New Atomic Decomposition for Besov Type Space ˙ B 01 , 1 Associated with Schrödinger Type Operators

Let ( X , d , μ) be a space of homogeneous type. Let L be a nonnegative self-adjoint operator on L 2 ( X ) satisfying certain conditions on the heat kernel estimates which are motivated from the heat kernel of the Schrödinger operator on R n . The main aim of this paper is to prove a new atomic decomposition for the Besov space ˙ B 0 , L 1 , 1 ( X ) associated with the operator L . As a consequence, we prove the boundedness of the Riesz transform associated with L on the Besov space ˙ B 0 , L 1 , 1 ( X )

We also assume further that (X , d, μ) satisfies the noncollapsing condition, i.e., there exists c 0 > 0 such that for all x ∈ X . From now on, for any measurable subset E ⊂ X , we denote V (E) := μ(E). For all x ∈ X and r > 0, we also denote V (x, r ) = μ(B(x, r )).
Note that the classical Hardy space H 1 (X ) is a suitable substitution for the space L 1 (X ) when we work with Calderón-Zygmund operators but the classical Hardy space might not be suitable for the study of certain operators that lie beyond the Calderon Zygmund class. This observation highlights the need for the development of new function spaces that adapt well to these operators. In recent times, there has been a remarkable progress in the field of function spaces associated with operators, reflecting the growing interest in understanding the behaviour of these operators and their associated function spaces. See for example [1,5,15,21,23,29] and the references therein.
Motivated by this ongoing research, we aim to study new atomic decomposition of Besov spaces associated to Schrödinger type operators. Throughout this paper, we assume that H is a non-negative self-adjoint operator on L 2 (X ) which generates the analytic semigroup {e −t H } t>0 . Denote by p t (x, y) and q t (x, y) the kernels of e −t H and t He −t H , respectively, we assume that the kernels p t (x, y) satisfy the following conditions: (H1) There exist positive constants C and c such that for all x, y ∈ X and t > 0; (H2) There exist positive constants δ 1 , c and C such that whenever d(x, x) ≤ √ t and t > 0; (H3) X p t (x, y)dμ(x) = 1 for y ∈ X .
In fact, the assumptions (H1) and (H2) can be assumed only for the kernel p t (x, y) since the estimates in (H1) and (H2) for p t (x, y) imply similar estimates for q t (x, y). However, for the sake of simplicity, we make the assumptions for both p t (x, y) and q t (x, y).
Standard examples of operators which satisfy conditions (H 1), (H 2) and (H 3) include the Laplacians on the Euclidean spaces R n , the Laplace-Beltrami operators on non-compact Riemannian manifolds with doubling property, the Bessel operators on (0, ∞) n , the sub-Laplacians on stratified Lie groups and certain degenerate elliptic operators on doubling spaces and domains.
Our motivation is to study the Schrödinger operator L = H + V which is a nonnegative self-adjoint operator on L 2 (X ). Under suitable conditions, the potential V induces a critical function ρ which appears on the upper bounds and regularity estimates of the heat kernels of L and its time derivative. We refer the reader to Sect. 2.1 for a general definition of critical functions and further details.
In this paper, without the assumption L = H + V , we assume that L is a nonnegative self-adjoint operator on L 2 (X ). Denote by p t (x, y) and q t (x, y) the kernels of e −t L and t Le −t L , respectively. Suppose that ρ is a critical function defined on X . See Sect. 2.1 for the precise definition of critical functions. We assume that the kernels p t (x, y) and q t (x, y) satisfy the following conditions: (L1) For all N > 0, there exist positive constants c and C so that for all x, y ∈ X and t > 0; (L2) There is a positive constant δ 2 so that for all N > 0, there exist positive constants c and C which satisfy √ t and t > 0; (L3) There is a positive constant δ 3 such that for all x, y ∈ X and t > 0.
(b) Note that the condition (L1) implies that for all N > 0, there exist positive constants c and C so that for all x, y ∈ X and t > 0. Since the proof of (4) is standard, we leave it to the interested reader. (c) As mentioned above, an example of the pairs of operators (H , L) which satisfy our assumptions are the operators H mentioned above and L = H + V for suitable potentials V . See Sect. 2.1, also [9, Section 6] and [34]. We remark that our work on the operator L in this paper only relies on the assumptions (L1), (L2), (L3) and does not use the representation L = H + V .
Our aim is to study the homogeneous Besov spaceḂ 0,L 1,1 (X ) associated with the operator L.
When L = − the Laplacian on R n , the Besov spaceḂ 0,L 1,1 (R n ) coincides with the classical Besov spaceḂ 0 1,1 (R n ). It is well known that the Besov spaceḂ 0 1,1 (R n ) is contained in the Hardy space H 1 (R n ) and is used in proving the dispersive estimates of the wave equations (see for example [3,8,13]) and the regularity of the Green functions on domains (see for example [20]). See also [17,18,[24][25][26] and the references therein for further discussion on the Besov space typeḂ 0 1,1 and the Besov spaces on spaces of homogeneous type. It is worth noticing that in the definition above we define the Besov space a subset of L 1 (X ). This is more advantageous than the approach using new distributions as in [5,26].
We are interested in atomic decompositions of the Besov spaceḂ 0,L 1,1 (X ). Note that atomic decompositions of Besov spaces associated to non-negative self-adjoint operators satisfying Gaussian upper bounds were obtained in [5] for homogeneous Besov spaces and in [27] for inhomogeneous Besov spaces. Adapting ideas in [5,27], we can define atoms for the Besov spacesḂ 0,L 1,1 (X ) as follows. Note that the atoms defined in [5,27] are supported in balls associated to dyadic cubes. See Lemma 2.2 for the definition of dyadic cubes. In this paper, we do not need the dyadic cubes in Definition 1.3 and we are able to prove the following result.
Then there exist a sequence of (L, M) atoms {a j } and a sequence of coefficients {λ j } ∈ 1 so that f = j λ j a j in L 1 (X ), .
where {a j } is a sequence of (L, M)-atoms and {λ j } ∈ 1 , then The proof of Theorem 1.4 will be presented later. In comparison with the atomic decomposition in Theorems 4.2 and 4.3 in [5], the main difference is that in Theorem 1.4, the convergence used in the atomic decomposition is in L 1 (X ) instead of in the space of new distributions associated with the operator L; moreover, Theorem 1.4 uses the atoms associated with balls rather than the dyadic cubes as in Theorems 4.2 and 4.3 in [5].
We now consider new atoms associated with the critical function ρ which will be defined in Sect. 2.1. Note that the idea of the atomic decomposition associated to the critical functions was used in the setting of Hardy spaces. In [16], the atomic decomposition associated to the critical functions was studied for the Hardy spaces associated to Schrödinger operators with potential satisfying certain reverse Hölder inequality. Then the results were extended to encompass a broader scope, incorporating Schrödinger operators in various contexts such as stratified Lie groups and doubling manifolds. See for example [9,34]. However, this is the first time the atomic decomposition associated to the critical functions was established for the Besov spaces. Definition 1.5 Let > 0 and ρ be a critical function. A function a is said to be an ( , ρ(·))-atom if there exists a ball B such that It is interesting that the atoms in Definition 1.5 depend on the critical function ρ only. This type of atoms can be viewed as an extended version of the atoms used for the inhomogeneous Besov type. In fact, in the particular case ρ = constant, the atoms in Definition 1.5 turn out to be the atoms which characterize the inhomogeneous Besov spaces. See for example [26]. Our main result is the following theorem.
Theorem 1.6 If f ∈Ḃ 0,L 1,1 (X ), then there exist a sequence of ( , ρ(·))-atoms {a j } for some > 0 and a sequence of coefficients {λ j } ∈ 1 so that Conversely, if where {a j } is a sequence of ( , ρ(·))-atoms with > 0 and {λ j } ∈ 1 , then The organization of the paper is as follows. In Sect. 2, we recall the definitions of critical functions and dyadic cubes, and prove some kernel estimates of the spectral multipliers of H . In Sect. 3, we will set up the theory of the inhomogeneous Besov space B 0 1,1 (X ) including atomic decomposition results. The proofs of the main results will be given in Sect. 4. Finally, Sect. 5 is devoted in the proof of the boundedness of the Riesz transform associated with L in Besov spaces.
Throughout the paper, we always use C and c to denote positive constants that are independent of the main parameters involved but whose values may differ from line to line. We write A B if there is a universal constant C so that A ≤ C B and A ≈ B if A B and B A. Given a λ > 0 and a ball B := B(x, r ), we write λB for the λ-dilated ball, which is the ball with the same center as B and with radius λr . For each ball B ⊂ X , we set S 0 (B) = B and S j (B) = 2 j B\2 j−1 B for j ∈ N.

Critical Functions
A function ρ : X → (0, ∞) is called a critical function if there exist positive constants C ρ and k 0 so that for all x, y ∈ X .
Note that the concept of critical functions was introduced in the setting of Schrödinger operators on R D in [19] (see also [30]) and then was extended to the spaces of homogeneous type in [34].
A simple example of a critical function is ρ ≡ 1. Moreover, one of the most important classes of the critical functions is the one involving the weights satisfying the reverse Hölder inequality. Recall that a non-negative locally integrable function w is said to be in the reverse Hölder class R H q (X ) with q > 1 if there exists a constant C > 0 so that for all balls B ⊂ X . Note that if w ∈ R H q (X ) then w is a Muckenhoupt weight. See [32]. Now suppose V ∈ R H q (X ) for some q > max{1, D/2} and, following [30,34], set Then it was proved in [30,34] that ρ is a critical function provided q > max{1, D/2}. The following result will be useful in the sequel which is taken from Lemma 2.3 and Lemma 2.4 of [34].

Lemma 2.1
Let ρ be a critical function on X . Then there exist a sequence of points {x α } α∈I ⊂ X and a family of functions {ψ α } α∈I satisfying the following: (ii) For every λ ≥ 1 there exist constants C and N 1 such that

Dyadic Cubes
We now recall an important covering lemma in [10].

Remark 2.3
Since the constants η and a 0 are not essential in the paper, without loss of generality, we may assume that η = a 0 = 1/2. We then fix a collection of open sets in Lemma 2.2 and denote this collection by D. We call open sets in D the dyadic cubes in X and x Q k τ the center of the cube Q k τ ∈ D. We also denote For the sake of simplicity we might assume that κ 0 = 1.

Kernel Estimates
Denote by E H (λ) a spectral decomposition of H . Then by spectral theory, for any bounded Borel funtion F : [0, ∞) → C we can define as a bounded operator on L 2 (X ). It is well-known that the kernel cos(t for somec 0 > 0. See for example [31].
In what follows, without loss of generality we may assume thatc 0 = 1. We have the following useful lemma.
(b) For any N > 0 and s = 2(N + 3D + 2) there exists C = C(N ) so that for all x, y, y ∈ X with d(y, y ) < λ, and for all functions Then we have This, along with Lemma 2.5, (H2) and the fact G W 2 On the other hand, Therefore, which implies (13). The estimate (14) can be proved similarly. This completes our proof.

Lemma 2.7
Let ϕ ∈ S(R) be an even function. Then for any N > 0 there exists C N such that and for all t > 0 and x, y, y ∈ X with d(y, y ) < t.
We need only to prove (16). Hence, By (14), for N > 0 we have for all t > 0 and x, y, y ∈ X with d(y, y ) < t.

Remark 2.8
The results in Lemmas 2.5, 2.6 and 2.7 hold true if we replace H by L since we do not use the assumption (H3) in the proofs.
for all x ∈ X and t > 0.
Proof Let ψ j be the function as in the proof of Lemma 2.7 for j = 0, 1, 2, . . .. Then we have Arguing similarly to the proof of Lemma 2.7, we also yield that for any N > n and j = 0, 1, 2, . . ., This, together with (19), implies that Using (20), and letting R → ∞, the above identity deduces that Therefore, due to Lemma 2.5, the upper bound of q t (x, y) and Fubini's theorem, In addition, from the conservation property (H3), we immediately have This completes our proof.

Inhomogeneous Besov Spaces B 0 1,1 (X) and Atomic Decomposition
In this section, we will introduce the Besov space B 0 1,1 (X ). Our approach relies on the function spaces associated to the "Laplace-like" operator. This is motivated from the classical case in which the classical Besov spaces can be viewed as Besov spaces associated with the Laplacian. In our setting, under the three conditions (H1), (H2) and (H3), the operator H satisfies important properties which are similar to the Laplacian on the Euclidean space.

Inhomogeneous Besov Spaces
In the sequel we will show that the Besov space B 0 1,1 (X ) is independent of the operator H . This is a reason why we do not include the operator H in the notation of the Besov space. In order to prove Lemma 3.2 we need the following technical lemmas.

Lemma 3.3 For each
Proof We first recall the following fact in [4] lim s→0 Assume that f ∈ B 0 1,1 (X ). It follows that f ∈ L 1 (X ). For each n ∈ N, define From the Gaussian upper bound condition (H1) and (3), By (21), Similarly, On the other hand, since e −s H is bounded on L 1 (X ), we have In addition, By the Dominated Convergence Theorem, It follows that This, along with the fact that f k ∈ L p (X ) for each n ∈ N and p ∈ [1, ∞), implies that L p (X ) is dense in B 0 1,1 (X ) for each p ∈ [1, ∞). This completes our proof.
We are now ready to prove Lemma 3.2.

Proof of Lemma 3.2
Assume that { f k } is a Cauchy sequence in B 0 1,1 (X ). Hence, this is also a Cauchy sequence in L 1 (X ) since B 0 1,1 (X ) → L 1 (X ). As a consequence, f k → f ∈ L 1 (X ) for some f ∈ L 1 (X ). On the other hand, we have Since { f k } is a Cauchy sequence in B 0 1,1 (X ), for any > 0 there exists N such that for m, k ≥ N , Fixing k, then using Fatou's Lemma we have It follows that This completes our proof.

Atomic Decomposition
In order to establish atomic decomposition for the Besov space, we need another Calderón reproducing formula.
Proof Similarly to the proof of Lemma 3.4, it suffices to prove the proposition for f ∈ L 2 (X ) ∩ B 0 1,1 (X ). Observe that which implies that This, along with spectral theory, yields in L 2 (X ). Set Then, by Lemma 3.9 and Corollary 3.5, This implies that for some g ∈ L 1 (X ). This, in combination with (25), implies that f = g for a.e.. Therefore, for f ∈ L 2 (X ) ∩ B 0 1,1 (X ). This completes our proof.
For any bounded Borel function ϕ defined on [0, ∞). We now define, for λ > 0, for some > 0, and Arguing similarly to the proof of Therem 1.2 in [28], we have:

Lemma 3.8 Let (ϕ, ϕ 0 ) be a pair of even functions in A(R).
Then, for λ > 2n, we have
We now introduce the notion of atoms for the Besov space B 0 1,1 (X ). Definition 3.10 Let > 0. A function a is said to be an -atom if there exists a ball B with r B ≤ 1 such that and (b) In particular, if f ∈ B 0 1,1 (X ) supported in a ball B with r B = 1, then there exist a sequence of -atoms {a j } supported in 3B for some > 0 and a sequence of numbers {λ j } such that (28) and (29) hold true.
Proof (a) Let , be as in Lemma 3.6 such that Moreover, according to Lemma 2.4, we have, for t > 0 and x, y ∈ X , and where F ∈ { , , , }.
We first decompose f 1 as follows: For each Q ∈ D 2 as in Remark 2.3, we set It is clear that For the part f 2 , we write For each Q ∈ D j with j ≥ 3, we set and Then we have Therefore, We next claim that a Q is an atom for each Q ∈ D j , j ≥ 2. Indeed, for j = 2 we have It follows, by (30) and Remark 2.
On the other hand, by Lemma 2.7, Hence, a Q is a multiple of an -atom associated to the ball B Q for each Q ∈ D j with j = 2.
Arguing similarly to above, we can verify that for Q ∈ D j , j ≥ 3, a Q satisfies (i)-(iii) in Definition 3.10 with the corresponding ball defined by The condition a Q (x)dμ(x) = 0 follows directly from Lemma 2.9 and the fact that and are even and (0) = (0) = 0. Hence, a Q is a multiple of an -atom associated to B Q with = δ for each Q ∈ D j , j ≥ 3.
It remains to show that ∞ j≥2 Q∈D j It remains to prove that 1 0 t He −t H a 1 dt t 1.
To do this, we write t He −t H a 1 dt t For the second term E 2 , using the Gaussian upper bound of q t (x, y), To estimate the term E 1 , using the fact that By the smoothness condition of the atom a and the Gaussian upper bound of q t (x, y), we have It follows that E 1 1.
It remains to estimate E 3 . Note that if r B = 1, then E 3 = 0. Hence, we need only to consider the case r B < 1. Due to the cancellation property of the atom a, we have It follows that E 3 1. This completes our proof of (a).
where {s Q } is a sequence of numbers satisfying (29) and {a Q } is a sequence of -atoms defined by (32) and (33). From (30), (32) and (33), we have This completes the proof of (b).
We now introduce a new variant of the inhomogeneous Besov spaces. For > 0, the Besov space B 0, 1,1 (X ) is defined as the set of functions f ∈ L 1 (X ) such that When = 1, we simply write B 0 1,1 (X ).

Definition 3.12
Let > 0 and > 0. A function a is said to be an ( , )-atom if there exists a ball B such that Using the approach in the proof of Theorem 3.11 and the scaling argument, we are also able to prove the following theorem.

Theorem 3.13
Let > 0 and f ∈ L 1 (X ). Then f ∈ B 0, 1,1 (X ) if and only if there exist a sequence of ( , )-atoms {a j } for some > 0 and a sequence of numbers {λ j } such that and In particular, if f ∈ B 0, 1,1 (X ) supported in a ball B with r B = , then there exist a sequence of ( , )-atoms {a j } j supported in 3B for some > 0 and a sequence of numbers {λ j } such that (34) and (35) hold true.

Proof of Theorem 1.4
We state the following results in which the proofs of Lemma 4.1 and Proposition 4.2 below are similar to those of Lemmas 3.4, 3.2, 3.3 and Corollary 3.5.

Lemma 4.1 Let ψ be an even function in
Then we have for f ∈Ḃ 0,L 1,1 (X ).

Proposition 4.2
The following properties hold true for the homogeneous Besov spacė B 0,L 1,1 (X ).
Proof Similarly to the proof of Lemma 3.4, it suffices to prove the proposition for f ∈ L 2 (X ) ∩Ḃ 0,L 1,1 (X ). By spectral theory, On the other hand, from Lemma 2.7, This implies that for some g ∈ L 1 (X ). This, in combination with (37), implies that f = g for a.e.. Therefore, for f ∈ L 2 (X ) ∩Ḃ 0,L 1,1 (X ). This completes our proof.

Proof of Theorem 1.4:
The proof of the atomic decomposition for functions f ∈ B 0,L 1,1 (X ) is similar to that of Theorem 4.2 in [5] and the proof of Theorem 3.11. Hence, we leave it to the interested reader.
For the reverse direction, it suffices to show that there exists C > 0 such that Suppose that a is an (L, M)-atom associated with a ball B. Then we have For the first term, we have For the second term, using a = L M b, This completes our proof.

Proof of Theorem 1.6
We refer the reader to Sect. 2.1 for the index set I, the family functions {ψ α } α and the family of balls {B α } α which will be used in this section.

Lemma 4.5 For each f
Proof Denote Then we write We estimate E 1 first. Owing to Lemma 2.1 and the upper bound of q t (x, y), we have This implies that where Since J 1,β is uniformly bounded in β ∈ I, using (39) we obtain If β ∈ I 2,α , then ψ α (y) = 0 for all y ∈ B β . Therefore, By the upper bound of q t (x, y) and the fact that d(x, y) > ρ(x α ) whenever x ∈ B α , y ∈ B β with β ∈ I 2,α , we further simplify to that On the other hand, invoking (5) we have Therefore, dt t dμ(y).
Since {B β } β∈I is a finite overlapping family and ∪ α∈J 2,β B α ⊂ X \B * β , we also obtain that This completes our proof.
We are ready to give the proof of Theorem 1.6.
Proof of Theorem 1.6: We first prove that each function f ∈Ḃ 0,L 1,1 (X ) admits an atomic decomposition as in the statement of the theorem.
Observe that by (5), for z ∈ 3B, This, together with (L1), yields that It follows that A 3 1. This completes our proof.

Application to Boundedness of Riesz Transforms Associated to Schrödinger Operators on R n
In this section, we show the boundedness of the Riesz transforms associated to Schrödinger operators L = − + V on R n on the new Besov spaceḂ 0,L 1,1 (R n ). It is worth noticing that although we restrict ourselves to consider the Schrödinger operators on R n , our approach works well in more general setting including settings listed in Remark 1.1.
Let L = − + V be a Schrödinger operator on R n , n ≥ 3 with V ∈ R H n/2 . Our main result in this section is the following theorem. We would like to remark that in the classical case, the Riesz transform ∇(− ) −1/2 is bounded on the classical Besov spacesḂ 0 1,1 (R n ). See for example [6,Proposition 2.4]. In the setting of Theorem 5.1, we have a better estimates for the Riesz transform ∇ L −1/2 since by Theorem 3.13,Ḃ 0 1,1 (R n ) →Ḃ 0,L 1,1 (R n ). Therefore, as a consequence of Theorems 3.13 and 5.1, we have:
In order to prove Theorem 5.1 we need the following technical lemma. To do this, we write