SOME OPERATORS ACTING ON WEIGHTED ... - Project Euclid

TAIWANESE JOURNAL OF MATHEMATICS Vol. 16, No. 4, pp. 1507-1530, August 2012 This paper is available online at http://journal.taiwanmathsoc.org.tw

SOME OPERATORS ACTING ON WEIGHTED SEQUENCE BESOV SPACES AND APPLICATIONS Po-Kai Huang and Kunchuan Wang* Abstract. In this article, we study the boundedness of matrix operators acting on weighted sequence Besov spaces b˙ α,q p,w . First we obtain the necessary and sufficient condition for the boundedness of diagonal matrices acting on weighted sequence ˙ α,q Besov space b˙ α,q p,w , and investigate the duals of bp,w , where the weight is nonnegative and locally integrable. In particular, when 0 < p < 1, we find a type of new sequence sapces which characterize the dual space of b˙ α,q p,w . α,q ˙ We also use the duals of bp,w to characterize an algebra of matrix operators acting on weighted sequence Besov spaces b˙ α,q p,w and find the necessary and sufficient conditions to such a characterization. Note that we do not require that the given weight satisfies the doubling condition in this situation. Using these results, we give some applications to characterize the boundedness of Fourier-Haar multipliers and paraproduct operators. In this situation, we need to require that the weight w is an Ap weight.

1. INTRODUCTION In order to study the boundedned of some kind of linear operators, such as Haar multipliers and paraproduct operators, one can do it by norm equivalence between function spaces and their corresponding sequence spaces. Precisely, for example, if we consider a linear operator T acting on homogeneous Triebel-Lizorkin space F˙pα,q , then one can use a discrete wavelet transform identity or the ϕ-transform identity introduced by Frazier and Jawerth [5] to deduce a linear operator T to a matrix A(T ) := {T ψP , ϕQ} and to consider the boundedness of A(T ) acting on sequence TriebelLizorkin space f˙pα,q . For simplicity, we only work with the ϕ-transform indetity, but let us emphasize that Meyer’s wavelet transform indetity could be used equally well in our development. Received December 15, 2010, accepted August 29, 2011. Communicated by Chin-Cheng Lin. 2010 Mathematics Subject Classification: 42B20, 42B35, 42C40. Key words and phrases: Almost diagonal matrix, Diagonal matrix, Double exponent, Duality, Matrix operator, Sequence space, Weight. Supported by NSC of Taiwan under Grant No. NSC 99-2115-M-259-001. *Corresponding author.

1507

1508

Po-Kai Huang and Kunchuan Wang

Let us start with recalling some definitions and properties. For some ν ∈ Z and k = (k1 , k2, · · · , kn) ∈ Zn , let Q = Qνk = {(x1 , · · · , xn ) ∈ Rn : 2−ν ki ≤ xi < 2−ν (ki + 1), i = 1, 2, · · · , n}, and xQ = 2−ν k the “lower left corner” of Q = Qνk . Also Q denotes the collection of all dyadic cubes in Rn and Qν denotes the subcollection of Q with side length 2 −ν for ν ∈ Z. For P ∈ Q, QP denotes the subcollection of Rn such that each cube in Q P is a subset of P . We choose a function ϕ ∈ S satisfying supp(ϕ) ⊆ {ξ : 1/2 ≤ |ξ| ≤ 2}; (1) |ϕ(ξ)| ≥ c > 0 if 3/5 ≤ |ξ| ≤ 5/3. Then there exists a function ψ ∈ S satisfying the same conditions as (1) such that −ν ξ) = 1 ϕ(2 −ν ξ)ψ(2 for ξ = 0. ν∈Z

Hence the ϕ-transform identity [5] is given by (2) f, ϕQψQ, f= Q∈Q

where gQ (x) := |Q|−1/2g((x − xQ )/(Q)) = 2νn/2 g(2ν x − k) if Q = Qνk for some ν ∈ Z and k ∈ Zn . Here |Q| is the usual Lebesgue measure of Q in Rn . Let P denote the class of all polynomials on Rn and S /P denote the tempered distributions on R n modulo polynomials. For ν ∈ Z, let ϕν (x) = 2νn ϕ(2ν x). For α ∈ R, 0 < p, q ≤ +∞ and f ∈ S /P, define the homogeneous Triebel-Lizorkin spaces F˙ pα,q via the norms  να q 1/q   q2 and d ∈ b˙

q1 q2 1 −q2

α2−α1 +γ, q p1 p2 ,w p1 −p2

;

1 +γ,∞ ; (b) p1 > p2 , q1 ≤ q2 and d ∈ b˙ αp21−α p2 ,w p1 −p2

(c) p1 ≤ p2 , q1 > q2 and d ∈ (d) p1 ≤ p2 , q1 ≤ q2 and d ∈

q q

α2−α1 +γ, q 1−q2 1 2 b˙ ∞,w ; 2 −α1 +γ,∞ ˙bα∞,w .

1 Proof. Without loss of generality, we may assume α1 = α2 = 0. Let s ∈ b˙ 0,q p1 ,w and suppose 0 < p1 , p2 , q1 , q2 ≤ ∞. For part (a), let δ = p1 /p2 and ρ = q1 /q2 . Applying Ho¨ lder’s inequality twice, then we have p2 p2 q2 1 p2 q2 (γ) − 12 − γn − 21 |dQ| w(Q) |Q| |Q| |sQ | Td s b˙ 0,q2 = p2 ,w

ν∈Z

≤

p2 δ q2 p2 δ − 12 − γn |dQ| w(Q) |Q|

Q∈Qν (w)

ν∈Z

×

Q∈Qν (w)

|Q|

− 21

|sQ|

p1

q2 q1 · 1 w(Q)

p1

q2 q1

Q∈Qν (w)

≤

ν∈Z

Q∈Qν (w)

×

p2 δ q2 ρ 1 p2 δ q2 ρ γ 1 w(Q) |Q|− 2 − n |dQ|

|Q|

− 21

|sQ|

p1

q1 1 w(Q)

p1

q1

Q∈Qν (w)

ν∈Z

= d ˙ γ,q2 ρ · s b˙ 0,q1 , bp

p1 ,w

2 δ ,w

p2 q2 and q2 ρ = q2 (q1 /q2 ) = qq11−q . where p2 δ = p2 (p1 /p2 ) = pp11−p 2 2 For part (b), let δ = p1 /p2 . Since q1 ≤ q2 , q1 /q2 ≤ 1. Applying Ho¨ lder’s inequality and triangle inequality, we obtain p2 p2 q2 1 p2 q2 (γ) − 12 − γn − 21 |dQ| w(Q) |Q| |Q| |sQ | Td s b˙ 0,q2 = p2 ,w

ν∈Z

≤

ν∈Z

Q∈Qν (w)

Q∈Qν (w)

p2 δ q2 p2 δ − 12 − γn |dQ| w(Q) |Q|

1514


×

|Q|

Q∈Qν (w)

ν∈Z

Q∈Qν (w)

×

≤ d b˙ γ,∞

p2 δ ,w

γ,∞ b˙ p δ ,w 2

·

q2 1 w(Q)

|Q|

− 21

|sQ |

Q∈Qν (w)

ν∈Z

= d

|sQ|

p1

p1

q2

p2 δ 1 p2 δ − 12 − γn |dQ| w(Q) |Q|

≤ sup

− 21

ν∈Z

Q∈Qν (w)

p1

q2 1 w(Q)

p1

q2

p1 q1 1 p1 q1 − 12 w(Q) |Q| |sQ |

· s b˙ 0,q1 . p1 ,w

For part (c), let ρ = q1 /q2 . Since p1 ≤ p2 , p1 /p2 ≤ 1. Applying triangle inequality and then Ho¨ lder’s inequality, we obtain p2 p2 q2 1 p2 q2 γ (γ) − 12 − n − 12 |dQ| w(Q) |Q| |Q| |sQ | Td s b˙ 0,q2 = p2 ,w

ν∈Z

≤ ≤

Q∈Qν (w)

ν∈Z

Q∈Qν (w)

ν∈Z

Q∈Qν (w)

×

sup

p1 q2 1 p1 q2 γ − 12 − 12 − n |dQ||sQ| w(Q) |Q| |Q| p1 q2 p1 γ − 12 − n |dQ| |Q|

|Q|

− 21

|sQ|

p1

q2 q1 · 1 w(Q)

Q∈Qν (w)

≤

ν∈Z

sup

Q∈Qν (w)

× ν∈Z

|Q|

γ − 21 − n

|Q|

|dQ|

− 21

p1

q2 ρ

|sQ |

q2 q1

1 q2 ρ

p1

q1 1 w(Q)

p1

q1

Q∈Qν (w)

= d ˙ γ,q2 ρ · s b˙ 0,q1 . b∞

p1 ,w

For part (d), since p1 ≤ p2 and q1 ≤ q2 ,we have p1 /p2 ≤ 1 and q1 /q2 ≤ 1. Applying triangle inequality twice, we obtain the result. p2 p2 q2 1 p2 q2 γ (γ) − 12 − n − 12 |dQ| w(Q) |Q| |Q| |sQ | Td s b˙ 0,q2 = p2 ,w

ν∈Z

Q∈Qν (w)

Some Operators Acting on Weighted Sequence Besov Spaces and Applications

≤ ≤

ν∈Z

Q∈Qν (w)

sup

|Q|

− 21

|Q|

− 21 − γn

Q∈Qν (w)

ν∈Z

×

|Q|

− 21

|Q|

|sQ |

− 12 − γn

|dQ|

|dQ||sQ|

1515

q2 1

p1 w(Q)

p1

q2

p1 q2

p1

p1

q2 1 w(Q)

p1

q2

Q

(Q)=2−ν

≤

sup |Q|

− 12 − γn

|dQ| ·

ν∈Z

Q∈Qν (w)

Q∈Q(w)

p1 q1 1 p1 q1 − 12 w(Q) |Q| |sQ |

= d b˙ γ,∞ · s b˙ 0,q1 . ∞,w p1 ,w

For the case pi = ∞ or qi = ∞, with modification, we could obtain the results. This completes the proof of Theorem 2.2. At the end of this section, let us consider the duals of weighted sequence Besov spaces. Proposition 2.3. [10, Theorem 1.3]). Let α ∈ R, 1 < p ≤ ∞, 0 < q ≤ ∞, and ˙ −α,q w be a weight. Then the dual of b˙ α,q p,w is bp ,w in the following sense. −α,q α,q (i) For t = {tQ }Q∈Q(w) ∈ b˙ p,w , the linear functional L t on b˙ p,w , given by Lt (s) = for s = {sQ }Q∈Q(w) ∈ b˙ α,q s, tw = Q∈Q(w) sQ tQ w(Q) p,w , is continuous with |Q| Lt ≤ C t b˙ −α,q . p ,w

(ii) Conversely, every continuous linear functional L on b˙ α,q p,w satisfies L = L t for −α,q ˙ some t = {tQ }Q∈Q(w) ∈ bp,w with t b˙ −α,q ≤ C L provided that w is a p ,w

“double measure”, i.e. w(2B) ≤ Cw(B) for every ball B in R n . In order to find the dual space of b˙ α,q p,w for 0 < p ≤ 1 and 0 < q ≤ ∞, we need to α,q define a sequence space c˙ p,w given in [10]. Definition 2.4. For α ∈ R, 0 < p ≤ 1, 0 < q ≤ ∞, and a weight w, we say that is finite, where t c˙α,q is defined by t = {tQ }Q ∈ c˙α,q p,w if t c˙α,q p,w p,w q 1 q α 1 1− p1 − − = . sup |Q| n 2 |tQ |w(Q) t c˙α,q p,w ν∈Z

Q∈Qν (w)

Remark 2.5. ˙ α,q (i) If p = 1, then c˙α,q 1,w = b∞,w .

1516


˙ (ii) If w ≡ 1, then c˙α,q p,1 = b∞

α−n+n/p,q

by a direct calculation.

Here is a characterization of the dual of b˙ α,q p,w for α ∈ R, 0 < p ≤ 1 and 0 < q ≤ ∞. Proposition 2.6. [10, Theorem 1.6]). Let α ∈ R, 0 < p ≤ 1, 0 < q ≤ ∞, and w −α,q in the following sense. be a weight. Then the dual of b˙ α,q p,w is c˙ p,w ˙ α,q (i) For t = {tQ }Q∈Q(w) ∈ c˙−α,q p,w , the linear functional L t on bp,w , given by Lt (s) = s, tw for s = {sQ }Q∈Q(w) ∈ b˙ α,q p,w , is continuous with L t ≤ C t c˙−α,q . p,w

b˙ α,q p,w

(ii) Conversely, every continuous linear functional L on with t c˙−α,q ≤ C L . some t = {tQ }Q∈Q(w) ∈ c˙−α,q p,w

satisfies L = L t for

p,w

Remark 2.7. Observe that s, tw =

sQ tQ

Q∈Q

where hQ = tQ

w(Q) |Q|

w(Q) = sQ hQ = s, h, |Q| Q∈Q

and h = {hQ }Q.

The characterization presented in Proposition 2.3 says that the dual of b˙ α,q p,w with −α,q respect to a weighted pairing can be identified with b˙ p,w for any doubling weight

w. Let us denote the dual with respect to the weighted pairing by b˙ α,q p,w . The

difference arises because the pairing used as above observation. In Roudenko’s case she has two sequences s = {sQ }Q∈Q, and h = {hQ }Q∈Q, indexed on the dyadic cubes and the pairing is: s, h = Q sQ hQ , whereas in this article the pairing is w(Q) s, tw = Q∈Q(w) sQ tQ |Q| . When dealing with any doubling weight, it may occur that w(Q) = 0 for some Q, which would imply w = 0 a.e. on Q; so two sequences will be equal in such space if and only if they coincide off those cubes (we are working with equivalence classes). In the case of the weighted pairing since there is no need to invoque the reciprocal of the weight (or a power of the weight), the above idetification work well, unlike when using the usual pairing. In the case when both w > 0 a.e, and w−1 > 0 a.e, it would be interesting to |Q| explicitly state that the map that takes a sequence h Q into the sequence tQ = hQ w(Q) is a one-to-one and continuous mapping from b˙ −α,q into b˙ −α,q for all weight. To p,w1−p

p ,w

see this, t b˙ −α,q = p ,w

ν∈Z

Q∈Qν (w)

|Q|

α/n−1/2

|Q| |hQ | w(Q)

p

q /p 1/q w(Q)


≤

ν∈Z

Q∈Qν (w)

= h b˙ −α,q

p ,w1−p

since

|Q| =

p

|Q|

α/n−1/2

|hQ |

w

1−p

1517

q /p 1/q (Q)

,

w 1/p(x)w −1/p(x)dx ≤ w(Q)

1/p

w 1−p (Q)]1/p ,

Q

by Ho¨ lder’s inequality. However the reverse embedding holds only if w ∈ Ap because 1/p 1−p (Q)]1/p ≤ C|Q|, where C is dependent only on the Ap w |Q| ≤ w(Q) constant. Effectively the duals with respective to the different pairings are different spaces when the weight w is not in A p , that explains the discrepancy. By Remark 2.5 and Proposition 2.6, we have a characterization of the dual space of unweighted sequence Besov space b˙ α,q p for α ∈ R, 0 < p ≤ 1 and 0 < q ≤ +∞. Corollary 2.8. Let α ∈ R, 0 < p ≤ 1 and 0 < q ≤ ∞. Then

α,q α,q −α−n+ np ,q = b˙ ≈ c˙−α,q = b˙ ∞ . b˙ p

p,1

p,1

3. ALMOST DIAGONAL MATRICES At the beginning of this section, let us recall a definition about “doubling condition”. Definition 3.1. A weight w is called a doubling measure, if there exists a constant C = Cn such that for any δ > 0 and any z ∈ Rn ,

(8) w(t)dt ≤ C w(t)dt, B2δ (z)

Bδ (z)

where Bδ (z) is an open ball in R n centered at z with radius δ. If C = 2β is the smallest constant for the inequality (8) holds, then β is called the doubling exponent of w. In this section, we always assume that w is a weight which is a doubling measure with doubling exponent β. For such a weight, we study the almost diagonality given by Roudenko [21] with matrix-weight for p ≥ 1 and by Bownik [1] for scalar case. Here we adopt Bownik’s definition, but we emphasize that, for p ≥ 1, both definitions are equivalent. Definition 3.2. Let w be a doubling measure with doubling exponent β. For α ∈ R, 0 < p, q ≤ ∞, let J = βp + max 0, n − np , we say that a matrix A = {aQP }Q,P is (α, p, w) almost diagonal, denoted by A ∈ ad αp(β), if there exist an ε > 0 and C > 0 such that for all dyadic cubes Q, P ,

1518


(Q) |aQP | ≤ C (P )

α

n+ε n+ε −J−ε |xQ − xP | (Q) 2 (P ) 2 +J−n min , . 1+ (P ) (Q) max((Q), (P ))

Remark 3.3. Note that if p ≥ 1 then J = n + (β − n)/p and if 0 < p < 1 then J = β/p. Also note that if the weight w ≡ 1, then β = n. Thus the definition of almost diagonality in Definition 3.2 is the same as the one given by M. Frazier and B. Jawerth in [5] under q ≥ 1 and w ≡ 1. Also note that in general the exponent J is independent of q for Besov case, while in case of the Triebel-Lizorkin spaces J = n/ min(1, p, q) in unweighted cases. Basically, the proof was showed by Roudenko for p ≥ 1 in [21] and showed by Bownik in more general setting in [2]. Proposition 3.4. ([2, 10, 21]). Let α ∈ R, 0 < p, q ≤ ∞, and w a doubling measure with exponent β. If A ∈ ad αp (β), then A is bounded on b˙ α,q p,w . Now we may state that the class of almost diagonal matrices is closed under composition. The class of all operators on the distribution space level, which corresponds to almost diagonal matrices, is then also an algebra under composition. For γ > 0, β δ > 0, J = p + max 0, n − np and P, Q dyadic, denote n+γ n+γ (Q) α (Q) 2 (P ) 2 +J−n min , wQP (δ, γ) : = (P ) (P ) (Q) −J−δ |xQ − xP | 1+ max((Q), (P ))

and WQP (δ, γ1, γ2 ) :=

wQR (δ, γ1 )wRP (δ, γ2).

R

Theorem 3.5. Suppose α ∈ R and 0 < p, q ≤ ∞. If A, B ∈ ad αp(β), then A ◦ B ∈ adαp (β). Consequently, ad αp (β) is an algebra. Before proving the Theorem 3.5, we need the following lemma, which is a modification of Theorem D.2 in [5]. Lemma 3.6. [5, Theorem D.2]). Suppose δ, γ 1 , γ2 > 0, γ1 = γ2 , and 2δ < γ1 +γ2 . Then there exists a constant C such that WQP (δ, γ1, γ2 ) ≤ CwQP (δ, min(γ1 , γ2 )). Proof. [Proof of Theorem 3.5.] By the proof for [5, Theorem 9.1], we have the desired result immediately by Lemma 3.6.


1519

4. AN ALGEBRA OF MATRIX OPERATORS ON WEIGHTED SEQUENCE BESOV SPACES In this section, we will treat more general matrices on special weighted sequence Besov space. Let b denote the class of matrices A such that |A| is bounded on f˙p0,q for all 1 ≤ p, q ≤ ∞, where |A| = {|aQP |}Q,P if A = {aQP }Q,P . Frazier and Jawerth [5] characterized the following result. Proposition 4.1. [5, Corollary 10.2]). A matrix {aQP }Q,P belongs to b if and only if {|a QP |}Q,P satisfies all conditions in the following: |aQP |(|Q|/|P |)1/2 < ∞; sup P ∈Q Q∈Q

|aQP |(|Q|/|P |)−1/2 < ∞; 1 1/2 |aQP ||P | sup ˙0,∞ < ∞; P0 ∈Q |P0 | f1 P ∈QP0 1 |aQP ||Q|1/2 sup ˙0,∞ < ∞. Q0 ∈Q |Q0 | f sup

Q∈Q P ∈Q

1

Q∈QQ0

The main purpose of this section is to characterize an algebra of bounded matrix operators acting on weighted sequence Besov spaces b˙ α,q p,w for all 1 ≤ p, q ≤ ∞, where α is fixed in R. Let us observe some special cases. Theorem 4.2. Suppose α ∈ R, 0 < p = q ≤ 1, w is a weight, and A = {a QP } is a matrix. Then A is bounded on b˙ α,q q,w if and only if α 1 1 q w(Q) q |Q| − n − 2 (9) |aQP | < ∞. sup |P | w(P ) P ∈Q(w) Q∈Q(w)

Proof. First let us suppose that A is bounded on b˙ α,q q,w . Fix a dyadic cube P ∈ Q(w) P and define a sequence s by α 1 −1 |P | n + 2 w(P ) q if Q = P P . (s )Q := 0 if Q = P α

1

= 1. Since (AsP )Q = aQP |P | n + 2 w(P ) Then sP b˙ α,q q,w

− 1q

for Q ∈ Q(w), we have

α 1 1 q w(Q) q |Q| − n − 2 |aQP | |P | w(P ) Q∈Q(w) 1 q q α 1 − 1q −α − 12 +2 n n = |aQP ||P | w(P ) w(Q) |Q| Q∈Q(w)

= AsP b˙ α,q ≤ A sP b˙ α,q = A . q,w q,w

1520


Thus, after taking supremum over all dyadic cubes P in Q(w), we have condition (9). Conversely, suppose that condition (9) holds. Since s ∈ b˙ α,q q,w , we have q 1 q 1 −α − n 2 |s | = w(P ) |P | s b˙ α,q P q,w P ∈Q(w)

and so As q˙ α,q bq,w

=

! !q ! ! −α − 12 ! ! w(Q) n a s |Q| QP P ! ! Q∈Q(w)

≤

P ∈Q(w)

α 1 q q |Q| − n − 2 w(Q) −α − 12 n · |P | |aQP | |sP | w(P ) |P | w(P )

P ∈Q(w) Q∈Q(w)

α 1 q w(Q) |Q| − n − 2 · s q˙ α,q , |aQP | ≤ sup bq,w |P | w(P ) P ∈Q(w) Q∈Q(w)

where we apply the triangle inequlaity in the first inequality for index q. Hence A is α,q bounded on b˙ q,w . By a duality argument, we have the following result. Corollary 4.3. Suppose α ∈ R, w is a weight, and A = {a QP } is a matrix. Then α,∞ A is bounded on b˙ ∞,w if and only if α 1 |P | n − 2 w(P ) (10) < ∞. |aQP | sup |Q| w(Q) Q P

∗ Proof. Note that a matrix A is bounded on b˙ α,∞ ∞,w if and only if its adjoint A is −α,1 α,∞ bounded on b˙ 1,w where a∗QP = aP Q . Thus, by Theorem 4.2, A is bounded on b˙ ∞,w if and only if condition (10) holds.

Theorem 4.4. Suppose α ∈ R, w is a weight and A = {a QP } is a matrix operator with aQP ≥ 0 for all dyadic cubes P and Q. Then A is bounded on b˙ α,∞ 1,w if and only if α 1 |Q| − n − 2 w(Q) (11) < ∞. sup aQP sup |P | w(P ) ν∈Z P ∈Qµ (w) µ∈Z

Proof.

Q∈Qν (w)

Suppose A is bounded on b˙ α,∞ 1,w . For each pair of µ and ν in Z, let α 1 |Q| − n − 2 w(Q) . aQP Kµ,ν := sup |P | w(P ) P ∈Qµ (w) Q∈Qν (w)


1521

Claim that Kµ,ν < ∞ for every pair of µ and ν. Suppose that there exist µ0 , ν0 ∈ Z such that K µ0 ,ν0 = ∞. For j ∈ N, there exists a dyadic cube Pj such that (Pj ) = 2−µ0 and α 1 |Q| − n − 2 w(Q) ≥ j. aQPj |Pj | w(Pj ) Q∈Qν0 (w)

For j ∈ N, let sj be a sequence defiend by, for P ∈ Q(w), α 1 |Pj | n + 2 w(Pj )−1 if P = Pj j . (s )P := 0 if P = Pj Then sj b˙ α,∞ = 1 and for Q ∈ Q(w) 1,w α 1 aQP (sj )P = aQPj |Pj | n + 2 w(Pj )−1 . (Asj )Q = P ∈Q(w)

Thus, Q∈Qν0 (w)

|Q| |Pj |

− α − 1 n

2

aQPj

w(Q) = w(Pj )

α

1

|Q|− n − 2 |(Asj )|w(Q)

Q∈Qν0 (w)

≤ Asj b˙ α,∞ ≤ C sj b˙ α,∞ = C, 1,w

1,w

where we apply the boundedness of the matrix A on b˙ α,∞ 1,w . This contradiction yields Kµ,ν < ∞ for each µ, ν ∈ Z. Fix ν ∈ Z and, for each µ ∈ Z, choose a dyadic cube Pµ satisfying (Pµ ) = 2−µ and α 1 |Q| − n − 2 1 w(Q) ≥ Kµ,ν . aQPµ |Pµ | w(Pµ ) 2 Q∈Qν (w)

Let sν be a sequence defiend by (s )P := ν

Then sν b˙ α,∞ = 1 and 1,w (Asν )Q =

α

1

|Pµ | n + 2 w(Pµ )−1 if P = Pµ 0

aQP (sν )P =

µ∈Z P ∈Qµ (w)

Since A is bounded on b˙ α,∞ 1,w , we have

if P = Pµ µ∈Z

α

1

.

aQPµ |Pµ | n + 2 w(Pµ )−1 .

1522


sup

µ∈Z P ∈Qµ (w) Q∈Qν (w)

|Q| |P |

− α − 1 n

2

aQP

w(Q) w(P )

α 1 w(Q) |Q| − n − 2 aQPµ ≤2 |Pµ | w(Pµ) µ∈Z Q∈Qν (w) α 1 −α − 12 +2 −1 n n |Q| aQPµ |Pµ | w(Pµ ) w(Q) =2

Q∈Qν (w) 2 Asν b˙ α,∞ 1,w

≤

µ∈Z

≤ C s b˙ α,∞ = C. ν

1,w

Thus, after taking the supremum over ν ∈ Z, we have condition (11). Conversely, suppose that condition (11) holds. Since s ∈ b˙ α,∞ 1,w , we have α 1 |P |− n − 2 |sP |w(P ) s b˙ α,∞ = sup 1,w

µ∈Z

P ∈Qµ (w)

and so As b˙ α,∞ ≤ sup 1,w

ν∈Z µ∈Z

|P |

−α − 12 n

|sP |w(P )

P ∈Qµ (w)

Q∈Qν (w)

− α − 1

|Q| sup ≤ sup |P | ν∈Z µ∈Z P ∈Qµ (w) Q∈Qν (w) 1 −α − |P | n 2 |sP |w(P ) × P ∈Qµ (w)

≤ sup ν∈Z

sup

µ∈Z P ∈Qµ (w) Q∈Qν (w)

|Q| |P |

n

2

− α − 1 n

2

|Q| |P |

− α − 1

w(Q) aQP w(P )

n

2

aQP

w(Q) w(P )

w(Q) · s b˙ α,∞ . aQP 1,w w(P )

Hence A is bounded on b˙ α,∞ 1,w . Corollary 4.5. Suppose α ∈ R, w is a weight and A = {a QP } is a matrix operator with aQP ≥ 0 for all dyadic cubes P and Q. Then A is bounded on b˙ α,1 ∞,w if and only if α 1 |P | n − 2 w(P ) (12) < ∞. sup aQP sup |Q| w(Q) µ∈Z Q∈Qν (w) ν∈Z

Proof.

P ∈Qµ (w)

By a duality argument, the result follows immediately.

Definition 4.6. Let w be a weight and α ∈ R. We say that a matrix operator A = {aQP } is an element of an algebra of bounded matrix operator, denoted by A ∈ amoα (w), if |A| is bounded on b˙ α,q p,w for all 1 ≤ p, q ≤ ∞.


1523

Here is a characterization of amoα (w). Theorem 4.7. Let α ∈ R, w be a weight, and A = {aQP } be a matrix operator. Then A ∈ amoα (w) if and only if A satisfies (9–10), α 1 |Q| − n − 2 w(Q) (13) < ∞, sup |aQP | sup |P | w(P ) ν∈Z P ∈Qµ (w) Q∈Qν (w)

µ∈Z

and (14)

sup

sup

µ∈Z ν∈Z Q∈Qν (w)

P ∈Qµ (w)

|P | |Q|

α−1 n

2

|aQP |

w(P ) < ∞. w(Q)

Proof. By Definition 4.6, the “if” part follows immediately by Theorem 4.2 with p = q = 1, Corollary 4.3, Theorem 4.4 and Corollary 4.5. Conversely, (a) by Theorem 4.2 with p = q = 1 and condition (9), A is bounded on b˙ α,1 1,w ; α,∞ (b) by Corollary 4.3 and condition (10), A is bounded on b˙ ∞,w ; (c) by Theorem 4.4 and condition (13), A is bounded on b˙ α,∞; 1,w

α,1 (d) by Corollary 4.5 and condition (14), A is bounded on b˙ ∞,w .

Hence it follows from interpolation theorem that A is bounded on b˙ α,q p,w for all 1 ≤ α p, q ≤ ∞, i.e., A ∈ amo (w). Remark 4.8. (a) It is routine to check that amoα(w) is an algebra with composition. (b) Because β/p + n/p ≤ β for p ≥ 1, it follows from Theorem 3.4 that we have adα1 (β) ⊆ amoα(w). (c) If the weight w ≡ 1, then β = n. So we have the following result: if a matrix A is almost diagonal then (i) the estimate w QP is independent of p for p ≥ 1, α (ii) the matrix A is bounded on b˙ α,q p for 1 ≤ p, q ≤ ∞ and (iii) A ∈ amo (1). 5. APPLICATIONS Consider that an operator T is linear from the Schwartz space S to its dual S and has a kernel K : Rn × Rn → C which gives the action of T away from the diagonal. The kernel K is a function which is locally integrable on R n × Rn \ {(x, y) : x = y} and there exist a constant C > 0 and a regularity exponent ε ∈ (0, 1] such that (15)

|K(x, y)| ≤ C|x − y|−n ε

for x = y;

(16)

|K(x, y) − K(x , y)| ≤ C

|x − x | |x − y|n+ε

for |x − x | ≤

|x − y| ; 2

(17)

|K(x, y) − K(x, y )| ≤ C

|y − y |ε |x − y|n+ε

for |y − y | ≤

|x − y| . 2

1524


That K gives the action of T away from the diagonal means that is for any two functions f and g in S and which have disjoint support, we have that

K(x, y)f (y)g(x)dxdy, T f (g) = T f, g = R2n

then T is called a singular integral operator, denoted by T ∈ SIO(ε). Next, we recall the definition of Ap weight in questions. Definition 5.1. (Ap Weights). For every cube Q in Rn , and a non-negative and locally integrable function w on Rn . We say that w ∈ Ap if w Ap is finte, where w Ap is defined by

 1 −1   sup ess sup w (y) w(t)dt if 0 < p ≤ 1   Q y∈Q |Q| Q , w Ap := p−1

 1 1  1−p  w(x)dx w(x) dx if 1 < p < ∞ sup |Q| Q |Q| Q Q " where 1p + p1 = 1. Also let A∞ = 0 q2 , the Fourier-Haar multiplier T t is bounded from B˙ p,w 2 ,q2 ρ if and only if t ∈ b˙ α∞,w , where ρ is the index conjugate of ρ = q 1 /q2 . α1 ,q1 into (b) For q1 ≤ q2 , the Fourier-Haar multiplier T t is bounded from B˙ p,w α +α ,q α ,∞ 1 2 2 2 ˙ ˙ if and only if t ∈ b∞,w . Bp,w

α1 ,q1 2 ,q2 ρ and f ∈ B˙ p,w where Proof. For part (a), suppose t = {tQ }Q ∈ b˙ α∞,w α1 ,q1 q1 q1 ˙ ρ = q2 and ρ = q1 −q2 . Then f, ϕQ Q ∈ bp,w , by Proposition 1.5. Thus, by (6), Propositions 1.5 and 2.1(a), we have 1 Ttf B˙ α1 +α2 ,q2 ≈ |Q|− 2 tQ f, ϕQ α1 +α2 ,q2 p,w Q b˙ p,w ≤ C t ˙ α2 ,q2 ρ · f, ϕQ ˙ α1 ,q1 b ∞,w

≤ C t Conversely, suppose that Tt is Proposition 1.5, we obtain 1 |Q|− 2 tQ f, ϕQ

Q bp,w

α2 ,q2 ρ b˙ ∞,w

· f

α1 ,q1 B˙ p,w

.

α1 ,q1 α1 +α2 ,q2 bounded from B˙ p,w into B˙ p,w . Then, by

α1 +α2 ,q2 ˙ α1 +α2 ,q2 ≈ Ttf B˙ p,w

Q bp,w

α1 ,q1 ≤ C f B˙ p,w ≤ C f, ϕQ α1 ,q1 . ˙

Q bp,w

1526


2 ,q2 ρ Thus, by Proposition2.1 (a), we get t ∈ b˙ α∞,w . α2,∞ α1 ,q1 and f ∈ B˙ p,w . Then for part (b), suppose t = {tQ }Q ∈ b˙ ∞,w Similarly, α1 ,q1 α1 +α2 ,q2 1 ,q1 f, ϕQ Q ∈ b˙ αp,w , and so Tt is bounded from B˙ p,w into B˙ p,w , by Propositions 1.5 and 2.1 (b). α1 ,q1 α1 +α2 ,q2 into B˙ p,w . Then Conversely, suppose that Tt is bounded from B˙ p,w 1 |Q|− 2 tQ f, ϕQ α1 +α2 ,q2 ≤ C f, ϕQ α1 ,q1 .

Q b˙ p,w

Q b˙ p,w

α2,∞ . Thus, by Proposition 2.1(b), we obtain t ∈ b˙ ∞,w

To prove the boundedness of paraproduct operators, we need the following lemma. Lemma 5.6. Let Φ be the function given in Definition 1.2. Define Φ Q (x) =

x−x |Q| Φ (Q)Q . Suppose G = {g QP }Q,P where gQP = ψP , ΦQ for all dyadic cubes P and Q. For α < 0, 0 < p, q ≤ ∞, G ∈ ad α(β), hence is bounded on b˙ α,q p,w . − 21

p

Proof. For (P ) ≤ (Q), since B.1], we have |ψP , ΦQ| ≤ C

(Q) α (P )

xγ ψP (x)dx = 0 for all γ, by [5, p. 150, Lemma

|xQ −xP | 1+ (Q)

−J−ε

+J−n (P ) n+ε 2 , (Q)

α ∈ R and ε > 0,

where C depends on J only. For (Q) < (P ), by [5, p. 152, Lemma B.2], we obtain |xQ − xP | −J−ε (Q) n2 |ψP , ΦQ| ≤ C 1 + (P ) (P ) (Q) α |xQ − xP | −J−ε (Q) n−2α 2 =C . 1+ (P ) (P ) (P )

So choosing ε = −2α, we obtain the result. Here is an application to paraproduct operators. Theorem 5.7. For α < 0, β ∈ R and 0 < p, q ≤ ∞, let w be an Ap -weight and Πg be the paraproduct operator defined in Definition 1.2. β,qr/(q−r) α,q α+β,r into B˙ r,w . (i) If 0 < r < p and g ∈ B˙ pr/(p−r),w , then Πg is bounded from B˙ p,w β,qr/(q−r)

(ii) If 0 < p ≤ r and S ϕ (g) = {g, ϕQ }Q∈Q ∈ c˙p/r,w α,q α+β,r B˙ p,w into B˙ r,w .

, then Πg is bounded from

1527


Proof.

α,q . By equation (4) and Proposition 1.5, we have Let f ∈ B˙ p,w r ψP , ΦQ f, ϕP Πg f r˙ α+β,r ≈ g, ϕQ |Q|−1/2 α+β,r Br,w

(18)

Q

P r −α/n−1/2+1/(2r) = |(Gs)Q | |Q|

b˙ r,w

Q∈Q

r w(Q) , |Q|−β/n−1/2+1/(2r)|g, ϕQ | |Q| where s = {f, ϕP }P ∈Q = Sϕ (f ). For case (i), by Proposition 2.3, the last inequality is dominated by a multiple of r |Q|−α/n−1/2+1/(2r)(Gs)Q 0,q/r Q∈Q b˙ p/r,w r × |Q|−β/n−1/2+1/(2r)g, ϕQ 0,(q/r) , Q∈Q b˙ (p/r) ,w

provided

r |Q|−β/n−1/2+1/(2r)g, ϕQ

0,(q/r)

Q∈Q

∈ b˙ (p/r),w . A calculation shows that

r −α/n−1/2+1/(2r) (Gs) |Q| Q = =

|Q|

Q∈Q bp/r,w

−α/n−1/2

ν∈Z Q∈Qν r Gs ˙ α,q ≤ C s r˙ α,q bp,w bp,w

by Proposition 1.5 and Lemma 5.6. Also r |Q|−β/n−1/2+1/(2r)g, ϕQ

˙ 0,q/r

|(Gs)Q |

q/p r/p

p w(Q)

≤ C f rB˙ α,q , p,w

˙ 0,(q/r)

Q∈Q b(p/r) ,w

qr/(q−r) (q−r)/r pr/(p−r) pr/(p−r) −β/n−1/2 = |g, ϕQ | w(Q) |Q| ν∈Z

Q∈Qν

r = g, ϕQ b˙ β,qr/(q−r) , pr/(p−r),w

r which is equivalent to g B˙ β,qr/(q−r) by Proposition 1.5. pr/(p−r),w

For case (ii), apply Proposition 2.6 to (18) to yield that r r −α/n−1/2+1/(2r) (Gs)Q Πg f B˙ α+β,r ≤ C |Q| 0,q/r r,w Q∈Q b˙ p/r,w r × |Q|−β/n−1/2+1/(2r)g, ϕQ 0,(q/r) . Q∈Q c˙p/r,w

1528


It is clear that r |Q|−β/n−1/2+1/(2r)g, ϕQ =

0,(q/r)

Q∈Q c˙p/r,w

|Q|

−β/n−1/2

ν∈Z

= g, ϕQ

Q∈Q

|g, ϕQ |w(Q)

β,qr/(q−r)

c˙p0 ,w

qr/(q−r) (q−r)/(qr)

1/r−1/p

,

where p0 satisfies 1 − 1/p 0 = 1/r − 1/p; that is p 0 = pr/(pr + r − p). Therefore Πg α,q α+β,r is bounded from B˙ p,w into B˙ r,w and the proof is finished. Remark 5.8. In 1989, M. Meyer [16] proved that a singular integral operator T is 0∞ , ΠT 1 is bounded on B˙ 10,1 , and T bounded on B˙ 10,1 if and only if T ∗ 1 = 0, T 1 ∈ B˙ ∞ satisfies the weak boundedness property. In 1995, Youssfi [27] showed that for β ∈ R, β,∞ 0,q β,p 1 < p < ∞, 1 ≤ q ≤ 2, and g ∈ B˙ ∞ , Πg is bounded from F˙ p into B˙ p if and β,q . In Theorem 5.7, we give a sufficient condition for the boundedness only if g ∈ F˙ ∞ of paraproduct operators acting on homogeneous weighted Besov spaces. ACKNOWLEDGMENT The authors wish to express their sincere appreciation to the referee for his/her very helpful and constructive comments and suggestions. REFERENCES 1. M. Bownik, Atomic and molecular decompositions of anisotropic Besov spaces, Math. Z., 250 (2005), 539-571. 2. M. Bownik, Duality and interpolation of anisotropic Triebel-Lizorkin spaces, Math. Z., 259 (2008), 131-169. 3. G. David and J.-L. Journe´ , A boundedness criterion for generalized Caldero´ n-Zygmund operators, Ann. Math., 120 (1984), 371-397. 4. M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J., 34 (1985), 777-799. 5. M. Frazier and B. Jawerth, A discrete transform and decomposition of distribution spaces, J. Funct. Anal., 93 (1990), 34-170. 6. M. Frazier and B. Jawerth, Applications of the ϕ and wavelet transforms to the theory of function spaces, Wavelets and Their Applications, Jones and Bartlett, Boston, MA, 1992, pp. 377-417. 7. M. Frazier, B. Jawerth and G. Weiss, Littlewood-Paley Theory and the Study of Function Spaces, CBMS Regional Conference Series, Vol. 79, A.M.S., Providence, RI, 1991.


1529

8. M. Frazier and S. Roudenko, Matrix-weighted Besov Space and Conditions of Ap type for 0 < p ≤ 1, Indiana Univ. Math. J., 53 (2004), 1225-1254. 9. J. Garcia-Cuerva and J. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, 116, North-Holland Publishing Co., Amsterdam, 1985. 10. P.-K. Huang and K. Wang, Duals of homogeneous weighted sequence Besov spaces and applications, J. Appl. Anal., 17 (2011), 181-205. 11. R. Hunt, B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc., 176 (1973), 227-251. 12. N. H. Katz and M. C. Pereyra, Haar multipliers, paraproducts, and weighted inequalities, Analysis of divergence, (Orono, ME, 1997), Appl. Numer. Harmon. Anal., Birkhauser Boston, Boston, MA, 1999, pp. 145-170. 13. C.-C. Lin and K. Wang, Triebel-Lizorkin spaces of para-accretive type and its T b theorem, J. Geom. Anal., 19 (2009) 667-694. 14. C.-C. Lin and K. Wang, Singular integral operators on Triebel-Lizorkin spaces of paraaccretive type, J. Math. Anal. Appl., 364 (2010), 453-462. 15. C.-C. Lin and K. Wang, The T 1 theorem for Besov spaces, preprint. 16. M. Meyer, Une classe d’espaces fonctionnels de type BM O. Application aux intg´ rales singulie` res, Ark. Mat., 27 (1989), 305-318. 17. Y. Meyer, Wavelets and Operators, Cambridge Studies in Advanced Mathematics 37, 1993. 18. F. Nazarov and S. Treil, The hunt for a Bellman function: applications to estimates for singular integral operators and to other classical problems of harmonic analysis, Algebra i Analiz, 8 (1996), 32-162; English transl., St. Petersburg Math. J., 8 (1997), 721-824. 19. F. Nazarov, S. Treil and A. Volberg, The Bellman functions and two-weight inequalities for Haar multipliers, J. Amer. Math. Soc., 12(4) (1999), 909-928. 20. F. Nazarov, S. Treil and A. Volberg, Two weight inequalitles for individual Haar multipliers and other well localized operators, Math. Res. Lett., 15 (2008), 583-597. 21. S. Roudenko, Matrix-weighted Besov spaces, Trans. Amer. Math. Soc., 355 (2002), 273-314. 22. S. Roudenko, Duality of matrix-weighted Besov spaces, Studia Math., 160 (2004), 129156. 23. H. Triebel, Theory of Function Spaces, Monographs in Math. 78, Birkh a¨ user Verlag, Basel, 1983. 24. K. Wang, The generalization of paraproducts and the full T 1 theorem for Sobolev and Triebel-Lizorkin spaces, J. Math. Anal. Appl., 209 (1997), 317-340. 25. K. Wang, The full T 1 theorem for certain Triebel-Lizorkin spaces, Math. Nachr., 197 (1999), 103-133.

1530


26. A. Volberg, Matrix Ap weights via S-functions, J. Amer. Math. Soc., 10 (1997), 445-466. 27. A. Youssfi, Regularity properties of commutators and BM O-Triebel-Lizorkin spaces, Ann. Inst. Fourier, (Grenoble) 45 (1995), 795-807.

Po-Kai Huang and Kunchuan Wang Department of Applied Mathematics National Dong Hwa University Hualien 970, Taiwan E-mail: [email protected] [email protected]