H\"ormander's conditions for vector-valued kernels of singular integrals and its commutators

In this paper we study Coifman type estimates and weighted norm inequalities for singular integral operators $T$ and its commutators, given by the convolution with a vector valued kernel $K$. We define a weaker H\"ormander type condition associated with Young functions for the vector valued kernels. With this general framework we obtain as an example the result for the square operator and its commutator given in [ M. Lorente, M.S. Riveros, A. de la Torre \emph{On the Coifman type inequality for the oscillation of the one-sided averages }, Journal of Mathematical Analysis and Applications, Vol 336, Issue 1,\ (2007) 577-592.]


Introduction
For several years, a classical problem in the harmonic analysis is the following: given a linear operator T , find the maximal operator M T such that T is controlled by M T in the following sense, for some 0 < p < ∞ and some 0 ≤ w ∈ L 1 loc (R n ). The maximal operator M T is related to the operator T which is normally easier to deal with. In general, M T is strongly related to the kernel of T .
The classical result of Coifman in [3] is, let T be a Calderón-Zygmund operator, then T is controlled by M, the Hardy-Littlewood maximal operator. In other words, for all 0 < p < ∞ and w ∈ A ∞ , More recently, in [12], Lorente, Riveros and de la Torre defined a L A -Hörmander condition where A is a Young function. If T is an operator such that satisfies this condition, then (1.1) holds for M A , the maximal operator associated to the Young function A.
As a consequence of the Coifman inequalities, one can prove weighted modular end-point estimates. In [11], Lorente, Martell, Riveros and de la Torre proved the following: if A is submultiplicative and λ > 0, then An example of this type of operator is the square operator S (see the precise definition below), by the results in [12] the following inequality holds, for all 0 < p < ∞ and w ∈ A ∞ . In [13] it was proved that the last inequality is not sharp in the sense that it can be replaced M 3 by M 2 .
In this paper we define a new Hörmander condition in the case of vector-valued kernels, weaker than the L A -Hörmander condition defined in [11]. We obtain inequality (1.1) improving results, for vector-valued operators, obtain in [11]. The applications of this results with the new condition are generalizations of ones for the square operator obtained [13]. In these applications, the maximal operators are of the form M L log L β , with some β ≥ 0. For instance, we obtain for all 0 < p < ∞ and w ∈ A ∞ , where X is an appropriate Banach space. If X = l 2 , S X = S the square operator, and in this case we obtain the same results as in [13].
In [2], Bernardis, Lorente and Riveros defined L A,α -Hörmander conditions for fractional integral operator. The authors obtain the inequality (1.1) with M A,α , the fractional maximal operator associated to A. In this paper, we also give a weaker condition for vector-valued kernels than L A,α -Hörmander condition and obtain similar kind of results and applications.
The plan of this paper is as follows. The next section contains some definitions and well known results. Later, in section 3, we introduced our condition and the main results. The applications are presents section 4. The proofs of the general results are in sections 5. Finally in the last section we present the Hörmander condition and the results for vector-valued fractional operators.

Preliminaries
In this section we present some notions needed to understand the main results and the applications. First we define the space in which we are going to work.
Let us consider the Banach spaces (X, · X ) where X = R Z and the norm in this space is monotone, i.e.
Observe that {a n } X = {|a n |} X for all {a n } ∈ X.
Remark 2.1. Some examples of this Banach spaces are the l p (Z) spaces, 1 ≤ p < ∞, and the space where the norm is associated to some Young function. Observe that not all Banach spaces satisfies the condition (2.1), for example, consider X = R Z with the norm Observe that |x n | ≤ |y n | for all n ∈ Z, and {x n } X = √ 13 y {y n } X = √ 10. Hence, the norm is not monotone.
Remark 2.2. If X is a Banach lattice, the norm is monotone by definition. Now, we define the notion of Young function, maximal operators related to Young function and generalized Hörmander condition. For more details see [15].
A function A : [0, ∞) → [0, ∞) is said to be a Young function if A is continuous, convex, no decreasing and satisfies A(0) = 0 and lim t→∞ A(t) = ∞. The average of the Luxemburg norm of a function f induced by a Young function A in the ball B is defined by Each Young function A has an associated complementary Young function A satisfying the generalized Hölder inequality Given f ∈ L 1 loc (R n ), the maximal operator associated to the Young function A is defined as For example, if β ≥ 0 and r ≥ 1, A(t) = t r (1 + log(t)) β is Young function then Remark 2.3. Let us observe that when D(t) = t, which gives L 1 , then D(t) = 0 if t ≤ 1 and D(t) = ∞ otherwise. Observe that D is not a Young function but one has L D = L ∞ . Besides, the inverse is D −1 ≡ 1 and the generalized Hölder inequality make sense if one of the three function is D.
Once the Luxemburg average has been defined, we can introduce the notion of the generalized Hörmander condition, for this we need to introduce some notation: |x| ∼ s means s < |x| ≤ 2s and given a Young function A, we write, In [11] and [12] were introduced the following classes, Definition A. Let K be a vector-valued function, A be a Young function and k ∈ N ∪ {0}, then K satisfies the L A,X,k -Hörmander condition (K ∈ H A,X,k ), if there exist c A > 1 and C A > 0 such that for all x and R > c A |x|: Remark 2.4. There exists a relation between the Hörmander classes, H A,X,k .
Next, we define the notions of singular integral operator and its commutator in the vector-valued sense.
The operator T will be a singular integral operator if it is strong (p 0 , p 0 ), for some p 0 > 1, and the kernel K = {K l } l∈Z ∈ H 1,X .
Remark 2.6. The operator T is strong (p 0 , p 0 ) in the sense of Bochner-Lebesgue spaces. Given a X Banach space, L p X (R n ) is called Bochner-Lebesgue spaces with the norm Remark 2.7. Since K = {K l } l∈Z ∈ H 1,X , then T is of weak type (1,1). Thus, using the fact that T is of strong type (p 0 , p 0 ) by interpolation and duality, T is of strong type (p, p) ∀ 1 < p < ∞. Moreover, since T is of weak type (1,1), T satisfies Kolmogörov's inequality where 0 < ǫ < 1 and supp(f ) = B ⊂ B.
Let us recall the BMO space and the sharp maximal function.
Remark 2.8. Some properties of BMO are the following. Given b ∈ BMO, a ball B, k ∈ N ∪ {0}, A(t) = exp(t 1/k ) and q > 0, by John-Nirenberg's Theorem we have On the other hand, for any j ∈ N and b ∈ BMO, we have Definition 2.9. Given T a singular integral operator and b ∈ BMO, it is define the k-th order commutator of T , k ∈ N ∪ {0}, by: Note that for k = 0, T k b = T and observe that We will consider weights in the Muckenhoupt classes A p , 1 ≤ p ≤ ∞. Let w be a non-negative locally integrable function. We say that w ∈ A p if there exists C p < ∞ such that for any ball B ⊂ R n , when 1 < p < ∞, and for p=1, Finally we set A ∞ = ∪ 1<p A p . It is well known that the Muckenhoupt classes characterize the boundedness of the Hardy-Littlewood maximal function on weighted In [12] and [11], the following results were proved, Let K be a vector-valued function that satisfies the L A,X -Hörmander condition and let T be the operator associated to K. Suppose T is bounded in some for any f ∈ C ∞ c and whenever the left-hand side is finite. For commutators of the operator T , there is the following result: whenever the left-hand side is finite. Furthermore, if A is sub-multiplicative, then for all w ∈ A ∞ and λ > 0,

Main results
In this section we will state a new condition weaker than the generalized Hörmander condition (Definition A). The previous Theorems B and C still remain true using this new condition.
Remark 3.2. The classes H † A,X,k satisfies the same inclusion of the classes H A,X,k , see remark 2.4. And the relation between this classes is the following, In section 4, we give an explicit example of a kernel K such that K ∈ H † A,X,k and K ∈ H A,X,k , (see Proposition F and Corollary 4.5).
Using Definition 3.1, the previous theorem are written as follows, for the case k = 0, whenever the left-hand side is finite.
And for the case k ∈ N, whenever the left-hand side is finite. Furthermore, if A is sub-multiplicative, then for all w ∈ A ∞ and λ > 0, Remark 3.5. These theorems are more general than Theorem B and C, since there exists a singular integral operator whose kernel K ∈ H † A,X,k and K ∈ H A,X,k for some appropriate Young function A.
In this case Theorems 3.3 and 3.4 can be written as follows: A,X,k , then for any 0 < p < ∞ and w ∈ A ∞ , there exists C such that whenever the left-hand side is finite. Furthermore, for all w ∈ A ∞ and λ > 0,

Applications and generalization
Now, we define the vector-valued singular integral operator,T , and its commutator, that will be an example of our results.
For this operatorT , the Banach space (X, · X ) will be (l 2 (Z), · l 2 ). The k-th order commutator is defined as, is the k-th order commutator ofT . The S k b is called the k-th commutator of the square operator.
In [17] and [11], the authors studied the kernel of the square operator for the onesided case, the results for the two-sided case are the following and the proof are analogous to the one-sided case.
In [12], using Proposition D the authors proved the following results Proposition E. [12] The kernel K ∈ H ∞, l 2 . Remark 4.3. As K ∈ H ∞, l 2 ,k we can not use Theorem 3.4 to conclude This inequality is still an open problem.
In [12] and [11], as an application of Theorems B and C the authors obtained the following result Theorem G. [12,11] Let b ∈ BMO and k ∈ N ∪ {0}. Let S k b be the k-th order commutator of the square operator. Then for any 0 < p < ∞ and w ∈ A ∞ , there exist C such that whenever the left-hand side is finite.
For the case of the kernel of the the square operator we obtain, Proposition 4.4. Let k ∈ N ∪ {0} and A be a Young function. Then,  Theorem H. [13] Let b ∈ BMO and k ∈ N∪{0}. Let S k b be the k-th order commutator of the square operator. Then, for any 0 < p < ∞ and w ∈ A ∞ , there exists C such that whenever the left-hand side is finite.

4.1.
Generalization of square operator. In this subsection, we will build a family of operators and we will prove that they satisfy Proposition 4.4. This operators are a generalization of the square operator. Let X be a Banach space with a monotone norm, see (2.1). We define S X f (x) := ||T f (x)|| X , whereT was define in Definition 4.1. Observe that if X = l 2 then S X = S, the square operator.
We can generalize Proposition 4.4 and Corollary 4.5, replacing the l 2 -norm by Xnorm.
In this context Proposition 4.4 affirm, for all k ≥ 0, and A be a Young function, Applying Theorem 3.6, we obtain whenever the left-hand side is finite. A interesting example is, given a Young function E, we denote X E = (R Z , · E ), the Banach space with Now we give an example of a family of Young functions, E, for which condition (4.2) holds. Let us consider, for t ≥ 0, the Young function E(t) = t r (log(t + 1)) β , where β ≥ 0 and r ≥ 1.

Proof of Proposition 4.4 and Corollary 4.5.
In this subsection we proceed to study the applications. Let K be the kernel of the square operator, defined above.
Definition. We define the sets, Proof of Proposition 4.4. Recall K ∈ H † A, l 2 ,k if there exist c A > 1 and C A > 0 such that for each x and R > c A |x|, Let us prove, x such that |x| < 2 i , I m := (−2 m+i , 2 m+i ) and −F m y F m as above. For l ∈ Z, using Proposition D we obtain Using that where the last inequality holds due A −1 is monotone. Then, for all |x| < 2 i , we obtain, In particular, the last inequality holds for all |x| < 2 i 4 . As |x| < 2 i 4 then 2 l+i+1 Then, by hypothesis, we obtain K ∈ H † A,l 2 ,k . Now let us prove that K ∈ H † A,l 2 ,k ⇒ m k A −1 (2 m 8) m∈Z l 2 < ∞. By hypothesis, there exist c A > 1 and C A > 0 such that for all R ∈ R and for all x, |x|c A < 2 i , then this holds for all |x| < 2 i . Then, taking supremum, we obtain, Hence, Proof of Corollary 4.5. Let A(t) = exp(t 1+k ) − 1. Using Proposition 4.4, is enough to prove that for any k ∈ N ∪ {0}, As

Proofs of the main results
For the proof of the main results we need the following, If T is a vector-valued singular integral operator with kernel K such that K ∈ H † B,X ∩ H † A,X,k , then for any b ∈ BMO, 0 < δ < ε < 1 we have a) if k = 0, B = A, then there exists C > 0 such that, for all x ∈ R n . b) If k ∈ N, then there exists C = C(δ, ε) > 0 such that, for all x ∈ R n .
Proof. The argument is similar to the proof of Lemma 5.1 in [11], we only give the main changes. Let consider the part (b), the part (a) is analog with k = 0.
Let K ∈ H † B,X ∩ H † A,X,k and k ∈ N. Then for any λ ∈ R, we can write Let us fix x ∈ R n and B a ball such that x ∈ B,B := 2B and c B := center of the ball B.
Let f = f 1 + f 2 , where f 1 := f χB and let a := T (bB − b) k f 2 (c B ) X . Using (5.1) and The estimates of I and II are analogous to corresponding in the Lemma 5.1 in [11]. Then , Now III. By Jensen's inequality and the property of the norm (2.1), we get For each coordinate l ∈ Z, we proceed as in the proof of Lemma 5.1 in [11]. Let B j := 2 j+1 B, for j ≥ 1 and we obtain Hence, where the last inequality holds since K ∈ H † B,X ∩ H † A,X,k and we have used that x ∈ B ⊂ B j and that |x B − y| < R since y ∈ B. Thus, Now we proceed to prove the main theorems.
Proof of Theorem 3.3. Let w ∈ A ∞ , and suppose the kernel K ∈ H † A,X , where A is a Young function and f ∈ C ∞ c (R n ). Let p > 0, we take ε such that 0 < δ = pε < 1. Then using the part (a) of Lemma 5.1 we obtain for the second inequality we need the left hand is finite, to prove this we use the fact that f ∈ C ∞ c imply R n M ε ( T f p x ) w < ∞. (see [6] and [11]). Thus, Since the space C ∞ c (R n ) is dense in L p (R n ) for all p, we prove the result.
Proof of Theorem 3.4. The proof is analogous to the proof of Theorem 3.3, part (a), in [11], using in this case Lemma 5.1.

Fractional integrals
For fractional integral operator there exist L A,α -Hörmander conditions defined in [2]. The authors obtain the inequality (1.1) with M A,α , the fractional maximal operator associated to A. In this section, we present a weaker condition for fractional vector-valued kernels and obtain similar results and applications.
Recall the notation: |x| ∼ s means s < |x| ≤ 2s and given a Young function A we write f A,|x|∼s = f χ |x|∼s A,B(0,2s) .
The new condition is the following, Definition 6.1. Let K α = {K α,l } l∈Z be a vector-valued function, A be a Young function, 0 < α < n and k ∈ N ∪ {0}. The function K α satisfies the L α,A,X,k † -Hörmander condition (K ∈ H † α,A,X,k ), if there exist c A > 1 and C A > 0 such that for all x and R > c A |x|, We say that K α ∈ H † α,∞,k if K α satisfies the previous condition with · L ∞ ,|x|∼2 m R in place of · A,|x|∼2 m R .
Also we need an extra condition that ensure certain control of the size, in this case is, Definition 6.2. Let A be a Young function and let 0 < α < n. The function K α = {K α,l } l∈Z is said to satisfy the S † α,A,X condition, denote it by K α ∈ S † α,A,X , if there exists a constant C > 0 such that K α,l A,|x|∼s l∈Z X ≤ Cs α−n . Remark 6.3. If A(t) ≤ cB(t) for t > t 0 , some t 0 > 0, then H † α,B,X,k ⊂ H † α,A,X,k and S † α,B,X ⊂ S † α,A,X . Remark 6.4. Observe that the M α,A is the fractional maximal operator associated to the Young function A, that is The results in this case are Theorem 6.5. Let A be a Young function and 0 < α < n.
whenever the left-hand side is finite.
Theorem 6.6. Let 0 < α < n, b ∈ BMO and k ∈ N. Let T α convolution operator with kernel K α = {K α,l } l∈Z such that T α is bounded from L p 0 X (dx) to L q 0 X (dx), for some 1 < p 0 , q 0 < ∞. Let A, B Young function such that with C k (t) = exp(t 1/k ) for t ≥ 1. If K α ∈ S † α,A,X ∩ H † α,A,X ∩ H † α,B,X,k , then for any 0 < p < ∞ and any w ∈ A ∞ , there exist c > 0 such that whenever the left-hand side is finite.
Remark 6.7. The proof of this results are analogous to the ones in [2] with the same changes of the results for the vector-valued singular integral operators above. Also for the proof of this results we need the following lemma and the proof is analogous to the Lemma 5.1 and the one Theorem 3.6 in [2].
Lemma 6.8. Let A be a Young function and 0 < α < n. Let T α f = K α * f with kernel K α ∈ S † α,A,X ∩ H † α,A,X , then for all 0 < δ < ε < 1 there exists c > 0 such that for all x ∈ R n and f ∈ L ∞ c . There exist relations between the kernels which satisfies the fractional conditions S † α,A,X and H † α,A,X,k and the kernels which satisfies the conditions S † A,X and H † A,X,k . The next proposition show this relation and also a form to define kernels such that satisfies the fractional condition. The proof is analogous to Proposition 4.1 in [2]. Proposition 6.9. Let K = {K l } l∈Z and K α = {K α,l } l∈Z defined by K α (x) = |x| α K(x). If K ∈ S † A,X ∩ H † A,X,k then K α ∈ S † α,A,X ∩ H † α,A,X,k . We know that, for certain X Banach space, the kernel of the square operator satisfies the conditions S † A,X and H † A,X,k , for example X = l p and A(t) = exp where K = {K l } l∈Z is the kernel defined in the Section 4. Let b ∈ BMO and k ∈ N, the commutator is defined by By Proposition 6.9, we have that S α,X f (x) satisfies the hypothesis of Theorem 6.6. Then, Theorem 3.6 for the fractional square operator is Theorem 6.10. Let b ∈ BMO, k ∈ N∪{0} and 0 < α < n. Let A(t) = exp(t 1 1+k )−1. If K ∈ S † A,X ∩ H † A,X,k i.e. 1 m m∈(Z−{0}) X = C A,X < ∞, then, for any 0 < p < ∞ and w ∈ A ∞ , there exists C such that R n |S k α,X,b f (x)| p w(x)dx C R n M α,L log L k+1 f (x) p w(x)dx.
In [1], the authors study the weights for fractional maximal operator related to Young function in the context of variable Lebesgue spaces. They characterized the weights for the boundedness of M α,A with A(t) = t r (1 + log(t)) β , r ≥ 1 and β ≥ 0.
For any 1 ≤ p, q < ∞, we define the A p,q weight class by, w ∈ A p,q if and only if w q ∈ A 1+ q p ′ The result in the classical Lebesgue spaces, that is the variable Lebesgue spaces with constant exponent, is the following, Theorem I. [1] Let w be a weight, 0 < α < n, 1 < p < n/α, and 1/q = 1/p − α/n. Let A(t) = t r (1 + log(t)) β , with 1 ≤ r < p and β ≥ 0. Then M α,A is bounded from L p (w p ) into L q (w q ) if and only if w r ∈ A p/r,q/r .
Applying this result to Theorem 6.10 we obtain that, if w ∈ A p,q then for all 1 < p < n/α, and 1/q = 1/p − α/n, So we have the following results, Corollary 6.11. Let 0 < α < 1, 1 < p < 1/α and 1/q = 1/p − α. If w ∈ A p,q then S k α,X,b is bounded from L p (w p ) into L q (w q ).