theorem 3.4] (see also [39,Proposition 3.4]), we know that there exists a positive Moreover, let ψ ∈ C ∞ c (R) be even and supp ψ ⊂ (−C −1 0 C −1 0 ), where C 0 is as in (5.1). Let Φ denote the Fourier transform of ψ. Then, for all ∈ N and ∈ (0 ∞), the kernel of ( 2 L) Φ( where ( ) and ( ) are respectively as in (2.6) and (2.5). Assume that Φ is as in (5.2). Then, for all ∈ N, ∈ L 2 ( + ) and ∈ , the operator π Φ L is defined by Using Minkowski's integral inequality and the quadratic estimates (see also [39, (3.14)]), we easily see that π Φ L can be continuously extended from T 2 ( + ) to L 2 ( ). Moreover, we have the following boundedness of π Φ L M , which can be viewed as an extension of [74, Proposition 4.6]. Proof. Without loss of generality, we may only prove Proposition 5.6 under the assumption that ∈ [2 L ). For the case when ∈ ( L 2), the following proof is still valid, only need to make a few modifications when using Hölder's inequality. Let ∈ T ( + ). From Proposition 3.3, Theorem 3.1, Corollary 3.2 and the fact that π Φ L M is bounded from T 2 2 ( + ) to L 2 ( ), we deduce that there exist a family {A } of (T ∞)-atoms associated respectively to the balls {B } and {λ } ⊂ C such that On the other hand, for any ∈ L (B ) ∩ L 2 (B ), By (5.6), Assumption (H 1 ), Fubini's theorem, the fact that supp A ⊂ B and Hölder's inequality, we conclude that, for all ∈ {0 M}, Indeed, if (5.7) holds, then by (5.5), we see immediately that, for all ∈ T ( + ) and λ ∈ (0 ∞), To estimate I , we write I into The estimate of is similar to that of and, via replacing Assumption (H 2 ) by the L ( )-boundedness of the family of operators {( 2 L) M − 2 L } >0 , we conclude that which, together with (5.10) and (5.11), shows immediately that Thus, from this, (5.9) and Lemma 2.5(vii), we deduce that . The estimate of J is similar to that of I . We only need to point out that, from Lemma 2.5(ii) and the fact that ( 2 ) < ( 1 ) , it follows that ∈ RH ( 2 ) ( ). Thus, we conclude that Combining (5.12) and (5.13), we immediately conclude that which completes the proof of (5.7) and hence Proposition 5.6.
Before turning to the proof of Theorem 5.4, we introduce a sufficient condition which guarantees a given operator to be bounded on the atomic Musielak-Orlicz-Hardy space.   with equivalent norms. We divide the proof of (5.16) into the following two steps.
Step 1. We first prove the inclusion L 2 ( ) ∩ H L ( ) ⊂ L 2 ( ) ∩ H M L at ( ). For any ∈ L 2 ( ) ∩ H L ( ), by the bounded functional calculus in L 2 ( ), we know that there exists a positive constant C in L 2 ( ). Moreover, from the fact that 2 L − 2 L ∈ T ( + ), we deduce that there exist {λ } ⊂ C and {A } of (T ∞)atoms, respectively, associated with {B } such that To prove Theorem 5.9, we need a few lemmas. The first one is a variant of Lemma 5.7, whose proof is similar. We omit the details. (ii) for all ∈ (0 ∞), the operator 2 L − 2 L , initially defined on L 2 ( ), extends to a bounded linear operator from L ( ) to T 2 ( + ).
Proof. We first prove (i). Let ∈ T 2 ( + ) ∩ T 2 2 ( + ). For any ∈ L ( ) ∩ L 2 ( ), by Fubini's theorem, Assumption (H 1 ), Hölder's inequality and the L ( )-boundedness of the square function S Φ L , we conclude that which, together with the dual representation of L ( ) norm and a density argument, implies that π Φ L extends to a bounded linear operator from T 2 ( + ) to L ( ). This shows that (i) is valid. We now turn to the proof of (ii). By the definition of the Hardy space H L ( ) (with ∈ (0 ∞)) associated with operators satisfying Assumptions (H 1 ) and (H 2 ) in [39], together with an argument similar to that used in the proof of [41, Proposition 9.1(v)], we see that, for all ∈ ( L L ), H L ( ) = L ( ). This, combined with the definition of H L ( ), immediately implies that the operator 2 L − 2 L extends to a bounded linear operator from L ( ) to T 2 ( + ). This shows (ii), which completes the proof of Lemma 5.11.
We now turn to the proof of Theorem 5.9.

A sufficient condition for the equivalence between the spaces H L (R ) and H (R )
In this section, we give a sufficient condition on the operator L, satisfying Assumptions (A) and (B), such that H L (R ) and H (R ) coincide with equivalent quasi-norms. We first recall some notions and properties of H (R ).
In what follows, we denote by (R ) the space of all Schwartz functions and by (R ) its dual space (namely, the space of all tempered distributions). For ∈ N, define Then for all ∈ R and ∈ (R ), the non-tangential grand maximal function * of is defined by setting * To introduce the molecular Musielak-Orlicz-Hardy space, we first introduce the notion of molecules associated with the growth function .

The Musielak-Orlicz-Hardy space associated with the second order elliptic operator in divergence form
In this section, we study the Musielak-Orlicz-Hardy space H L (R ) associated with the second order elliptic operator in divergence form on R with complex bounded measurable coefficients. By making full use of the special structure of the divergence form elliptic operator, we establish the radial and non-tangential maximal function characterizations of H L (R ) based respectively on the heat and Poisson semigroups of L. Moreover, we establish the boundedness of the associated Riesz transform on H L (R ).

Maximal function characterizations of H L (R )
We begin this subsection by recalling some necessary notions and notation. Let A be an × matrix with entries { } =1 ⊂ L ∞ (R C) satisfying the ellipticity condition, namely, there exist constants 0 < λ A ≤ Λ A < ∞ such that, for all ξ ζ ∈ C and almost every ∈ R , where · · denotes the inner product in C and ξ denotes the real part of the complex number ξ. Then the second order elliptic operator L in divergence form is defined by L := −div(A∇ ) (7.1) interpreted in the weak sense via a sesquilinear form. It is well known that there exists a positive constant ω ∈ [0 π/2) such that the operator L is of type ω on L 2 (R ) and L has a bounded H ∞ -functional calculus on L 2 (R ) (see, for example, [1,41]  We also recall the definitions of some maximal functions associated with L from [40]. Let ∈ L 2 (R ) and ∈ R , the radial maximal functions, α and α P , respectively associated with the heat semigroup and Poisson semigroup generated by L are defined by setting, for all α ∈ (0 ∞), ∈ L 2 (R ) and ∈ R , Similarly, the non-tangential maximal functions, α and α P , respectively associated with the heat semigroup and Poisson semigroup generated by L are defined by setting, for all α ∈ (0 ∞), ∈ L 2 (R ) and ∈ R , In what follows, when α = 1, we remove the superscript α for simplicity. We also define the Lusin-area functions, S and S P , associated respectively to the heat semigroup and Poisson semigroup by setting, for all ∈ L 2 (R ) and ∈ R , We first introduce the Musielak-Orlicz-Hardy space, defined via the above maximal functions, as follows.

Definition 7.1.
Let and L be respectively as in Definition 2.2 and (7.1), and S P as in (7.7 For the operator S P , we have the following boundedness.

Lemma 7.2.
Let S and S P be respectively as in (7.6) and (7.7). Then, for all ∈ ( − (L) + (L)), both S and S P are bounded on L (R ).
The proof of Lemma 7.2 follows from a similar method used for the vertical Lusin-area function associated with the heat semigroup (see [4, Theorem 6.1]). We omit the details here. and || || L (E) (7.9) Proof. We first prove (7.8). To this end, by the change of variable, we write which, together with (7.10) and (7.11), shows immediately that (7.8) holds. The proof of (7.9) is similar to that of (7.8).
We omit the details here. Now, we are in the position to state our first main result in this section.

Proposition 7.4.
Let and L be respectively as in Definition 2.2 and (7.1), S and S P respectively as in (7.6) and (7.7). Assume further that ∈ RH From this, Definition 4.3 and Lemma 2.5(vii), we deduce that We now estimate I in the case when ≥ 5. Similar to the case when ≤ 4, we first have , since ≥ 2, by the dual norm representation of the L 2 (R )-norm, we know that there exists ∈ L ( 2 ) where denotes the classical Hardy-Littlewood maximal function. To estimate , we need the following subordination formula, where C is a positive constant. By using Hölder's inequality, (7.19), Minkowski's integral inequality, the L ( 2 ) (R )boundedness of the Hardy-Littlewood maximal function and Lemma 7.2, we conclude that We continue to estimate . Similar to the estimates for , we first conclude that By the L off-diagonal estimates (similar to the estimates used in (7.11)) and the change of variable (let := (2 + B ) 2 1+ 1 ), we further find that which, together with (7.17), (7.18), (7.20) and Definition 4.3, implies that, when ≥ 5, Similar to the estimates for I , we see that which, combining with (7.15), implies that I B |λ| ||χ B || L (R ) (7.21) Also, by following the same way as the estimates for I, we know that J (B |λ| ||χ B || L (R ) ), which, together with (7.14) and (7.21), shows that (7.13) holds true. The proof for the equivalence of H S (R ) and H L (R ) is similar. We omit the details here. This finishes the proof of Proposition 7.4. Now, we state the maximal function characterizations of H L (R ) as follows.

Theorem 7.5.
Let and L be respectively as in Definition 2.2 and (7.1), , P , and P respectively as in (7.2), (7.3), (7.4) and (7.5). Assume further that ∈ RH To prove Theorem 7.5, we need a good-λ inequality concerning the non-tangential maximal function and the truncated Lusin-area function associated with the heat semigroup. More precisely, let α ∈ (0 ∞) and 0 < < R < ∞. For all ∈ L 2 (R ) and ∈ R , the truncated Lusin-area function S R α , associated with the heat semigroup, is defined by setting, We have the following good-λ inequality.
With the help of Lemma 7.7, we now prove Theorem 7.5.
Proof of Theorem 7.5. We first prove the following equivalence relationships The proof is divided into the following three steps. Step . For any 0 < < R < ∞ and γ ∈ (0 1], by Lemma 2.4(ii), Fubini's theorem and Lemma 7.7, we conclude that By the change of variables and the fact that is of uniformly upper type 1, we further see that Moreover, using an argument similar to that used in the proof of [74,Lemma 7.7], we conclude that, for all 0 < α < β < ∞ and ∈ (0 ∞), From this and (7.26) with γ sufficient small, it follows that Letting → 0 and R → ∞, we immediately know that . By their definitions (see (7.2) and (7.3)), we see that 1 2 ( ) ≤ ( ). Moreover, similar to [46, Lemma 5.3], we conclude that, for any 0 < α < β < ∞, which immediately implies that . By Theorem 4.9 and Lemma 5.7, it suffices to prove that, for any λ ∈ C and ( M ) L -molecule α associated with the ball B, where ∈ (0 ∞) and M ∈ N can be chosen sufficient large. The estimate (7.27) can be proved by using Assumption (B); see, for example, the proof of (4.8). We omit the details. From Steps 1 though 3, we deduce that with equivalent quasi-norms, which, together with the fact that are, respectively, dense in H L (R ), H (R ) and H (R ), and a density argument, then implies that the spaces H L (R ), H (R ) and H (R ) coincide with equivalent quasi-norms. The proof for the equivalent relationships that H L (R ) = H P (R ) = H P (R ) is similar, we omit the details here. This finishes the proof of Theorem 7.5.

Boundedness of the Riesz transform ∇L −1/2
In this subsection, we study the boundedness of the Riesz transform ∇L −1/2 associated with L on H L (R ) for ( ) ∈ ( +1 1], and the associated weak boundedness at the endpoint ( ) = +1 . Our main result is as follows.   The proof of Proposition 7.9 is similar to that of [42,Theorem 4.11]. We omit the details here. Observe that, in [42,Theorem 4.11], the ranges of the exponents may be different from these of Proposition 7.9. More precisely, in [42,Theorem 4.11], the authors want to relax the range of the Musielak-Orlicz function , by narrowing the range of the exponent . However, in the present case, we need more wider range of .  (7.28) Using the L (R )-boundedness of ∇L −1/2 for all ∈ ( − (L) + (L)) and the following off-diagonal estimates that and for closed sets E F ⊂ R with (E F ) > 0, we conclude (7.28) by using the same method as in (5.7). This shows (i). ) and Proposition 7.9, implies that ∇L −1/2 is bounded from H L (R ) to H (R ), which completes the proof of (ii) and hence Theorem 7.8. Now, we establish the weak boundedness of ∇L −1/2 at the endpoint ( ) = +1 . Before stating our conclusions, we first recall some necessary definitions.
We now turn to the proof of Theorem 7.11. Proof of Theorem 7.11. To prove this theorem, let ∈ ( − (L) min{ + (L) + (L)}). We first claim that it suffices to show that, for all λ ∈ C and each ( M) L -molecules α associated with the ball B (with and M large enough) and all η ∈ (0 ∞), Indeed, if (7.32) holds true, then for all ∈ L 2 (R ) ∩ H L (R ), by Theorem 4.8, we know that, there exist a sequence {λ } ⊂ C and a sequence {α } of ( M ) L -molecules associated with the balls {B } such that = λ α in L 2 (R ) and || || H L (R ) ∼ Λ({λ α } ). Moreover, by the assumption that I( ) < 1, the change of variable and Lemma 7.13, we infer that which, together with a density argument, implies that ∇L −1/2 is bounded from H L (R ) to W H (R ). This shows the claim. Now, we turn to the proof of (7.32 which, together with (7.29), further implies that Thus, ∈ (2 + 2 )B. This, together with the mean value theorem, Hölder's inequality and (7.29), implies that, there exists a sufficiently large constant 0 such that where C 0 is a positive constant independent of and . Let 0 := max ∈ N : We know that, for all ∈ 2 + 2 0 B , Thus, from the assumption that ( ) is attainable and ∈ A 1 (R ), we infer that This, combined with (7.33) and (7.34), implies that (7.32) holds true, which completes the proof of Theorem 7.11.

The Musielak-Orlicz-Hardy space associated with the Schrödinger operator
In this section, we establish several equivalent characterizations of the Musielak-Orlicz-Hardy space H L (R ) associated with the Schrödinger operator L := −∆ + V , where 0 ≤ V ∈ L 1 loc (R ), in terms of the Lusin-area function associated with the Poisson semigroup of L, the non-tangential and the radial maximal functions associated with the heat semigroup generated by L, and the non-tangential and the radial maximal functions associated with the Poisson semigroup generated by L. Moreover, we also consider the boundedness of the associated Riesz transform on H L (R ). Let L := −∆ + V (8.1) be a Schrödinger operator, where 0 ≤ V ∈ L 1 loc (R ). Since V is a nonnegative function, from the Feynman-Kac formula, we deduce that the kernel of the semigroup − L , , satisfies that, for all ∈ (0 ∞) and ∈ R , Similar to Definition 4.1, we introduce the space H S P (R ) as follows.

Definition 8.1.
Let be as in Definition 2.2 and L as in (8.1). A function ∈ L 2 (R ) is said to be in H S P (R ) if S P ( ) ∈ L (R ); moreover, define || || H S P (R ) := ||S P ( )|| L (R ) := inf λ ∈ (0 ∞) : The S P -adapted Musielak-Orlicz-Hardy space H S P (R ) is defined to be the completion of H S P (R ) with respect to the quasi-norm || · || H S P (R ) .
For any ∈ L 2 (R ) and ∈ R , let From the L (R )-boundedness of , similar to the proof of (5.7), it follows that the above estimate holds true. We omit the details here.
Step 3. H (R ) ⊂ H P (R ), whose proof is similar to that of Step 3 in the proof of [74,Theorem 7.4]. We omit the details here.
Step 4. H P (R ) ⊂ H P (R ), whose proof is similar to that of Step 4 in the proof of [74,Theorem 7.4], and hence we omit the details here.
Step 5. H P (R ) ⊂ H S P (R ), whose proof is similar to that of [74,Proposition 7.6]. We omit the details here.
Let ∈ H S P (R ). Then Proof. The proof of Theorem 8.5(i) is similar to that of Theorem 7.8(i). We omit the details here. Now we prove (ii).
Moreover, similar to Theorem 7.11, for the Riesz transform ∇L −1/2 associated with the Schrödinger operator L, we also have the following endpoint boundedness.

Theorem 8.6.
Let and L be respectively as in Definition 2.  The proof of Theorem 8.6 is similar to that of Theorem 7.11. We omit the details here.