Hardy spaces associated to generalized Hardy operators and applications

In this paper, we will study the Hardy and BMO spaces associated to the generalized Hardy operator Lα=(-Δ)α/2+a|x|-α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{\alpha }= (-\Delta )^{\alpha /2}+a|x|^{-\alpha }$$\end{document}. Similarly to the classical Hardy and BMO spaces, we will prove that our new function spaces will enjoy some important results such as molecular decomposition and duality. As applications, we show the boundedness of the spectral multiplier of Laplace transform type and the Sobolev norm inequalities involving the generalized Hardy operator.

The main aim of this paper is twofold. Firstly, we develop the theory of Hardy spaces H p Lα (R n ) associated to the operator L α for n n+α < p ≤ 1. We show that the Hardy spaces admit molecular decomposition which is similar to the classical Hardy space. Then we also prove the duality result of the Hardy spaces H p Lα (R n ) and new BMO spaces. See Sect. 3. Secondly, as applications we obtained our result to prove the boundedness of the spectral multiplier of Laplace transform type and the Sobolev norm equivalence involving the generalized Hardy operators L α .

Tent spaces
The tent spaces, which were first introduced in [8], become an effective tool in the study of function spaces in harmonic analysis including Hardy spaces. In this section we will recall the definitions of the tent spaces and their fundamental properties in [8]. We begin with some notations in [8]. • For x ∈ R n and β > 0, we denote When β = 1, we briefly write Γ(x) instead of Γ 1 (x). • For any closed subset F ⊂ R n , define Let R n+1 The following definition is taken from [8].

Definition 2.4. (The tent spaces)
The tent spaces are defined as follows.
One of the most important properties of the tent spaces is atomic decomposition. We now recall the definition of atoms in the tent space T p 2 .
Then we have: Furthermore, if F ∈ T p 2 ∩ T 2 2 , then the sum also converges in T 2 2 . Proof. See Proposition 5 in [8] and Corollary 3.1 in [16].
We also recall duality results of the tent spaces. Proposition 2.7. (i) The following inequality holds, whenever f ∈ T 1 2 and g ∈ T ∞ 2 : More precisely, the pairing as the dual of T p 2 . Proof. We refer Theorems 1 and 2 in [8] for the proof of (i) and (ii), and Proposition 3.2 in [19] for the proof of (iii).

Square functions
For f ∈ L 2 (R n ) and x ∈ R n , we define the vertical square function and the area square function by Theorem 2.8. Let L α be as in (1) with σ defined by (2). For all (n σ ) < p < n σ , we have For the boundedness of the area square function S Lα we have the following result. Theorem 2.9. The area square function S Lα is bounded on L p (R n ) for all (n σ ) < p < n σ .
In order to prove the theorem, we need the following result in [1]. Theorem 2.10. Let 1 ≤ p 0 < 2 and T be sublinear operator which is bounded on L 2 (R n ). Assume that there exists a family of operators {A t } t>0 satisfying that for every ball B and for all f supported in B Proof of Theorem 2.9. For (n σ ) < p < 2, we apply Theorem 2.10 with where M is a fixed constant such that M ∈ N and M > n α . We have (2) from Theorem 2.10 is a direct result of Theorem 2.2. Hence, we have to show that for all j ≥ 3, balls B = B(x B , r B ) and f ∈ L p (R n ) supported in B, it holds that . . .
We will now study A 1 . Let . Hence, using the following identity where 1 2 In order to estimate A 1 , we need only to estimate the term since the estimates of the integrals over S j−1 and S j+1 can be done similarly. Let τ = t α + s 1 + · · · + s M . By using Minkowski's inequality we have In addition, by using Theorem 2.2 and the fact τ (2 j r B ) α in this situation, we havê Then, by a straightforward calculation we havê In case of Hardy spaces associated to generalized Hardy operators Page 11 of 40 40 Hence, we have In addition, by using Theorem 2.2 and Minkowski's inequality, Therefore, Hence, using Theorem 2.10 we conclude that S Lα is bounded on L p (R n ) for all p ∈ (n σ , 2). Let M be the Hardy-Litllewood maximal operator. By using Fubini's theorem and Hölder inequality, and for p ∈ (2, n σ ), for any By using Theorem 2.8 we have G Lα is bounded on L p . In addition, we have p which imply that S Lα is bounded on L p (R n ) for all p ∈ 2, n σ . Thus, Theorem 2.9 is proved.

Hardy spaces and BMO spaces associated to generalized Hardy operators
In this section, we always assume that L α is the operator defined by (1) with σ defined by (2). Our approach based on the approaches in [2,3,[9][10][11]14,16,19]. However, since the heat kernel estimates of L α are weaker than those in existing settings, new ideas and modifications are required.

Hardy spaces associated to generalized Hardy operators
We now follow [2] to define the new Hardy spaces via the area square function associated to L α .
Similarly to the classical case, we will show that our new Hardy spaces admit molecular decomposition property. We adapt ideas in [9,13,14] to introduce a notion of molecules in our setting. (ii) For every k = 0, 1, .., M and j ∈ N one has Then we can define the Hardy spaces associated to L α via molecular decomposition. Furthermore, we have Definition 3.3. Given 0 < p ≤ 1 and > 0, we say that To begin the proof of Theorem 3.4, we state the following lemma. The proof of this lemma is similar to that of Lemma 3.3 in [13] and we omit it here. Lemma 3.5. Fix n n+α < p ≤ 1, > 0 and M > n αp . Assume that T is a linear operator , or non-negative sublinear operator of weak-type (2,2), and that for every Proof. Fix p ∈ n n+α , 1 , > 0 and M > n αp . By Lemma 3.5, it suffices to prove that there exists C > 0 such that Let m be a (p, 2, M, ) Lα -molecule and B(x B , r B ) be the ball associated to m. Then we have Thus it suffices to show that there is a > 0 such that is supported in 2 j+2 B. Hence, by using Holder's inequality we havê We will consider four cases corresponding to possibilities of X j . Case 1: Using the inequalityˆ| x−y|<t dx t n 1 and (7), we havê where in the second inequality we used the L 2 -boundedness of the square function G Lα . This proves (8).
In this case, we have ˆR By using Theorem 2.2 and the definition of molecules, we have Furthermore, using Theorem 2.2 we have Collecting the estimates of E 1 , E 2 and E 3 , we obtain (8) for this case.
Using the inequalityˆ| x−y|<t dx t n 1, we can dominate the left hand side of (8) by In addition, by using Theorem 2.2 and the similar argument used to estimate E 1 , we have where in the first inequality we used Definition 3.2.
This proves (8). Hence, we have proved (8). As a consequence, which implies that This completes our proof. Now we will prove that H SL α (R n )∩L 2 (R n ) ⊆ H p Lα,mol,M, (R n ). The proof of this inclusion bases on the following results.
define a multiple of (p, 2, M, ) Lα -molecule associated to the ball B with = α + n − n p .
Proof. Let A be a T p 2 -atom supported in B with some ball B. Then we have, We now define In addition, for a fixed i ∈ N, let f ∈ L 2 (S i (B)) such that supp f ⊆ S i (B) and f L 2 (Si(B)) = 1. Then Using Theorem 2.2, we have ¨ Combining the two inequalities we have ˆR Taking supremum over all f such that f L 2 (2 i B) = 1 we obtain In a similar way we can show that for all integer k such that k ≤ M , where = α + n − n p > 0. This ensures Lemma 3.7 and this completes the proof.
Proof. Let From the definition of H p SL α and Theorem 2.9, we have F (x, t) ∈ T p 2 ∩ T 2 2 . Thus, by using Lemma 2.6, F can be represented in the form 2 -atom and the sum converges in T p 2 and T 2 2 . In addition, Using the L 2 -functional calculus, there is C > 0 such that This, in combination with the fact that the operator π L,M defined by is a bounded from T 2 2 to L 2 (R n ), implies that where the convergence in L 2 (R n ).

NoDEA
Hardy spaces associated to generalized Hardy operators Page 21 of 40 40 In Lemma 3.7 we have proved that each m i is multiple of a (p, 2, M, ) Lαmolecule with a harmless constant. Therefore, f ∈ H p Lα,mol,M, (R n ) and This completes our proof.
Remark 3.9. Due to the coincidence between H p Lα,mol,M, (R n ) and H p SL α (R n ), in the sequel we will write H p Lα (R n ) for either H p Lα,mol,M, (R n ) or H p SL α (R n ) with n n+α < p ≤ 1, = α + n − n p and M ≥ nα/p.

BMO spaces associated to generalized Hardy operators
In this section we will develop the theory of BMO spaces associated to L α . This function space plays an important role in proving the duality of the Hardy spaces H p Lα (R n ).
We denote the set of all functions of type (L α , β) by M β . For f ∈ M β , we define It is easy to see that M β is a Banach space and M β ⊂ M β for β < β .
for almost all x ∈ R n .
Proof. We only prove the lemma for M = 0. The case M ∈ N can be done similarly. By Hölder's inequality and the fact that σ ∈ (−α, n−α 2 ), it is easy to verify that if f ∈ M β for any β > 0, then Let E 1 = y ∈ R n : |y| > t 1 α and E 2 = y ∈ R n : |y| ≤ t For the term I, we have Hence, Hence, For the term II, we have Hardy spaces associated to generalized Hardy operators Page 23 of 40 40 Let B 2 = y ∈ E 2 : |x − y| > max t 1 α , 1 + |x| . Similarly to inequality (11), we have 1 + |y| ≤ 2|x − y|, whenever y ∈ C. Therefore, Taking the estimates of I and II into account, we get This completes our proof.
where the supremum is taken over all balls B = B(x B , r B ) in R n .

Note that BM O γ
Lα,M (R n ) is a seminormed vector space, with the semi norm vanishing on the space K Lα,M , defined by f (x) = 0, for a.e x ∈ R n , for all t > 0 .
In this paper, BM O γ Lα,M space is understood to be modulo K Lα,M .
Proof. The proof of this lemma is simple and we omit the details.
Recall that a measure ν is a Carleson measure of order β ≥ 1, if there is a positive constant c such that for each ball B on R n ν B ≤ c|B| β .
The smallest constant in (13) is define to be the norm of ν, and denoted ν V β . See chapter XV of [18].
Proof. It suffices to prove that for any ball We By using Theorem 2.2, Lemma 3.13 and that γ < α n , we have Hence, ν(x, t) is a Carlson measure V 2γ+1 .

Duality of Hardy spaces associated to generalized Hardy operators
The main result of this section is the following theorem.
initially defined on H p Lα (R n ) ∩ L 2 (R n ) the dense subspace of H p Lα (R n ), has a unique bounded extension to H p Lα .

Applications
In this section, we will apply our results to prove the boundedness of the spectral multiplier of Laplace transform type of the operator L α and the Sobolev norm inequalities involving the generalized Hardy operator L α .

Spectral multipliers
Let a(t) : [0, ∞) → C be a bounded Borel function. We define which is bounded on L 2 (R n ). Note that when a(t) = − t is Γ(is) , the spectral multiplier turns out to be the imaginary power operator F (L α ) = L is α .
Particularly, the imaginary power operator L is α , s ∈ R is bounded on L p (R n ) for n σ < p < n σ , and for fixed n σ < p < n σ its operator norm does not exceed C p e |s| .
Proof. Once we have proved that F (L α ) is bounded from H p Lα (R n ) to L p (R n ) for n n+α < p ≤ 1, by using the fact that F (L α ) is bounded on L 2 (R n ) and Theorem 3.19, F (L α ) is bounded on L p (R n ) for all n σ < p < 2. Then by the duality, F (L α ) is bounded on L p (R n ) for all 2 < p < n σ . The boundedness of the imaginary power operator L is is a direct consequence by taking a(t) = − t is Γ(is) . Therefore, we need only to prove that F (L α ) is bounded from H p Lα (R n ) to L p (R n ) for n n+α < p ≤ 1. To do this, suppose that m = L M α b is a (p, 2, M, ) Lαmolecule associated to a ball B with = α + n − n p . We need to show that F (L α )m p 1.
We now split F (L α )m into where C N k are constants. This follows that It suffices to prove that E + F 1.
We will take care of F first. We need only to show that since the other terms can be done similarly.
To do this we write For each j, by Hölder's inequality we have For k = 0, 1, 2, using the definition of (p, 2, M, ) Lα -molecule and the L 2boundedness of F (L) and (r α For k ≥ 3, using (22), Minkowski's inequality and Theorem 2.2 we have We now break the integral into small parts corresponding to integrals over (0, r α B ], (r α B , (2 j r B ) α ], ((2 j r B ) α , (2 j+k r B ) α ] and [(2 j+k r B ) α , ∞)] and by using a simple calculation we come up with For the first term on the RHS of (24), using the identity At this stage, arguing similarly to the estimate of F we also have E 1.
This completes our proof.

Sobolev norm inequalities
The main result of this section is the following theorem.
Note that the estimate (25) was proved in [17] for s = 2 and s ∈ (0, 2) with a ≥ 0. We now fill the gap to prove the general case s ∈ (0, 2] and a ≥ a * . It is important to note that the general case s ∈ (0, 2] and a ≥ a * was also proved in [6] but using a different approach.