Maximal Function Characterizations of Musielak-Orlicz-Hardy Spaces Associated to Non-negative Self-adjoint Operators Satisfying Gaussian Estimates

Let $L$ be a non-negative self-adjoint operator on $L^2(\mathbb{R}^n)$ whose heat kernels have the Gaussian upper bound estimates. Assume that the growth function $\varphi:\,\mathbb{R}^n\times[0,\infty) \to[0,\infty)$ satisfies that $\varphi(x,\cdot)$ is an Orlicz function and $\varphi(\cdot,t)\in {\mathbb A}_{\infty}(\mathbb{R}^n)$ (the class of uniformly Muckenhoupt weights). Let $H_{\varphi,\,L}(\mathbb{R}^n)$ be the Musielak-Orlicz-Hardy space introduced via the Lusin area function associated with the heat semigroup of $L$. In this article, the authors obtain several maximal function characterizations of the space $H_{\varphi,\,L}(\mathbb{R}^n)$, which, especially, answer an open question of L. Song and L. Yan under an additional mild assumption satisfied by Schr\"odinger operators on $\mathbb{R}^n$ with non-negative potentials belonging to the reverse H\"older class, and second-order divergence form elliptic operators on $\mathbb{R}^n$ with bounded measurable real coefficients.


Introduction
The real-variable theory of Hardy spaces on the n-dimensional Euclidean space R n , initiated by Stein and Weiss [27] and then systematically developed by Fefferman and Stein [10], plays an important role in various fields of analysis (see, for example, [10,26]). It is well known that the Hardy space H p (R n ), with p ∈ (0, 1], is a suitable substitute of the Lebesgue space L p (R n ); for example, the classical Riesz transform is bounded on H p (R n ), but not on L p (R n ) when p ∈ (0, 1]. Moreover, the Hardy space H p (R n ) is essentially related to the Laplace operator ∆ := n i=1 ∂ 2 ∂x 2 i on R n . However, in many settings, these classical function spaces are not applicable; for example, the Riesz transforms ∇L −1/2 may not be bounded from the Hardy space H 1 (R n ) to L 1 (R n ) when L is a second-order divergence form elliptic operator with complex bounded measurable coefficients (see [13]). Motivated by this, the study for the real-variable theory of various function spaces associated with different differential operators has inspired great interests in recent years (see, for example, [1,2,4,8,12,13,15,16,17,21,24,25,28,29,31]).
Let L be a second-order divergence form elliptic operator on R n with bounded measurable complex coefficients. The Hardy space H 1 L (R n ) associated with L was characterized by Hofmann and Mayboroda [13] in terms of the molecule, the Lusin area function, the nontangential maximal function (N L (f ) or N L, P (f )) or the radial maximal function (R L (f ) or R L, P (f )), respectively, associated with its heat semigroup or its Poisson semigroup generated by L. Meanwhile, the same equivalent characterizations of the Orlicz-Hardy space associated with L as those of H 1 L (R n ) were independently obtained in [15]. Recall that, for any f ∈ L 2 (R n ) and x ∈ R n , the non-tangential maximal function N L (f ) and the radial maximal function R L (f ), associated with the heat semigroup of L, are defined by (1.1) N L (f )(x) := sup Moreover, let L be a non-negative self-adjoint operator on L 2 (R n ) whose heat kernels satisfy the Davies-Gaffney estimates. The equivalent characterizations of the Hardy space H 1 L (R n ) associated with L, including the atom, the molecule or the Lusin area function associated with L, were established in [12], which were extended to the Orlicz-Hardy space in [16]. As a special case of this kind of operators, when L := −∆ + V is the Schrödinger operator with 0 ≤ V ∈ L 1 loc (R n ), the non-tangential maximal function (f * L or f * L, P ) or the radial maximal function (f + L or f + L, P ) characterizations, associated with its heat semigroup or its Poisson semigroup, of the Hardy space H 1 L (R n ), the Orlicz-Hardy space H Φ, L (R n ) and the Musielak-Orlicz-Hardy space H ϕ, L (R n ) were, respectively, obtained in [12], [16] and [4,31]. Furthermore, the same maximal function characterizations of the Hardy space H p L A (R n ), with p ∈ (0, 1], and the Musielak-Orlicz-Hardy space H ϕ, L A (R n ) associated with the magnetic Schrödinger operator L A := −(∇ − iA) 2 + V were, respectively, established in [17] and [29], where A ∈ L 2 loc (R n , R n ) and 0 ≤ V ∈ L 1 loc (R n ). Recall that, for any f ∈ L 2 (R n ) and x ∈ R n , the non-tangential maximal function f * L and the radial maximal function f + L , associated with the heat semigroup of L, are defined by respectively. Furthermore, the non-tangential maximal function f * L, P and the radial maximal function f + L, P , associated with the Poisson semigroup of L, are defined via a similar way. Observe that the maximal functions in (1.3) and (1.4) are different from those in (1.1) and (1.2). The main reason for adding the averaging for the spatial variable in (1.1) and (1.2) is that we need to compensate for the lack of pointwise estimates of the heat semigroup and the Poisson semigroup in that case (see [13] for more details). Recall that, when L := −∆ + V with 0 ≤ V ∈ L 1 loc (R n ), its heat semigroup and its Poisson semigroup have pointwise estimates (see, for example, [12, (8.4)]).
From now on, assume that L is a densely defined operator on L 2 (R n ) satisfying the following two assumptions: (A1) L is non-negative and self-adjoint; (A2) the kernels of the semigroup {e −tL } t>0 , denoted by {K t } t>0 , are measurable functions on R n × R n and satisfy the Gaussian upper bound estimates, namely, there exist positive constants C and c such that, for all t ∈ (0, ∞) and x, y ∈ R n , The typical examples of operators L, satisfying both the assumptions (A1) and (A2), include the Schrödinger operator L := −∆ + V with 0 ≤ V ∈ L 1 loc (R n ) and the secondorder divergence form elliptic operator L := −div(A∇) with A := {a ij } n i, j=1 satisfying that, for any i, j ∈ {1, . . . , n}, a ij is a real measurable function on R n and there exists a constant λ ∈ (0, 1] such that, for all i, j ∈ {1, . . . , n} and x, ξ ∈ R n , a ij (x) = a ji (x) and λ|ξ| 2 ≤ n i, j=1 Denote by S(R n ) the space of all Schwartz functions on R n . Let p ∈ (0, 1], α ∈ (0, ∞), φ ∈ S(R) be an even function and φ(0) = 1. Recently, the characterizations of H p L (R n ) in terms of the non-tangential maximal function (φ * L, α (f )) or the grand maximal function (G * L (f )) were obtained by Song and Yan [25] via some essential improvements of techniques due to Calderón [6]. Recall that, for any f ∈ L 2 (R n ) and x ∈ R n , the non-tangential maximal function φ * L, α (f ) is defined by (see (2.1) below for the definition of φ(t √ L)) and the grand maximal function G * L (f ) is defined by with N being a large positive integer. It is easy to see that, when φ(x) := e −x 2 for all x ∈ R and α := 1, the maximal function φ * L, α (f ) in (1.6) coincides with the maximal function f * L in (1.3). Let the operator L satisfy both the assumptions (A1) and (A2). In this article, motivated by [4,25,31], we characterize the Musielak-Orlicz-Hardy space associated with L via the non-tangential maximal function in (1.6) or the grand maximal function in (1.7). Under an additional assumption for L (see Assumption (A3) below for the details), which is satisfied by Schrödinger operators on R n with non-negative potentials belonging to the reverse Hölder class and second-order divergence form elliptic operators on R n with bounded measurable real coefficients, we obtain the equivalent characterization of the Musielak-Orlicz-Hardy space associated with L in terms of the radial maximal function in (1.4). As a special case, under the additional mild assumption (A3) for L, we give an answer for the open question in [25,Remark 3.4] whether or not the Hardy space H p L (R n ), with p ∈ (0, 1], can be characterized via the radial maximal function in (1.4).
Then ϕ is said to satisfy the uniformly Muckenhoupt condition for some q ∈ [1, ∞), denoted by ϕ ∈ A q (R n ), if, when q ∈ (1, ∞), where the first suprema are taken over all t ∈ (0, ∞) and the second ones over all balls B ⊂ R n . The function ϕ is said to satisfy the uniformly reverse Hölder condition for some q ∈ (1, ∞], denoted by ϕ ∈ RH q (R n ), if, when q ∈ (1, ∞), where the first suprema are taken over all t ∈ (0, ∞) and the second ones over all balls B ⊂ R n .
is called a growth function if the following hold true: (i) ϕ is a Musielak-Orlicz function, namely, (a) ϕ(x, ·) is an Orlicz function for all x ∈ R n ; (b) ϕ(·, t) is a measurable function for all t ∈ [0, ∞).
The function ϕ is of uniformly lower type p for some p ∈ (0, 1] and of uniformly upper type 1.
For a Musielak-Orlicz function ϕ as in Definition 1.1, a measurable function f on R n is said to be in the Musielak- Clearly, is a growth function if ω ∈ A ∞ (R n ) and Φ is an Orlicz function of lower type p for some p ∈ (0, 1] and upper type 1. Here and hereafter, A q (R n ) with q ∈ [1, ∞] denotes the class of Muckenhoupt weights (see, for example, [11]). A typical example of such functions Φ is Φ(t) := t p , with p ∈ (0, 1], for all t ∈ [0, ∞) (see, for example, [14,19,20,31] for more examples of such Φ). Another typical example of growth functions is given by for all x ∈ R n and t ∈ [0, ∞); more precisely, ϕ ∈ A 1 (R n ), ϕ is of uniformly upper type 1 (indeed, I(ϕ) = 1, which is attainable) and i(ϕ) = 1 which is not attainable (see [19] for the details). Now we recall the definition of the Musielak-Orlicz-Hardy space H ϕ, L (R n ) introduced in [4,31]. Definition 1.3. Let L be an operator on L 2 (R n ) satisfying the assumptions (A1) and (A2), and ϕ as in Definition 1.2. For f ∈ L 2 (R n ) and x ∈ R n , the Lusin area function, S L (f ), associated with L is defined by Moreover, we recall the following definitions of (ϕ, q, M ) L -atoms and atomic Musielak- where the infimum is taken over all the atomic (ϕ, q, M ) L -representations of f as above and The atomic Musielak-Orlicz-Hardy space H M, q ϕ, L, at (R n ) is then defined as the completion of the set H M, q ϕ, L, at (R n ) with respect to the quasi-norm · H M, q ϕ, L, at (R n ) . Now we introduce Musielak-Orlicz-Hardy spaces via maximal functions associated with the operator L. (i) Assume that φ ∈ S(R) is an even function with φ(0) = 1 and α ∈ (0, ∞). For any for all x ∈ R and α := 1, denote φ * L, α (f ) simply by f * L and, in this case, denote the space H φ, α ϕ, L, max (R n ) simply by H ϕ, L, max (R n ).
Then the first main result of this article reads as follows.
The following chains of inequalities give the strategy of the proof of Theorem 1.6(i).
where q ∈ ([r(ϕ)] ′ I(ϕ), ∞) and the implicit constants are independent of f . We show the first inequality in (1.15) via borrowing some ideas from the proof of [25,Theorem 1.4]. By the definitions of the spaces H M, q ϕ, L, at (R n ), H φ, α ϕ, L, max (R n ) and H A ϕ, L, max (R n ), we find that the second and the fourth inequalities in (1.15) are obvious. Moreover, we obtain the third inequality in (1.15) by establishing a pointwise estimate for α * L , where α is a (ϕ, q, M ) Latom (see (2.26) below for the details). Furthermore, (ii) of Theorem 1.6 is obtained by (i) and the fact that the spaces H ϕ, L (R n ) and H M, q ϕ, L, at (R n ), with q ∈ ([r(ϕ)] ′ I(ϕ), ∞), coincide with equivalent quasi-norms, which was established in [4,Theorem 5.4]. Let be a magnetic Schrödinger operator on R n with n ≥ 3, where A ∈ L 2 loc (R n , R n ) and the potential V belongs to the Kato class, namely, Moreover, the Kato norm of V is defined by For the potential V , let V + := max{V, 0} and V − := min{V, 0}. Under the assumption that L A is as in (1.16) with V + belonging to the Kato class and V − K < π n/2 /Γ(n/2 − 1), it was showed in [5] that L satisfies the assumptions (A1) and (A2). Thus, as a corollary of Theorem 1.6, we have the following several equivalent characterizations of the Musielak-Orlicz-Hardy space H ϕ, L A (R n ) associated with L A . Corollary 1.7. Let L A be as in (1.16), with V + belonging to the Kato class and V − K < π n/2 /Γ(n/2 − 1), and ϕ as in Definition 1.2.
(i) Assume that q, M and α are as in Theorem 1.6(i). Then the spaces Assume that q, M and α are as in Theorem 1.6(ii). Then the spaces H ϕ, Let L satisfy the assumption (A2) and ϕ be as in Definition To answer this question, we need to introduce another assumption for the operator L as follows: (A3) There exist positive constants C and µ ∈ (0, 1] such that, for all t ∈ (0, ∞) and x, y 1 , y 2 ∈ R n , We point out that there are lots of operators on R n satisfying the assumption (A3); for example, Schrödinger operators on R n with non-negative potentials belonging to the reverse Hölder class (see, for example, [9]) and second-order divergence form elliptic operators on R n with bounded measurable real coefficients (see, for example, [3]). Theorem 1.9. Let L be an operator on L 2 (R n ) satisfying the assumptions (A2) and (A3), and ϕ as in Definition 1.2. Then the spaces H ϕ, L, max (R n ) and H ϕ, L, rad (R n ) coincide with equivalent quasi-norms. As a corollary of Theorems 1.6 and 1.9, we have the following conclusion. The layout of this article is as follows. Sections 2 and 3 are, respectively, devoted to the proofs of Theorems 1.6 and 1.9.
Finally we make some conventions on notation. Throughout the whole article, we always 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 ≤ CB. If A B and B A, then we write A ∼ B. For each ball B := B(x B , r B ) ⊂ R n , with some x B ∈ R n and r B ∈ (0, ∞), and α ∈ (0, ∞), let αB := B(x B , αr B ). For any measurable subset E of R n , we denote by χ E its characteristic function. We also let N := {1, 2, . . .} and Z + := N ∪ {0}. For any ball B in R n and j ∈ Z + , let S j (B) := (2 j+1 B) \ (2 j B) with j ∈ N and S 0 (B) := 2B. Finally, for q ∈ [1, ∞], we denote by q ′ its conjugate exponent, namely, 1/q + 1/q ′ = 1.

Proof of Theorem 1.6
In this section, we give out the proof of Theorem 1.6. To this end, we first recall some auxiliary conclusions. For a non-negative self-adjoint operator L on L 2 (R n ), denote by E L the spectral measure associated with L. Then, for any bounded Borel function F : [0, ∞) → C, the operator F (L) : Then we have the following lemma, which was obtained in [12, Lemma 3.5].
Lemma 2.1. Assume that the operator L satisfies the assumptions (A1) and (A2). Let φ ∈ C ∞ c (R) be even and supp (φ) ⊂ (−1, 1). Denote by Φ the Fourier transform of φ. Then, for any k ∈ Z + , the kernels satisfy that there exists a positive constant C, depending on n, k and Φ, such that, for all t ∈ (0, ∞) and x, y ∈ R n , Denote by M the Hardy-Littlewood maximal operator on R n , namely, for all f ∈ L 1 loc (R n ) and x ∈ R n , M(f )(x) := sup where the supremum is taken over all balls B ∋ x. Moreover, we have the following properties of growth functions, which were obtained in [ . Then ϕ is equivalent to ϕ, namely, there exists a positive constant C such that, for all (x, t) ∈ R n × [0, ∞), (iii) If p ∈ (1, ∞) and ϕ ∈ A p (R n ), then there exists a positive constant C such that, for all measurable functions f on R n and t ∈ [0, ∞), Moreover, to show Theorem 1.6, we need to establish the following conclusion.
Furthermore, to show Theorem 1.6, we also need the following atomic characterization of the Musielak-Orlicz-Hardy space H ϕ, L (R n ) obtained in [4,Theorem 5.4]. Proof of Theorem 1.6. We first show (i) of Theorem 1.6. To this end, we begin with proving that To prove (2.10), via Proposition 2.3, it suffices to show that, for any f ∈ H ϕ, L, max (R n )∩ L 2 (R n ), f ∈ H M, ∞ ϕ, L, at (R n ) and where the implicit positive constant depends on n, M and ϕ. Let Ψ(x) := x 2M Φ(x) for all x ∈ R n , where Φ is as in Lemma 2.1. Then, by the spectral calculus, we know that there exists a constant C (Ψ) such that Then η ∈ S(R) is an even function and, for any a, b ∈ R, which further implies that For any f ∈ L 2 (R n ) and x ∈ R n , let Then, from Proposition 2.3, it follows that here and hereafter, e := (1, . . . , 1) ∈ R n . It is easy to prove that, for all i ∈ Z, Then By this, we conclude that (y Furthermore, notice that, for any Now we prove that the summation (2.13) converges in L 2 (R n ). Indeed, it is well known that, for any f ∈ L 2 (R n ), (see, for example, [12, (3.14)]), which, together with (2.13), implies that as N 1 → ∞ and N 2 → ∞. Thus, the summation (2.13) converges in L 2 (R n ). Now we claim that there exists a positive constant C, depending on n, M , Φ and ϕ, such that, for all i and j, C −1 α i,j is a (ϕ, ∞, M ) L -atom associated with the ball 30B i,j , where B i,j denotes the ball with the center being the same as Q i,j and the radius r B i,j := √ nℓ(Q i,j )/2. Here and hereafter, ℓ(Q i,j ) denotes the side length of Q i,j . Once this claim is proved, by Lemma 2.2(iv), we then know that, for all λ ∈ (0, ∞), Moreover, similar to the proof of [19,Lemma 5.4] (see also [31,Lemma 3.4]), we conclude that which, combined with (2.12) and (2.14), further implies that (2.11) holds true. Now we prove the above claim. We first show that, for any k ∈ {0, 1, . . . , M }, From the definition of T i,j , it follows that, if (y, t) ∈ T i,j , then B(y, 4 √ nt) ⊂ O i . Let y := y + 3te. Then y ∈ Q i,j and B( y, √ nt) ⊂ O i . Moreover, by the fact that Q i,j is the Whitney cube of O i , we know that 5Q i,j ∩ O ∁ i = ∅ and hence t ≤ 3ℓ(Q i,j ), which, together with y + 3te ∈ Q i,j , further implies that y ∈ 20Q i,j . Furthermore, from Lemma 2.1, we deduce that supp (K (t 2 L) k Φ(t √ L) ) ⊂ {(x, y) ∈ R n × R n : |x − y| ≤ t}, which, combined with y ∈ 20Q i,j , implies that (2.15) holds true. To finish the proof of the above claim, it remains to prove that, for any k ∈ {0, 1, . . . , M }, By [25, (3.12)], we find that, for any k ∈ {0, 1, . . . , M − 1}, where the implicit positive constant depends on n, M and Φ, which further implies that Furthermore, it follows, from [25, (3.13) where the implicit positive constant depends on n and Ψ, which implies that By this and (2.17), we conclude that (2.16) holds true, which completes the proof of the above claim and hence (2.10). Now we prove that, for any M ∈ N ∩ (nq(ϕ)/2i(ϕ), ∞) and q ∈ ([r(ϕ)] ′ I(ϕ), ∞], For any φ ∈ A and x ∈ R, let ψ(x) := [φ(0)] −1 φ(x) − e −x 2 . Repeating the proof of [25, (3.4)], we know that, for any λ ∈ (0, 2M ), there exists a positive constant C, depending on n, Ψ and λ, such that, for all φ ∈ A, sup |w|<t, t∈(0,∞) R n+1 where Ψ is as in (2.6). Via this estimate and repeating the proof of (2.9), we find that where the implicit constant depends on n, Ψ, λ and ϕ, which, combined with the fact that G * L (f ) sup φ∈A ψ * L, 1 (f ) + f * L and Lemma 2.2(i), further implies that Via (2.19), to finish the proof of (2.18), it suffices to show that, for any λ ∈ C and (ϕ, q, M ) L -atom α associated with the ball B := B(x B , r B ) with x B ∈ R n and r B ∈ (0, ∞), 20) where the implicit positive constant depends on n and ϕ. Indeed, let f ∈ H M, q ϕ, L, at (R n ) ∩ L 2 (R n ). Then there exist {λ j } j ⊂ C and a sequence {α j } j of (ϕ, q, M ) L -atoms, associated with the balls {B j } j , such that which, together with (2.20), further implies that, for all λ ∈ (0, ∞), .
Furthermore, (ii) of Theorem 1.6 is deduced from (i) and Proposition 2.4, which completes the proof of Theorem 1.6.
3 Proof of Theorem 1.9 In this section, we show Theorem 1.9. We first introduce some notation. Let f ∈ L 2 (R n ). For all t ∈ (0, ∞) and x ∈ R n , let For all ε ∈ (0, ∞), N ∈ N and x ∈ R n , define and U * ε, N (x) := sup where µ is as in (1.17) and y, y 1 , y 2 ∈ R n . Lemma 3.1. Let L be an operator on L 2 (R n ) satisfying the assumptions (A2) and (A3), and ϕ as in Definition 1.2. Then there exists a positive constant C, depending on n and ϕ, such that, for all u as in (3.1), ε ∈ (0, ∞) and N ∈ N, where u * ε, N and U * ε, N are, respectively, as in (3.2) and (3.3).
Proof of Theorem 1.9. By the definitions of the spaces H ϕ, L, max (R n ) and H ϕ, L, rad (R n ) and the fact that H ϕ, L, max (R n ) ∩ L 2 (R n ) and H ϕ, L, rad (R n ) ∩ L 2 (R n ) are, respectively, dense in H ϕ, L, max (R n ) and H ϕ, L, rad (R n ), to show Theorem 1.9, it suffices to show that Let f ∈ H ϕ, L, rad (R n ) ∩ L 2 (R n ) and u be as in (3.1). By (1.5), we conclude that f + L M(f ), which, combined with the fact that, for all ε ∈ (0, ∞) and N ∈ N, u * ε, N f + L , implies that u * ε, N M(f ). From this and the boundedness of M on L 2 (R n ), we deduce that, for all ε ∈ (0, ∞) and N ∈ N, u * ε, N ∈ L 2 (R n ). Define where E is a positive constant determined later. By (3.4), we know that where C is as in (3.4). Let p 0 ∈ (0, i(ϕ)) be a uniformly lower type of ϕ. Take E ∈ (1, ∞) large enough such that which, together with the definition of G ε, N and the uniformly lower type p 0 property of ϕ and the increasing property of ϕ about the variable t, implies that From (3.8), it follows that For all x ∈ R n , let M r (f + L )(x) := {M ([f + L ] r )(x)} 1/r with r ∈ (0, 1). Then, for almost every x ∈ G ε, N , we have u * ε, N (x) M r (f + L )(x), (3.10) where the implicit positive constant depends on n, µ and E. Indeed, let x ∈ G ε, N such that u * ε, N (x) < ∞. By the definition of u * ε, N , we know that there exist y ∈ R n and t ∈ (0, ∞) such that |y − x| < √ t < ε −1 and Since x ∈ G ε, N , if |z 1 − x| < √ t < ε −1 and |z 2 − x| < √ t < ε −1 , then, from the definition of U * ε, N (x) and (3.11), it follows that Take E t := {w ∈ R n : |w − y| < √ t 2 C 1 }, where C 1 := (4E) 1/µ /2. Obviously, C 1 ≥ 1. Taking z 1 := y and z 2 ∈ E t , by (3.12), we find that √ t |y − z 2 | µ |u(y, t) − u(z 2 , t)| ≤ 2E|u(y, t)|.