A note on the boundedness of discrete commutators on Morrey spaces and their preduals
aa r X i v : . [ m a t h . F A ] J un A note on the boundednessof discrete commutators on Morrey spaces and theirpreduals
Yoshihiro SawanoAugust 11, 2018
Abstract
Dyadic fractional integral operators are shown to be bounded on Morrey spacesand their preduals. It seems that the proof of the boundedness by means of dyadicfractional integral operators is effective particularly on the preduals. In the presentpaper the commutators are proved to be bounded as well.
In the present paper, we consider the dyadic analysis of Morrey spaces and their pre-duals. The Haar wavelet, which plays a central role in this field, is given as follows:First, we write h ε i ( t ) := χ [0 , (2 t ) + ( − ε i χ [1 , (2 t ) ( t ∈ R ) (1)for ε i ∈ Z / Z . Given ε ∈ E := ( Z / Z ) n \ { (0 , , . . . , } , we define h ε := h ε ⊗ h ε ⊗ . . . ⊗ h ε n , that is, h ε ( x , x , . . . , x n ) = n Y i =1 h ε i ( x i ) . (2)By D we mean the set of all dyadic cubes. If we write Q jm := n Y ν =1 (cid:20) m ν j , m ν + 12 j (cid:19) for j ∈ Z and m ∈ Z n , then we have D = { Q jm : j ∈ Z , m ∈ Z n } . The set D j is thesubset of D made up of the cubes of volume 2 − jn : D j = { Q jm : m ∈ Z n } . Given adyadic cube Q = Q jm ( j ∈ Z , m ∈ Z n ), we define the corresponding Haar functionby h εQ ( x ) := 2 jn/ h ε (2 j x − m ) . (3)The idea of discretizing I α dates back to Lacey’s 2007 paper [5]. ≤ q ≤ p < ∞ . Then let us define the Morrey norm k f k M pq by k f k M pq := sup Q ∈D | Q | p − q (cid:18)Z Q | f ( y ) | q dy (cid:19) q , (4)where f ∈ L q, loc . We will also use the dyadic BMO space. Given a cube Q ∈ D and f ∈ L , loc , we can write m Q ( f ) := 1 | Q | Z Q f ( x ) dx . The dyadic sharp maximal operatorhere is defined by M ♯, dyadic f ( x ) := sup x ∈D m Q ( | f − m Q ( f ) | ) . (5)A function a ∈ L , loc is said to belong to the dyadic BMO, which we will write asBMO dyadic , if M ♯, dyadic a ∈ L ∞ . We define the dyadic BMO norm by k a k BMO dyadic := k M ♯, dyadic a k ∞ .The present paper, based upon Theorem 1.1, considers the boundedness of com-mutators. Throughout the paper, for A, B >
0, we write A . B to indicate that thereexists a constant c > A ≤ c B and that this constant depends only on p, q, s, t, α which will appear in each theorem. We also use A & B to denote B . A and A ∼ B to denote the two-sided inequality A . B . A . Theorem 1.1.
Let < q ≤ p < ∞ . ( i ) Let f ∈ M pq . Then we have equivalence k f k M pq ∼ X ε ∈ E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = −∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X Q ∈D j h f, h εQ i h εQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) M pq . (6)( ii ) If a locally integrable function f satisfies X ε ∈ E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = −∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X Q ∈D j h f, h εQ i h εQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) M pq < ∞ , (7) then the limit g := lim M →∞ X ε ∈ E M X j = − M X Q ∈D j h f, h εQ i h εQ (8) exists in the topology of L q, loc and defines an M pq -function. Furthermore, k g k M pq ∼ X ε ∈ E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = −∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X Q ∈D j h f, h εQ i h εQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) M pq . Theorem 1.2.
Let a ∈ BMO dyadic and < q ≤ p < ∞ . Then we have X ε ∈ E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = −∞ X Q ∈D j h f, χ Q i · h a, h εQ i h εQ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) M pq . k a k BMO dyadic k f k M pq . One formally defines I α, dyadic f ( x ) := X ε ∈ E ∞ X j = −∞ X Q ∈D | Q | αn h f, h εQ i h εQ ( x ) . (9)We can justify the definition of I α, dyadic . In particular, we can also justify the conver-gence of the sum (9) in the next theorem. Theorem 1.3.
Let < α < n, < q ≤ p < ∞ , < t ≤ s < ∞ . Assume s = 1 p − αn , ts = qp . (10) Then, for every f ∈ M pq , I α, dyadic f ( x ) = lim M →∞ X ε ∈ E M X j = − M X Q ∈D j | Q | αn h f, h εQ i h εQ ( x ) (11) converges for almost every x ∈ R n and we have k I α, dyadic f k M st . k f k M pq . (12) Theorem 1.4.
Let < α < n, < q ≤ p < ∞ , < t ≤ s < ∞ and a ∈ BMO dyadic .Assume s = 1 p − αn , ts = qp . (13) Then, for every f ∈ M pq , the limit [ a, I α, dyadic ] f ( x ) = a ( x ) I α, dyadic f ( x ) − I α, dyadic [ a · f ]( x ):= lim M →∞ X ε ∈ E M X j = − M X Q ∈D j h a · I α, dyadic f − I α, dyadic [ a · f ] , h εQ i h εQ ( x ) exists in the topology of L q, loc and we have k [ a, I α, dyadic ] f k M st . k a k BMO dyadic k f k M pq . (14)Next, we prove that the operator norm is characterized by the dyadic BMO norm.3 heorem 1.5. Let a ∈ BMO dyadic . Suppose that we are given parameters p, q, s, t, α satisfying < q ≤ p < ∞ , < t ≤ s < ∞ , < α < n and pq = ts , s = 1 p − αn . Then we have k [ a, I α, dyadic ] k B ( M pq , M st ) ∼ k a k BMO dyadic . Needless to say, it is significant to prove that k [ a, I α, dyadic ] k B ( M pq , M st ) & k a k BMO dyadic in view of Theorem 1.4. In the usual setting of p = q and s = t , Theorem 1.4 is knownas the result due to S. Chanillo [1].All the results above carry over to predual spaces. Recall that the predual space H pq of the Morrey space M p ′ q ′ is given as follows: Let 1 < p ≤ q < ∞ .( i ) A function A ∈ L q is said to be a ( p, q )-block, if there exists a dyadic cube Q such that k A k L q ≤ | Q | q − p and that A is supported on Q .( ii ) The predual space H pq is given by H pq := ∞ X j =1 λ j a j : ∞ X j =1 | λ j | < ∞ and each a j is a ( p, q )-block (15)and the norm is given by k f k H pq := inf ∞ X j =1 | λ j | : f = ∞ X j =1 λ j a j and each a j is a ( p, q )-block (16)for f ∈ H pq .A well-known fact is that the dual of H p ′ q ′ is M pq (see [13]). Therefore, it seems easyto prove this theorem by duality. Theorem 1.6.
Let < α < n , < r ≤ r < ∞ and < p ≤ p < ∞ . Assume inaddition r = 1 p − αn , rr = pp . ( i ) The fractional integral operator I α, dyadic , which is originally defined on L r ′ , isbounded from H r ′ r ′ to H p ′ p ′ . That is, k I α, dyadic f k H p ′ p ′ ≤ C k f k H r ′ r ′ for all f ∈ H r ′ r ′ ii ) The commutator [ a, I α, dyadic ] , which is originally defined on L r ′ , is bounded from H r ′ r ′ to H p ′ p ′ . Actually, we invoke dualtiy to prove this theorem. However, we need to pay atten-tion to perform duality argument. Here is a “wrong” proof for I α, dyadic . The same canbe said for [ a, I α, dyadic ] or I α . Wrong proof of Thoerem 1.6.
By duality argument, we have k I α, dyadic f k H p ′ p ′ = sup (cid:26)(cid:12)(cid:12)(cid:12)(cid:12)Z R n I α, dyadic f ( x ) h ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) : h ∈ M p p , k h k M p p = 1 (cid:27) . In view of the definition of I α, dyadic , we have Z R n I α, dyadic f ( x ) h ( x ) dx = Z R n f ( x ) I α, dyadic h ( x ) dx. If we invoke the boundedness of I α, dyadic obtained in Theorem 1.3 and we denote by k I α, dyadic k B ( M p p , M r r ) the operator norm, then we have k I α, dyadic f k H p ′ p ′ ≤ k I α, dyadic k B ( M p p , M r r ) k f k H r ′ r ′ . The proof is now complete.Here is some gap in the proof: There is no guarantee for I α, dyadic f to be a memberof H p ′ p ′ . So to overcome this trouble, we need to take full advantage of the dyadicfractional integral operator I α, dyadic : I α, dyadic h εR = | R | αn h εR .Next, we investigate the compactness of the commutator [ a, I α, dyadic ]. To this endwe define VMO dyadic as the closure of Span( { h εQ } ε ∈ E, Q ∈D ), where Span( A ) denotes alinear subspace generated by a set A . Theorem 1.7.
Let < α < n , < q ≤ p < ∞ and < t ≤ s < ∞ . Assume s = 1 p − αn , ts = qp . (17) Then a ∈ BMO dyadic generates a compact commutator [ a, I α, dyadic ] : M pq → M st if andonly if a ∈ VMO dyadic . We remark that the “if” part of Theorem 1.7 is investigated in [7].All the theorems above are proved in Section 3 after collecting some auxiliary factsin Section 2. 5
Preliminaries
Here we collect some preliminary facts. For the proof of Proposition 2.1 we refer to [4,Chapter 2].
Proposition 2.1.
Let < q < ∞ . ( i ) For f ∈ L q , the following equivalence holds : k f k L q ∼ X ε ∈ E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = −∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X Q ∈D j h f, h εQ i h εQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q . (18)( ii ) For f ∈ L q and k ∈ Z , the following equivalence holds : k f k L q ∼ X ε ∈ E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X Q ∈D j h f, h εQ i h εQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X Q ∈D k h f, χ Q i χ Q | Q | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q . (19)( iii ) For f ∈ L q, loc and R ∈ D , the following equivalence holds : k f − m R ( f ) k L q ( R ) ∼ X ε ∈ E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = − log ℓ ( R ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X Q ∈D j , Q ⊂ R h f, h εQ i h εQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q . (20) Here the implicit constant in (19) does not depend on k . A counterpart of Proposition 2.1 for Herz spaces was proved in [6]. So, it seemspossible to extend the results to these spaces.This is the only propositions whose proof we omit in the present paper.When n = 1, the next proposition is [5, Theorem 2.6.]. Proposition 2.2.
Let a ∈ BMO dyadic and < q < ∞ . Then the following is anequivalent norm of k a k BMO dyadic :sup X ε ∈ E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = −∞ X Q ∈D j h f, χ Q i · h a, h εQ i h εQ | Q | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q : f ∈ L q , k f k L q = 1 . This theorem is motivated by the results due to Coifman and Meyer. (See [2, 3].)
Proof.
This is somehow well known [5, Theorem 2.6.]. The proof ofsup X ε ∈ E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = −∞ X Q ∈D j h f, χ Q i · h a, h εQ i h εQ | Q | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q : f ∈ L q , k f k L q = 1 & k a k BMO dyadic T p = 2.The situation resembles that in [12, p.302 (64)].Here and below, for k ∈ N ∪ { } and R ∈ D , we write R + k for the unique dyadiccube containing R and of volume 2 kn | R | . | R + k | = 2 kn | R | , R ⊂ R + k , R + k ∈ D . (21)Finally before we prove Theorems 1.1–1.7, we shall obtain a counterpart of Propo-sition 2.2 for Morrey spaces. Proposition 2.3.
Let < q ≤ p < ∞ . Then we have sup f X ε ∈ E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = −∞ X Q ∈D j h f, χ Q i · h a, h εQ i h εQ | Q | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) M pq ∼ k a k BMO dyadic , (22) where the supremum is taken over all f ∈ M pq such that k f k M pq = 1 .Proof. Observe that by Proposition 2.1 (see (20)) we have k a k BMO dyadic ∼ sup Q ∈D | Q | Z Q ∞ X j = −∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X R ∈D j , R ⊂ Q h a, h εR i h εR ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q dx q . Consequently the inequality & in (22) follows by considering f = | Q | − /p − / h εQ for acube Q .Let us prove the inequality . in (22). Let S be a fixed dyadic cube. Then we needto show that | S | p − q Z S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j = −∞ X Q ∈D j h f, χ Q i · h a, h εQ i h εQ ( x ) | Q | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q dx q . k a k BMO dyadic for all f ∈ M pq with norm 1.By Proposition 2.2, we have | S | p − q Z S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j = −∞ X Q ∈D j h χ S f, χ Q i · h a, h εQ i h εQ ( x ) | Q | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q dx q ≤ | S | p − q Z R n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j = −∞ X Q ∈D j h χ S f, χ Q i · h a, h εQ i h εQ ( x ) | Q | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q dx q . k a k BMO dyadic | S | p − q k χ S f k L q .
7y the definition of the Morrey norm k f k M pq we have | S | p − q Z S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j = −∞ X Q ∈D j h χ S f, χ Q i · h a, h εQ i h εQ ( x ) | Q | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q dx q . k a k BMO dyadic . Meanwhile, a geometric observation shows that | S | p − q Z S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j = −∞ X Q ∈D j h χ R n \ S f, χ Q i · h a, h εQ i h εQ ( x ) | Q | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q dx q = | S | p − q Z S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X k =0 h χ R n \ S f, χ S + k i · h a, h εS + k i h εS + k ( x ) | S + k | !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q dx ! q , (23)where S + k is given by (21) with R replaced with S . With the definition of the Morreynorm (4), a crude estimate |h a, h εQ i| ≤ C | Q | / k a k BMO and this observation (23) in mind, we obtain | S | p − q Z S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j = −∞ X Q ∈D j h χ R n \ S f, χ Q i · h a, h εQ i h εQ ( x ) | Q | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q dx q . k a k BMO dyadic . The inequality . in (22) is proved and the proof is therefore complete. We shall prove an auxiliary inequality which is interesting of its own right.
Lemma 3.1.
Let ≤ q ≤ p < ∞ . Let f ∈ M pq . Then we have equivalence k f k M pq ∼ X ε ∈ E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = −∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X Q ∈D j h f, h εQ i h εQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) M pq + k f k M p . (24) Proof.
Let R ∈ D be fixed throughout the proof.By virtue of a crude estimate and the H¨older inequality |h f, h εR + m i h εR + m | ≤ | R + m | p − k f k M p (25)8e obtain | R | p − q Z R ∞ X j = −∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X Q ∈D j , Q ⊃ R h f, h εQ i h εQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q q . k f k M p ≤ k f k M pq . (26)Keeping in mind (26), let us first prove that k f k M pq & X ε ∈ E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = −∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X Q ∈D j h f, h εQ i h εQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) M pq + k f k M p . (27)By Proposition 2.1 ( i ) and (26) we have | R | p − q Z R ∞ X j = −∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X Q ∈D j h f, h εQ i h εQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q q . | R | p − q (cid:18)Z R | f ( x ) | q dx (cid:19) q + | R | p − q Z R ∞ X j = −∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X Q ∈D j h χ R n \ R f, h εQ i h εQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q q . k f k M pq . Thus, (27) is established.Now let us prove the converse inequality of (27). First by the triangle inequalityand the definition of the Morrey norm (4), we have | R | p − q (cid:18)Z R | f ( x ) | q dx (cid:19) q ≤ | R | p − q (cid:18)Z R | f ( x ) − m R ( f ) | q dx (cid:19) q + k f k M p . By Proposition 2.1 ( iii ) we obtain | R | p − q (cid:18)Z R | f ( x ) | q dx (cid:19) q . | R | p − q Z R ∞ X j = −∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X Q ∈D j h h εQ f i h εQ ( x ) − m R X Q ∈D j h h εQ f i h εQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q dx q + k f k M p = | R | p − q Z R ∞ X j = −∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X Q ∈D j , Q ⊂ R h h εQ f i h εQ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q dx q + k f k M p .
9f we use (26) again, then we have | R | p − q Z R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j = −∞ X Q ∈D j , Q ⊂ R h h εQ f i h εQ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q dx q . | R | p − q Z R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j = −∞ X Q ∈D j h h εQ f i h εQ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q dx q + k f k M p Thus, the proof of Lemma 3.1 is complete.Let us now prove Theorem 1.1. In view of Lemma 3.1, for the proof of ( i ) it sufficesto establish | R | p m R ( | f | ) = | R | p − Z R | f ( x ) | dx . X ε ∈ E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = −∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X Q ∈D j h f, h εQ i h εQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) M pq . (28)By the triangle inequality, we have | R | p m R ( | f | ) = lim k →∞ | R | p − Z R | f ( x ) − m R + k ( f ) | dx . ∞ X k =0 − k (cid:16) p − (cid:17) | R + k | p − Z R + k | f ( x ) − m R + k ( f ) | dx We calculate, by using Proposition 2.1 and the fact that p > | R | p m R ( | f | ) . X ε ∈ E ∞ X k =0 − k (cid:16) p − (cid:17) | R + k | p − q Z R + k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j = −∞ X Q ∈D j h h εQ f i h εQ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q dx q . X ε ∈ E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = −∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X Q ∈D j h f, h εQ i h εQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) M pq . As a conseqeunce (28) is proved.Therefore, the proof of ( i ) is complete.For the proof of ( ii ) we fix a compact set K ⊂ R n and prove thatlim M →∞ X ε ∈ E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = −∞ X Q ∈D j h f, h εQ i h εQ − M X j = − M X Q ∈D j h f, h εQ i h εQ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q ( K ) = 0 . (29)However, since K can be covered by 3 n dyadic cubes of the same size, we have only toprove (29) with K replaced by a dyadic cube R ∈ D k , where k ∈ Z is a fixed integer.Let us denote f εj := X Q ∈D j h f, h εQ i h εQ (30)10or ε ∈ E and j ∈ Z . If x ∈ R and M ≥ | k | , then we have X ε ∈ E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j = −∞ f εj ( x ) − M X j = − M f εj ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = X ε ∈ E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j = M +1 f εj ( x ) + − M − X j = −∞ f εj ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X ε ∈ E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j = M +1 X Q ∈D j Q ⊂ R h f, h εQ i h εQ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + X ε ∈ E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − M − X j = −∞ f εj ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Recall that R ∈ D k . Consequently, we can write f εj ( x ) = X Q ∈D j h f, h εQ i h εQ ( x ) = h f, h εR +( − j + k ) i h εR +( − j + k ) ( x ) ( x ∈ R k ) , (31)if j is negative enough, that is, j ≤ − M − < −| k | . If we use (31), then we obtain X ε ∈ E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j = −∞ f εj ( x ) − M X j = − M f εj ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X ε ∈ E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j = M +1 X Q ∈D j , Q ⊂ R h f, h εQ i h εQ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + X ε ∈ E ∞ X m = M + k +1 |h f, h εR + m i h εR + m ( x ) | . Thus, by the triangle inequality, we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = −∞ f εj − M X j = − M f εj (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q ( R ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = M +1 X Q ∈D j , Q ⊂ R h f, h εQ i h εQ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q ( R ) + ∞ X m = M + k +1 (cid:13)(cid:13)(cid:13) h f, h εR + m i h εR + m (cid:13)(cid:13)(cid:13) L q ( R ) . A geometric observation shows that (cid:13)(cid:13)(cid:13) h f, h εR + m i h εR + m (cid:13)(cid:13)(cid:13) L q ( R ) = 2 − mn/q (cid:13)(cid:13)(cid:13) h f, h εR + m i h εR + m (cid:13)(cid:13)(cid:13) L q ( R + m ) = 2 − mn/p + kn (1 /p − /q ) | R + m | /p − /q (cid:13)(cid:13)(cid:13) h f, h εR + m i h εR + m (cid:13)(cid:13)(cid:13) L q ( R + m ) .
11f we use this equality, then we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = −∞ f εj − M X j = − M f εj (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q ( R ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = M +1 X Q ∈D j , Q ⊂ R h f, h εQ i h εQ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q ( R ) + ∞ X m = M + k +1 − mn/q (cid:13)(cid:13)(cid:13) h f, h εR + m i h εR + m (cid:13)(cid:13)(cid:13) L q ( R + m ) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = M +1 | f εj | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q ( R ) + 2 − nM/p − nk/q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = −∞ | f εj | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) M pq . Thus, we obtain (29), which shows that (8) holds in the topology of L q, loc . As aconsequence Theorem 1.1 is proved completely. Remark 3.2.
It may be interesting to compare Theorem 1.1 and Lemma 3.1 with theresult in [11, Theorem 1.3]. In [11, Theorem 1.3], we have proved that k f k M pq ∼ k M ♯ f k M pq + k f k M p (1 < q ≤ p < ∞ ) . Here M ♯ denotes the sharp maximal operator due to Fefferman, Stein and Stromberg. Let ε ∈ E be fixed. We also take a dyadic cube R . Then it suffices from Theorem 1.1I := | R | p − q Z R ∞ X j = − log ℓ ( R ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X Q ∈D j , Q ⊂ R h f, χ Q i · h a, h εQ i h εQ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q dx q by C k a k BMO dyadic k f k M pq with constants independent of R , a and f . By using Proposi-tion 2.2 we obtainI = | R | p − q Z R ∞ X j = − log ℓ ( R ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X Q ∈D j , Q ⊂ R h χ R f, χ Q i · h a, h εQ i h εQ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q dx q . Note that { Q : Q ∈ D j } partitions R n . Consequently, we haveI . k a k BMO dyadic | R | p − q Z R ∞ X j = − log ℓ ( R ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X Q ∈D j , Q ⊂ R h χ R f, χ Q i| Q | h εQ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q dx q from the definition of k a k BMO dyadic . If we use the definition of the Morrey norm (4)crudely, then we haveI . k a k BMO dyadic | R | p − q (cid:18)Z R | f ( x ) | q dx (cid:19) q . k a k BMO dyadic k f k M pq . Therefore, since R is arbitrary, the proof of Theorem 1.2 is complete.12 .3 Proof of Theorem 1.3 We shall make use of the following estimate in the proof of Theorem 1.4 as well asTheorem 1.3. Actually Proposition 3.3 is a little stronger than Theorem 1.3.
Proposition 3.3.
Let < α < n , < q ≤ p < ∞ and < t ≤ s < ∞ . Assume s = 1 p − αn , ts = qp . (32) Then k I α, dyadic f k M st . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = −∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X Q ∈D j | Q | αn h f, h εQ i h εQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) M st . k f k M pq . (33) Proof.
The proof is simple: Let x ∈ R n and j ∈ Z be fixed and choose Q ∈ D j so that x ∈ Q . If we use a simple inequality (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X Q ∈D j | Q | αn h f, h εQ i h εQ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ℓ ( Q ) α m Q ( | f | ) ≤ ℓ ( Q ) α − np k f k M pq = 2 − j (cid:16) α − np (cid:17) k f k M pq , (34)we have ∞ X j = −∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X Q ∈D j | Q | αn h f, h εQ i h εQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ∞ X j = −∞ − jα min (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X Q ∈D j h f, h εQ i h εQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , jnp k f k M pq ≤ ∞ X j = −∞ − jα min sup l ∈ Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X Q ∈D l h f, h εQ i h εQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , jnp k f k M pq . k f k − ps M pq sup l ∈ Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X Q ∈D l h f, h εQ i h εQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ps . If we use this pointwise estimate, then we obtain the desired estimate.
Remark 3.4.
It may be interesting compare Proposition 3.3 with the following result.Let ϕ ∈ S be chosen so that ϕ ( ξ ) = 1 if 2 ≤ | ξ | ≤ ϕ ( ξ ) = 0 if | ξ | ≤ | ξ | ≥
8. Then in [8], we have established (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = −∞ | jα F − [ ϕ (2 − j · ) F f ] | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) M st . k f k M pq . (35)Estimate (35) admits an extension to Triebel-Lizorkin-Morrey spaces as we did in [8].13 .4 Proof of Theorem 1.4 We freeze ε ∈ E for a while. By definition of I α, dyadic f , we obtain ∞ X j = −∞ X Q ∈D j h a, h εQ i{ h εQ ( x ) I α, dyadic f ( x ) − I α, dyadic [ h εQ f ]( x ) } = ∞ X j = −∞ X Q ∈D j X ε ′ ∈ E ∞ X l = −∞ X R ∈D l h a, h εQ ih f, h ε ′ R i{ h εQ ( x ) I α, dyadic h ε ′ R ( x ) − I α, dyadic [ h εQ h ε ′ R ]( x ) } . Observe that, if Q ) R , then h εQ h ε ′ R = m R ( h εQ ) h ε ′ R and hence h εQ ( x ) I α, dyadic h ε ′ R ( x ) = I α, dyadic [ h εQ h ε ′ R ]( x ) . (36)Therefore, we have ∞ X j = −∞ X Q ∈D j h a, h εQ i{ h εQ ( x ) I α, dyadic f ( x ) − I α, dyadic [ h εQ f ]( x ) } = ∞ X j = −∞ X Q ∈D j X ε ′ ∈ E j X l = −∞ X R ∈D l h a, h εQ ih f, h ε ′ R i{ h εQ ( x ) I α, dyadic h ε ′ R ( x ) − I α, dyadic [ h εQ h ε ′ R ]( x ) } . If Q = R and ε = ε ′ , then we obtain h εQ ( x ) I α, dyadic h ε ′ R ( x ) = I α, dyadic [ h εQ h ε ′ R ]( x ) . (37)Let us writeI ( x ) := ∞ X j = −∞ X Q ∈D j X ε ′ ∈ E j − X l = −∞ X R ∈D l | R | αn h a, h εQ ih f, h ε ′ R i h εQ ( x ) h ε ′ R ( x ) , I ( x ) := ∞ X j = −∞ X Q ∈D j X ε ′ ∈ E j − X l = −∞ X R ∈D l | Q | αn h a, h εQ ih f, h ε ′ R i h εQ ( x ) h ε ′ R ( x ) , II( x ) := ∞ X j = −∞ X Q ∈D j h a, h εQ ih f, h εQ i| Q | αn | h εQ ( x ) | , III( x ) := ∞ X j = −∞ X Q ∈D j h a, h εQ ih f, h εQ i I α, dyadic [ | h εQ | ]( x ) . Note that both I and III have another expression:I ( x ) = ∞ X j = −∞ X Q ∈D j h I α, dyadic f, χ Q i · h a, h εQ i h εQ | Q | , III( x ) = I α, dyadic ∞ X j = −∞ X Q ∈D j h a, h εQ ih f, h εQ i| h εQ | ( x ) . ∞ X j = −∞ X Q ∈D j h a, h εQ i ( h εQ ( x ) I α, dyadic f ( x ) − I α, dyadic [ h εQ f ]( x ))= ∞ X j = −∞ X Q ∈D j X ε ′ ∈ E j − X l = −∞ X R ∈D l h a, h εQ ih f, h ε ′ R i{ h εQ ( x ) I α, dyadic h ε ′ R ( x ) − I α, dyadic [ h εQ h ε ′ R ]( x ) } + ∞ X j = −∞ X Q ∈D j h a, h εQ ih f, h εQ i{ h εQ ( x ) I α, dyadic h εQ ( x ) − I α, dyadic [ h εQ h εQ ]( x ) } = I ( x ) − I ( x ) + II( x ) − III( x ) . Let us start with dealing with I . If we invoke again Theorem 1.2, then we have k I k M st . k a k BMO dyadic k I α, dyadic f k M st . k a k BMO dyadic k f k M pq . Since I ( x ) = X ε ′ ∈ E ∞ X j = −∞ j − X l = −∞ X R ∈D l X Q ∈D j , Q ( R | R | αn h a, h εQ ih f, h ε ′ R i h εQ ( x ) h ε ′ R ( x )= ∞ X j = −∞ X Q ∈D j h a, h εQ ih I α, dyadic f, χ Q i h εQ ( x ) , we have by Theorem 1.2 k I k M st = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = −∞ X Q ∈D j h a, h εQ ih I α, dyadic f, χ Q i h εQ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) M st . k a k BMO dyadic k I α, dyadic f k M st If we invoke Theorem 1.3, then we have k I k M st . k a k BMO dyadic k f k M pq . By Proposition 3.3 we obtain k II k M st = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = −∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X Q ∈D j | Q | αn h a, h εQ ih f, h εQ i h εQ · h εQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) M st ≤ k a k BMO dyadic (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = −∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X Q ∈D j | Q | αn h f, h εQ i h εQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) M st . k a k BMO dyadic k f k M pq . Next, by Proposition 3.3 and equality I α, dyadic [ h εQ h εQ ]( x ) = c α | Q | αn − χ Q ( x ) , we have k III k M st . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = −∞ X Q ∈D j |h a, h εQ ih f, h εQ i| Q | αn − | χ Q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) M st . k a k BMO dyadic k f k M pq . Thus, the proof of Theorem 1.4 is complete.15 .5 Proof of Theorem 1.5
Let U ∈ D be fixed. Then we have[ a, I α, dyadic ] h ε ′′ U ( x ) = X ε ∈ E ∞ X j = −∞ X Q ∈D j , Q ( U ( | Q | αn − | U | αn ) h a, h εQ i h εQ ( x ) h ε ′′ U ( x )+ h a, h ε ′′ U ih f, h ε ′′ U i ( | U | αn h ε ′′ U ( x ) h ε ′′ U ( x ) − I α, dyadic [ h ε ′′ Q h ε ′′ U ]( x )) . By virtue of the non-homogeneous wavelet expansion (see (19)), we obtain k [ a, I α, dyadic ] k B ( M st , M pq ) ≥ k [ a, I α, dyadic ] h ε ′′ U k M st | U | − p + & (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = −∞ X Q ∈D j , Q ( U ( | Q | αn − | U | αn ) h a, h εQ i h εQ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) M st | U | − p . By Theorem 1.1 we have k [ a, I α, dyadic ] k B ( M st , M pq ) & | U | αn − p (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = −∞ X Q ∈D j , Q ( U h a, h εQ i h εQ h ε ′′ U (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) M st = 1 | U | s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = −∞ X Q ∈D j , Q ( U h a, h εQ i h εQ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) M st ≥ | U | t Z U (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j = −∞ X Q ∈D j , Q ( U h a, h εQ i h εQ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t dx t . By the H¨older inequalty we have1 | U | Z U (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j = −∞ X Q ∈D j , Q ( U h a, h εQ i h εQ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx . k [ a, I α, dyadic ] k B ( M st , M pq ) . Therefore, we obtain m U (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j = −∞ X Q ∈D j , Q ( U h a, h εQ i h εQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . k [ a, I α, dyadic ] h ε ′′ U k M st k h ε ′′ U k M pq (38)for all U ∈ D . Denote by U ∗ the dyadic parent of U , that is, the smallest dyadic cubeengulfing U . With U replaced by U ∗ above, we obtain m U (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j = −∞ X Q ∈D j , Q ⊂ U h a, h εQ i h εQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . k [ a, I α, dyadic ] h ε ′′ U k M st k h ε ′′ U k M pq . (39)It follows from the definition of the operator norm k [ a, I α, dyadic ] k M pq →M st that m U (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j = −∞ X Q ∈D j , Q ⊂ U h a, h εQ i h εQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . k [ a, I α, dyadic ] k M pq →M st . (40)16f we take the supremum over U ∈ D in (40), then we obtain k a k BMO dyadic . k [ a, I α, dyadic ] k B ( M st , M pq ) . Thus, the proof is complete.
By Theorem 1.1 ( iii ), we see that { h εQ } Q ∈D ,ε ∈ E is dense in H r ′ r ′ . Thus, to check ( i ) and( ii ), we need only to prove k I α, dyadic f k H p ′ p ′ + k [ a, I α, dyadic ] f k H p ′ p ′ . k f k H r ′ r ′ for all f ∈ Span( { h εQ } Q ∈D ,ε ∈ E ). If we assume f ∈ Span( { h εQ } Q ∈D ,ε ∈ E ), then from thedefinition we have I α, dyadic f, [ a, I α, dyadic ] f ∈ H p ′ p ′ . (41)Observe that (41) counts in that we can obtain (41) only by using the discrete fractionalintegral operators. Consequently, if we invoke Theorem 1.1, we obtain k I α, dyadic f k H p ′ p ′ = sup g ∈M p p \{ } k g k M p p (cid:12)(cid:12)(cid:12)(cid:12)Z R n g ( x ) I α, dyadic f ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) = sup g ∈M p p \{ } k g k M p p (cid:12)(cid:12)(cid:12)(cid:12)Z R n I α, dyadic g ( x ) f ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ sup g ∈M p p \{ } k g k M p p k I α, dyadic g k M r r k f k H r ′ r ′ ≤ k I α, dyadic k M p p →M r r k f k H r ′ r ′ . Thus, the proof is now complete.
Remark 3.5.
A usual averaging procedure yields the following corollaries. For thistechnique we refer to [5].
Corollary 3.6.
Maintain the same conditions on the parameters p, p , r, r , α . If afunction a belongs to BMO , then the following boundedness is true : k I α f k H p ′ p ′ + k [ a, I α ] f k H p ′ p ′ . k f k H r ′ r ′ , where I α is given by I α f ( x ) = Z R n f ( y ) | x − y | n − α dy. As for I α we made an alternative approach in [9, Theorem 3.1] and [10, Theorem3.1]. 17 .7 Proof of Theorem 1.7 “If part” is a direct consequence of Theorem 1.5. Let us prove the converse. To thisend, we need the following fundamental lemma. Lemma 3.7.
Let X and Y be Banach spaces. Suppose that we are given a compactlinear operator T : X → Y . If { f j } j ∈ N is a sequence in X ∗ that is weak-* convergentto . Then { T ∗ f j } j ∈ N is norm-convergent to . Now let us prove a ∈ VMO dyadic assuming that [ a, I α, dyadic ] is compact.Let us set[ a, I α, dyadic ] ≥ L := lim M →∞ X ε ∈ E M X j = L X Q ∈D j h a · I α, dyadic f − I α, dyadic [ a · f ] , h εQ i h εQ . (42)Then we have k [ a, I α, dyadic ] ≥ L k M pq →M st . X ε ∈ E sup U ∈D m U (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j = L X Q ∈D j , Q ⊂ U h a, h εQ i h εQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (43)by virtue of Theorem 1.4. The triangle inequality yieldssup U ∈D m U (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j = L X Q ∈D j , Q ⊂ U h a, h εQ i h εQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = sup U ∈ S ∞ ν = L D ν m U (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j = L X Q ∈D j , Q ⊂ U h a, h εQ i h εQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . According to (39) and Lemma 3.7, we havelim L →∞ sup U ∈D m U (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j = L X Q ∈D j , Q ⊂ U h a, h εQ i h εQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . Thus, assuming that [ a, I α, dyadic ] is compact, we havelim L →∞ [ a, I α, dyadic ] ≥ L = 0 (44)in the operator topology. Also, we set[ a, I α, dyadic ] ≤− L := lim M →∞ X ε ∈ E − L X j = − M X Q ∈D j h a · I α, dyadic f − I α, dyadic [ a · f ] , h εQ i h εQ . (45)Then we have k [ a, I α, dyadic ] ≤− L k M pq →M st . X ε ∈ E sup U ∈D m U (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − L X j = −∞ X Q ∈D j , Q ⊂ U h a, h εQ i h εQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (46)18y virtue of Theorem 1.4. Note thatsup U ∈D m U (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − L X j = −∞ X Q ∈D j , Q ⊂ U h a, h εQ i h εQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = sup m U (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − L X j = −∞ X Q ∈D j , Q ⊂ U h a, h εQ i h εQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) : U ∈ − L [ ν = −∞ D ν in order that Q ⊂ U actually happens. Thus, assuming that [ a, I α, dyadic ] is compact,we have lim L →∞ [ a, I α, dyadic ] ≤− L = 0 (47)again in the opertor topology. From (44) and (47) we havelim L →∞ k a − a ( L ) k BMO dyadic = 0 , (48)if we write a ( L ) := X ε ∈ E L X j = − L X Q ∈D j h a, h εQ i h εQ . It is not so hard to see thatlim R →∞ sup {|h a, h εQ νm i| : m ∈ Z n , | m | ≥ R } = 0 (49)for all ν ∈ Z if [ a, I α, dyadic ] is compact. Thus, it follows thatlim R →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X m ∈ Z n , | m | >R h a, h εQ jm i h εQ jm (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) BMO dyadic = 0 (50)for all j ∈ Z .From (50) we learn that a ( L ) ∈ VMO dyadic , which in turn yields a ∈ VMO dyadic byvirtue of (48).
The author is supported by Grant-in-Aid for Young Scientists (B) No. 24740085 fromthe Japan Society for the Promotion of Science. The author is also indebted to ZenisCo. Ltd for the check of the presentation in English in Section 1.
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