Boundedness of Some Paraproducts on Spaces of Homogeneous Type

: Let ( X , d , µ ) be a space of homogeneous type in the sense of Coifman and Weiss. In this article, the author develops a partial theory of paraproducts { Π j } 3 j = 1 deﬁned via approximations of the identity with exponential decay (and integration 1), which are extensions of paraproducts deﬁned via regular wavelets. Precisely, the author ﬁrst obtains the boundedness of Π 3 on Hardy spaces and then, via the methods of interpolation and the well-known T ( 1 ) theorem, establishes the endpoint estimates for { Π j } 3 j = 1 . The main novelty of this paper is the application of the Abel summation formula to the establishment of some relations among the boundedness of { Π j } 3 j = 1 , which has independent interests. It is also remarked that, throughout this article, µ is not assumed to satisfy the reverse doubling condition.


Introduction
Classical paraproducts defined via convolutions are kinds of non-commutative bilinear operators, which are useful tools in the decompositions of products of functions. The prototypes of paraproducts can be found, for examples, in the work of Fujita and Kato [1] and Kato [2] on the study of mild solutions of Navier-Stokes equations and in the investigation of pseudo-differential operators and para-differential operators by Meyer and Coifman [3][4][5]. The formal notion of paraproducts has been introduced in 1981 by Bony for the study of the nonlinear hyperbolic partial differential equations in [6]. Since then the theory of papraproducts has been developed rapidly, which plays an essential role in both harmonic analysis and partial differential equations. For applications of paraproducts in harmonic analysis, we refer the reader to [7][8][9][10][11][12][13][14][15][16]. See also [17,18] for more applications of paraproducts in mathematical physics. The paraproducts defined via wavelets was first investigated by Grafakos and Torres [19] and then studied by Bonami et al. [20], which play crucial roles in both the bilinear decompositions of products of functions in [20,21], the (sub-)bilinear decompositions of commutators and the endpoint estimates of commutators in [22,23]. See the survey [24] and the monographs [25,26] for more information.
In 1970s, Coifman and Weiss [27,28] introduced the notion of the space of homogeneous type which has been proven to be a natural background for extensions of many classical results on Euclidean spaces. Recall that a quasi-metric space (X , d) is a non-empty set X equipped with a quasi-metric d such that, for any x, y, z ∈ X , (i) d(x, y) = 0 if and only if x = y; (ii) d(x, y) = d(y, x); (iii) the quasi-triangle inequality d(x, y) ≤ A 0 [d(x, z) + d(z, y)] holds true, where A 0 ∈ [1, ∞) is called the quasi-triangle constant which is independent of x, y and z.
A space of homogeneous type, (X , d, µ), is called a metric measure space of homogeneous type if the quasi-triangle constant A 0 = 1. In this setting, Fu et al. [48] proved that f × g of f ∈ H 1 at (X ) and g ∈ BMO(X ) can be written into a sum of three bilinear operators {Π j } 3 j=1 , which are also called paraproducts. These paraproducts play important roles in the study on the endpoint boundedness of the (sub-)linear commutator [b, T] of a (sub-)linear operator T and b ∈ BMO (X ) on (local) Hardy spaces in [29,57,58]; see also the survey [63] for more details. A natural question is whether there exists a relatively complete boundedness theory for paraproducts {Π j } 3 j=1 in [48] which enjoy the same boundedness as the paraproducts in [30,62].
In this article, we give a partial affirmative answer to this question with the paraproducts {Π j } 3 j=1 in [48] replaced by more general forms via the exp-ATIs and 1-exp-ATIs from [53]. We obtained the boundedness of Π 3 on Hardy spaces and its endpoint estimates, and the endpoint estimates for Π 1 and Π 2 . The boundedness of Π 1 and Π 2 on Hardy spaces may need different approaches and was left as an unsolved question.
In what follows, we always assume that (X , d, µ) is a space of homogeneous type. The remainder of this article is organized as follows.
Section 2 is devoted to some preliminary notions and results which are needed to the proof of the main results Theorems 2-4 below. In particular, we recall the T(1) theorem from ( [32], Section 12) (see Lemma 3 below), and use the Abel summation formula to build some relations among the boundedness of {Π j } 3 j=1 (see Theorem 1 below). In Section 3, we prove Theorems 2-4 below. In precise, Theorem 2 is an easy consequence of the Hölder inequality and the definition of H p (X ). To show (i)-(iv) of Theorem 3, we first fix an f ∈ BMO (X ) and express the paraproduct Π 3 by an integral operator K (3) f . Then, via the methods of interpolation and the crucial estimates (11) and (12), we show that K (3) f has the weak boundedness property WBP(η) with η as in Lemma 2 below. Next we prove that the kernel of K (3) f is an η-Calderón-Zygmund kernel, which also relies on estimates (11) and (12). Moreover, we point out that K which, together with the T(1) theorem from ( [32], Theorem 12.2) and the boundedness of Calderón-Zygmund operators, we finally finish the proof of (i)-(iv) of Theorem 3. In order to prove (v) and (vi) of Theorem 3, we first fix g ∈ L ∞ (X ) and write Π 3 as an integral operator K (3) g . By the fact that L ∞ (X ) ⊂ BMO (X ) and some arguments used in the proof of (i)-(iv) of Theorem 3, we obtain the desired results and finish the proof of Theorem 3. The proof of (i)-(iv) of Theorem 4 is a consequence of the arguments and ideas from the proof of (i)-(iv) of Theorem 3. The main novelty of this paper lies in the proof of (v)-(vi) of Theorem 4, where we use the Abel summation formula to build some relations among the boundedness of {Π j } 3 j=1 and then transform the same boundedness of Π 1 from L 2 (X ) × L ∞ (X ) into L 2 (X ) into the same boundedness of Π 2 and Π 3 . We also remark that, throughout this article, µ is not assumed to satisfy the reverse doubling condition (2).
Finally, we list some notation used throughout this article. Let N := {1, 2, . . .} and Z + := {0} ∪ N. We use C or c to denote a positive constant which may be different from line to line, but is independent of main parameters. In addition, we also use C (ρ, α, ...) or c (ρ, α, ...) to denote a positive constant depending on the indicated parameters ρ, α, . . .. For any two real functions f and g, we write f g when f ≤ Cg and f ∼ g when f g f . For any subset E of X , denote by 1 E its characteristic function. For any x, y ∈ X , r, ρ ∈ (0, ∞) and ball B := B(x, r) := {y ∈ X : d(y, x) < r}, define ρB := B(x, ρr), V(x, r) := µ(B(x, r)) =: V r (x), and V(x, y) := µ(B(x, d(x, y))). For any p ∈ [1, ∞], let p denote its conjugate index, namely, 1/p + 1/p = 1. For any a, b ∈ R, let a ∧ b := min{a, b} and a ∨ b := max{a, b}. Finally, for any linear integral operator T, we keep the notation T for its integral kernel.

Preliminary Notions and Results
In this section, we mainly state some preliminary notions and results which are needed to the proof of the main results Theorems 2-4 below. In particular, we investigate some relations among the boundedness of {Π j } 3 j=1 . We first recall the notions of some function spaces. Let q ∈ (0, ∞]. The Lebesgue space L q (X ) is defined to be the set of all µ-measurable functions f on X such that, if q ∈ (0, ∞), supremum of | f | on X . Denote by L 1 loc (X ) the space of all locally integrable functions. Let s ∈ (0, 1] and denote by C(X ) the space of all continuous functions on X . Then the homogeneous and inhomogeneous spaces C s (X ) andĊ s (X ) of s-Hölder continuous functions on X are, respectively, defined by setting Moreover, the space C s b (X ) of all s-Hölder continuous functions with bounded support on X is defined by setting where we equip C s b (X ) with the usual strict inductive limit topology (see, for instance, ( [36], p. 273) and ( [33], p. 23)). A useful subspaceC s ] is defined to be the set of all linear functionals on C s b (X ) [resp., onC s b (X )] equipped with the weak- * topology.
called an s-Calderón-Zygmund kernel if there exists a positive constant C (K) , depending on K, such that (i) for any x, y ∈ X with x = y, and A linear operator T : C s b (X ) → (C s b (X )) is called an s-Calderón-Zygmund operator if T can be extended to a bounded linear operator on L 2 (X ) and if there exists an s-Calderón-Zygmund kernel K such that, for any f ∈ C s b (X ) and x / ∈ supp f , T f (x) := X K(x, y) f (y) dµ(y). A function f ∈ ( Lip 1/p−1 (X )) when p ∈ (0, 1), or f ∈ L 1 (X ) when p = 1, is said to belong to the atomic Hardy space H p, q at (X ) if there exist (p, q)-atoms {a j } ∞ j=1 and numbers {λ j } ∞ j=1 ⊂ C such that ∑ ∞ j=1 |λ j | p < ∞ and f = ∑ ∞ j=1 λ j a j in ( Lip 1/p−1 (X )) when p ∈ (0, 1), or in L 1 (X ) when p = 1. Moreover, the quasi-norm of f in H p, q at (X ) is defined by setting where the infimum is taken over all decompositions of f as above.
Let p ∈ (0, 1]. It was shown in ( [28], Theorem A) that H p,q at (X ) is independent of the choice of q ∈ [1, ∞] ∩ (p, ∞] and hence simply denoted by H p at (X ).
. Let x 1 ∈ X be fixed, r, ϑ ∈ (0, ∞) and κ ∈ (0, 1]. The space G(x 1 , r, κ, ϑ) of test functions is defined to be the set of all measurable functions f on X such that there exists a positive constant C such that (T1) for any x ∈ X , Moreover, the norm of f in G(x 1 , r, κ, ϑ) is defined by setting f G(x 1 , r, κ, ϑ) := inf{C : C satisfies (T1) and (T2)}.
A k being a countable set of indices for any k ∈ Z, satisfies the following properties: for any k ∈ Z, Then there exists a family of sets, {Q k α : k ∈ Z, α ∈ A k }, which is called the system of half-open dyadic cubes, satisfying In what follows, for any k ∈ Z, let and, for any (6)] of points as in Lemma 1 and its related dyadic cubes, Auscher and Hytönen ([32], Theorem 7.1) constructed the following notable system {ψ k β } k∈Z, β∈G k of regular wavelets on X , which is an orthonormal basis of L 2 (X ).

Definition 5 ([53], Definition 2.8).
A sequence {P k } k∈Z of bounded linear integral operators on L 2 (X ) is called an approximation of the identity with exponential decay and integration 1 (for short, 1-exp-ATI) if {P k } k∈Z has the following properties: (i) for any k ∈ Z, P k satisfies (ii), (iii), and (iv) of Definition 4, but without the exponential decay factor with Y k as in (7); (ii) X P k (x, y) dµ(y) = 1 = X P k (y, x) dµ(y) for any k ∈ Z and x ∈ X ; (iii) Let Q k := P k − P k−1 for any k ∈ Z. Then {Q k } k∈Z is an exp-ATI.

Remark 4.
In Theorems 2 and 3 below, we prove that Π 3 ( f , g) in Definition 7 is well defined for Definition 8. Let κ, ϑ ∈ (0, η) with η be as in Lemma 2. Let {P j } j∈Z be a 1-exp-ATI and Q j := P j − P j−1 for any j ∈ Z. Then the paraproducts Π 1 and Π 2 are formally defined, respectively, by setting ) and x ∈ X , ) and x ∈ X , where the above two series converge in G η 0 (κ, ϑ) .

Remark 5.
(i) In Theorem 4 below, we show that for any proper functions f and g , we conclude that Π 2 shares corresponding boundedness to Π 1 as in Theorem 4 below.
To prove Theorem 3 below, we need to recall the T(1) theorem from ( [32], Section 12). Let σ ∈ (0, 1) and s ∈ (0, σ]. A linear continuous operator T : C s b (X ) → (C s b (X )) is said to have weak boundedness property WBP(σ) if there exists a positive constant C 1 such that, for any f , g ∈ C σ b (X ) normalized by f L ∞ (X ) + r σ f Ċσ (X ) ≤ 1 and g L ∞ (X ) + r σ g Ċσ (X ) ≤ 1, with support in some ball B(x, r) (x ∈ X and r ∈ (0, ∞)), As for T(1) with T associated with the s-Calderón-Zygmund kernel, it is defined as a continuous linear functional onC s b (X ) by setting where g : X → R satisfies that there exists a ball B(x 0 , r) ⊃ supp f such that, for any . It is not difficult to show that both of the two terms in the right hand side of (8) are well defined.
At the end of this section, we use the Abel summation formula to make some links among the boundedness of Π 1 , Π 2 and Π 3 in some sense, which plays an important role in the proof of Theorem 4 below. In what follows, for any N ∈ Z and suitable functions f and g, Theorem 1. Assume that there exists a positive constant C such that, for any N ∈ N, f ∈ L 2 (X ) and g ∈ L ∞ (X ), Then Π 1 defined as in Definition 8 is bounded from L 2 (X ) × L ∞ (X ) into L 2 (X ).
Proof. Let f ∈ L 2 (X ) and g ∈ L ∞ (X ). For any N ∈ N, by the Abel summation formula, we know that From this, (9) and Remark 2(ii), we deduce that which, combined with the Fatou lemma, implies that This completes the proof of Theorem 1.

Boundedness of Paraproducts
This section is devoted to the proofs of the main results of this article on the boundedness of paraproducts {Π j } 3 j=1 . We now state the first main result of this article as follows.
(ii) It is still unclear whether Π 1 and Π 2 can be extended to bounded operators from H p (X ) × H q (X ) into H r (X ) or not.
The following result is an easy consequence of Theorem 2, we omit the details here.
Then we state other two main results of this article, which give various endpoint estimates of Π 3 and Π 1 . In what follows, the weak Lebesgue space L 1,∞ (X ) is defined to be the set of all µ-measurable functions f on X such that and the space BMO (X ) the set of all locally integrable functions f on X such that where the supremum is taken over all balls of X and, here and thereafter, for any locally integrable function f and a ball B ⊂ X , m B ( f ) := 1 µ(B) B f (y) dµ(y).

As in ([30], Remark 3.3) or ([62]
, Remark 1.8), the following estimates are important to escape the dependence on the RD-condition (2). For any given a, c ∈ (0, ∞), and, for any r ∈ (0, ∞) and x ∈ X , (see ( [32], Lemma 8.3)) and, for any x, y ∈ X with x = y, where the implicit positive constant is independent of x and y (see ( [54], Lemma 4.9)), which essentially connect the geometrical properties of X expressed via its equipped quasi-metric d, dyadic reference points and dyadic cubes. Now, we are ready to prove Theorem 3.

Proof of Theorem 3.
Without loss of generality, we may assume that the sum ∑ j∈Z in Π 3 ( f , g) is a finite sum ∑ N j=−N for any fixed N ∈ N, see ( [66], pp. 302-305) for some details. We first prove (i)-(iv) of this theorem. To this end, we temporarily fix an f ∈ BMO (X ).
For any x ∈ X , we write f (x, y)g(y) dµ(y) =: K f is an integral operator associated with the kernel defined by setting, for any x, y ∈ X , To prove (i)-(iv) of this theorem, the key point is the proof of the boundedness of K (3) f on L 2 (X ), where we need some ideas from ( [67], Remark 4.4.5).
We first claim that K (3) f has WBP(η) and hence maps from , supported on some ball B(x 0 , r 0 ) with x 0 ∈ X and r 0 ∈ (0, ∞), be normalized by Then, by the fact from ( [62], (2.3)) that and the Hölder inequality, we conclude that Thus, to prove the above claim, it suffices to show that We further consider the following two cases.

by (v) and (ii) of Definition 4 and (1), we have
which implies that On the other hand, from (15), we deduce that Thus, which, combined with (11), further implies that V(x 0 , r 0 ).
Indeed, by Definition 4(ii) and (1), we have Combining the estimates of I 1 and I 2 , we obtain Now we estimate Q j (g) L 1 (X ) . Indeed, we know that For any fixed x ∈ X , we further consider the following two cases.
Observe that, by Definition 4(ii) and (1), In this case, we observe that, for any y ∈ B(x 0 , r 0 ), and hence, by Definition 4(ii) and (1), Combining Cases 1 and 2, we find that, for any x ∈ X , which implies that Q j (g) L 1 (X ) V(x 0 , r 0 ). (17), it follows that

From this and and
By this and (16), we conclude that ∑ j∈Z Q j (g) L 2 (X ) V(x 0 , r 0 ), which further completes the proof of the above claim.
Combining the Cases (1) and (2), we have which further proves that K f (·, ·) satisfies (4). By the arguments similar to those used in the proof of (20), we conclude that We further show that K  (21), we deduce that To estimate A, we deal with the following two cases.
Combining Cases (i) and (ii), we know that K f (·, ·) satisfies (5). These complete the proof of (3) through (5) for K (3) f (·, ·). Next we show that K By X h(y) dµ(y) = 0, we conclude that We observe that if d(y, x 0 ) < r 0 and d(x, Then we need to prove lim N→∞ I N = f , h . Indeed, for any h ∈C η b (X ), observe that, by ([29], Corollary 3.14) and the boundedness of Q j on L 2 (X ), we know that Q j ( f )η N ∈ L 2 (X ), which, combined with the assumptions (a) and (b) in Theorem 3, the fact that h is a multiple of a (1, 2)-atom, and the Lebesgue dominated convergence theorem, further implies that where in the fifth inequality of this equation, we need to show that the series ∑ j∈Z Q j ( f )Q j (h) absolutely converges in L 1 (X ). Indeed, from (13), ([62], (2.4)) and the fact that C η b (X ) ⊂ G(x 1 , r, η, ϑ) for any given x 1 ∈ X and r, ϑ ∈ (0, ∞) (see ([33], p. 19)), it follows that which proves the desired result. This shows lim N→∞ I N = f , h , which, together with the estimate of II N , implies that (K Moreover, from the T(1) theorem (see Lemma 3) ( [32], Theorem 12.2), we deduce that K Now we begin to show (v) and (vi) of Theorem 3. To this end, we temporarily fix a g ∈ L ∞ (X ). From the fact that g ∈ L ∞ (X ) ⊂ BMO (X ), and the arguments used in the proof of (i)-(iv) of Theorem 3, it follows that the kernel of the operator K g (·) := Π(·, g), defined by setting, for any (x, y) ∈ X × X , g (x, y) := ∑ j∈Z Q j (x, y)Q j (g)(y) satisfies (3) through (5) and WBP(η) with f BMO (X ) replaced by g L ∞ (X ) , K g is bounded on L 2 (X ). Thus, K g is an η-Calderón-Zygmund operator, which, combined with the fact that (K (3) g ) * (1) = g ∈ L ∞ (X ) ⊂ BMO (X ) and the T(1) theorem (see Lemma 3) and ( [27], Theorem 2.4 in Chapter III), further completes the proof of (v) and (vi) of Theorem 3 and hence of Theorem 3.
Proof of Theorem 4. Similar to the proof of Theorem 3, without loss of generality, we may assume that the sum ∑ j∈Z in Π 1 ( f , g) is a finite sum ∑ N j=−N for any fixed N ∈ N. We first prove (i) through (iv) of Theorem 4. Fix f ∈ L ∞ (X ), we consider the operator K (1) f and its kernel, which is still denoted by K (1) f , defined by setting, for any x ∈ X ,

Now we show that K
(1) f (·, ·) satisfies (3) through (5). To this end, we first prove that K (1) f (·, ·) satisfies (3). From Remark 2(ii), it follows that there exists a positive constant C such that, for any f ∈ L ∞ (X ), which, together with (13) and some arguments used in the proof of (18), further implies that, for any x, y ∈ X , which completes the proof of (3) for K (1) f (·, ·). Then we prove that K (1) From (20) and (23), we deduce that Moreover, by the fact that L ∞ (X ) ⊂ BMO (X ) and some arguments similar to those used in the proof of (22), we know that which completes the proof of (4) for K (1) (18) and (21), it follows that which completes the proof of (5) for K (1) f (·, ·). This completes the proof of (3) through (5) for K (1) f (·, ·).
Then we claim that K (1) f has WBP(η) and hence maps from C η b (X ) into (C η b (X )) . Indeed, let g, h ∈ C η b (X ), supported on some ball B(x 0 , r 0 ) with x 0 ∈ X and r 0 ∈ (0, ∞), normalized by Then, by (23), the Hölder inequality and (14), we conclude that Next we show that K f (1) = 0 ∈ BMO (X ). Now we prove that (K (1) f ) * (1) = 0 ∈ BMO (X ). It is easy to see that By the same arguments used in the proof of (K (3) f ) * (1) ∈ BMO (X ), we conclude that lim N→∞ II N = 0. Then we show that lim N→∞ I N = 0. Indeed, for any h ∈C η b (X ), observe that, by ( [29], Corollary 3.14) and the boundedness of P j on L 2 (X ), we know that P j ( f )η N ∈ L 2 (X ), which, combined with the assumption (10), and the Lebesgue dominated convergence theorem, further implies that where in the third to the last inequality of this equation, we have used the fact that the series ∑ j∈Z P j ( f )Q j (h) absolutely converges in L 1 (X ), which is similar to that of ∑ j∈Z Q j ( f )Q j (h). This shows lim N→∞ I N = 0, which, together with the estimate of II N , implies that (K (1) f ) * (1) = 0 on (C η b (X )) and hence (K (1) f ) * (1) = 0 ∈ BMO (X ).
Moreover, from the T(1) theorem (see Lemma 3), we deduce that K (1) f is bounded on L 2 (X ). Then, by the boundedness of the Calderón-Zygmund operator (see, for instance, ( [27], Theorem 2.4 in Chapter III)), we find that (i) through (iv) of Theorem 4 hold true.
Now we begin to prove (v) and (vi) of Theorem 4. To this end, we temporarily fix a g ∈ L ∞ (X ). By the fact that L ∞ (X ) ⊂ BMO (X ) and checking the proofs of Theorem 3 and (i) and (ii) of Theorem 4 carefully, we conclude that there exists a positive constant C such that, for any f ∈ L ∞ (X ), g ∈ L 2 (X ), and N ∈ N, which, further implies that, for any g ∈ L ∞ (X ) and f ∈ L 2 (X ), From this and Theorem 1, we deduce that Π 1 ( f , g) is bounded from L 2 (X ) × L ∞ (X ) into L 2 (X ), which, combined with the fact that L ∞ (X ) ⊂ BMO (X ) and some arguments used in the proof of (i) through (iv) of Theorem 3, implies that the kernel of the operator K (1) g (·) := Π 1 (·, g), defined by setting, for any (x, y) ∈ X × X , K (1) g (x, y) := ∑ j∈Z P j (x, y)Q j (g)(x) satisfies (3) through (5), and hence K (1) g is an η-Calderón-Zygmund operator which is bounded on L 2 (X ). By these and the boundedness of Calderón-Zygmund operators (see, for example, ( [27], Theorem 2.4 in Chapter III)), we finish the proof of (v) and (vi) of Theorem 4 and hence of Theorem 4.

Remark 9.
We observe that the proofs of Theorems 3 and 4 do not use the second difference regularity condition of {Q j } j∈Z in Definition 4. Thus, the results in Theorems 3 and 4 hold true for more general approximations of identity.