Equivalent norms for the Morrey spaces with non-doubling measures
aa r X i v : . [ m a t h . F A ] J un Equivalent norms for the Morrey spaceswith non-doubling measures
June 25, 2018
Yoshihiro Sawano, Hitoshi Tanaka Graduate School of Mathematical Sciences, The University of Tokyo,3-8-1 Komaba, Meguro-ku Tokyo 153-8914, JAPAN
E-mail: [email protected], [email protected]
Abstract
In this paper under some growth condition we investigate the con-nection between RBMO and the Morrey spaces. We do not assume thedoubling condition which has been a key property of harmonic analysis.We also obtain another type of equivalent norms.
KEYWORDS :
Morrey space, Campanato space, equivalent norms
AMS Subject Classification :
Primary 42B35, Secondary 46E35.
In this paper we discuss equivalent norms for the (vector-valued) Morreyspaces with non-doubling measures. We consider the connection between theMorrey spaces and the Campanato spaces with underlying measure µ non-doubling.The Morrey spaces appeared in [5] originally in connection with the partial dif-ferential equations and the Campanato spaces in [1] and [2]. We refer to [6] for The first author is supported by Research Fellowships of the Japan Society for the Pro-motion of Science for Young Scientists. The second author is supported by the 21st centuryCOE program at Graduate School of Mathematical Sciences, the University of Tokyo and byF¯ujyukai foundation. R d .We say that a (positive) Radon measure µ on R d satisfies the growth conditionif µ ( Q ( x, l )) ≤ C l n for all x ∈ supp ( µ ) and l > , (1)where C and n ∈ (0 , d ] are some fixed numbers. A measure µ is said to satisfythe doubling condition if µ ( Q ( x, l )) ≤ C µ ( Q ( x, l )) for all x ∈ R d and l > C >
0. A measure µ which satisfies the growth condition willbe called growth measure while a measure µ with the doubling condition will becalled the doubling measure.By a “cube” Q ⊂ R d we mean a closed cube having sides parallel to the axes.Its center will be denoted by z Q and its side length by ℓ ( Q ). By Q ( x, l ) we willalso denote the cube centered at x of sidelength l . For ρ > ρ Q means a cubeconcentric to Q with its sidelength ρ ℓ ( Q ). Let Q ( µ ) denote the set of all cubes Q ⊂ R d with positive µ -measures. If µ is finite, we include R d in Q ( µ ) as well. In[7], the authors defined the Morrey spaces M pq ( k, µ ) for non-doubling measuresnormed by k f : M pq ( k, µ ) k := sup Q ∈Q ( µ ) µ ( k Q ) p − q (cid:18)Z Q | f | q dµ (cid:19) q , ≤ q ≤ p < ∞ , k > . The fundamental property of this norm is k f : M pq ( k , µ ) k ≤ k f : M pq ( k , µ ) k ≤ C d (cid:18) k − k − (cid:19) d k f : M pq ( k , µ ) k for 1 < k < k < ∞ . With this relation in mind, we will denote M pq ( µ ) = M pq (2 , µ ). The aim of this paper is to find some norms equivalent to this Morreynorm. In this section we investigate an equivalent norm related to the doubling cubes.Although we now envisage the non-homogeneous setting, we are still able to placeourselves in the setting of the doubling cubes. In [10], Tolsa defined the notionof doubling cubes. Let k, β >
1. We say that Q ∈ Q ( µ ) is a ( k, β )-doublingcube, if µ ( kQ ) ≤ β µ ( Q ). It is well-known that, if β > k d , then for µ -almostall x ∈ R d and for all Q ∈ Q ( µ ) centered at x , we can find a ( k, β )-doubling2ube from k − Q, k − Q, . . . . In what follows we denote by Q ( µ ; k, β ) the set of all( k, β )-doubling cubes in Q ( µ ). We fix k, β > β > k d . Let 1 ≤ q ≤ p < ∞ .For f ∈ L loc ( µ ) define k f : M pq ( µ ) k d := sup Q ∈Q ( µ ; k,β ) µ ( Q ) p − q (cid:18)Z Q | f ( y ) | q dµ ( y ) (cid:19) q . Now we present the main theorem in this section.
Theorem 2.1.
Let µ be a Radon measure which does not necessarily satisfy thegrowth condition nor the doubling condition and let ≤ q < p < ∞ . If β > k dpqp − q ,then C − k f : M pq ( µ ) k d ≤ k f : M pq ( µ ) k ≤ C k f : M pq ( µ ) k d , f ∈ M pq ( µ ) , for some constant C > . Before we come to the proof of Theorem 2.1, two clarifying remarks may bein order.
Remark 2.2. If p = q , this theorem fails in general. However, if we assumethe growth condition or the doubling condition, the theorem is still available for p = q . In fact, under the growth condition or the doubling condition for any cube Q ∈ Q ( µ ) we can find a large integer j ≫ j Q ∈ Q ( µ ; k, β ). Remark 2.3.
This theorem readily extends to the vector-valued version. Let1 ≤ q ≤ p < ∞ and r ∈ (1 , ∞ ). We define the vector-valued Morrey spaces M pq ( l r , µ ) by the set of sequences of µ -measurable functions { f j } j ∈ N for which k f j : M pq ( l r , µ ) k := sup Q ∈Q ( µ ) µ (2 Q ) p − q (cid:18)Z Q k f j : l r k q dµ (cid:19) q < ∞ . The theorem can be extended to the vector valued version. Let k f j : M pq ( l r , µ ) k d := sup Q ∈Q ( µ ; k,β ) µ ( Q ) p − q (cid:18)Z Q k f j ( y ) : l r k q dµ ( y ) (cid:19) q . Then C − k f j : M pq ( l r , µ ) k d ≤ k f j : M pq ( l r , µ ) k ≤ C k f j : M pq ( l r , µ ) k d . Thesame proof as the scalar-valued spaces works for the vector-valued spaces, so inthe actual proof we concentrate on the scalar-valued cases.
Proof.
Given k >
1, we shall prove C − k f : M pq ( µ ) k d ≤ k f : M pq ( k, µ ) k , k f : M pq ( µ ) k ≤ C k f : M pq ( µ ) k d β >
0. The left inequality is obvious, so let us prove the right inequality.We have only to show that, for every cube Q ∈ Q ( µ ), µ (2 Q ) p − q (cid:18)Z Q | f ( y ) | q dµ ( y ) (cid:19) q ≤ C k f : M pq ( µ ) k d . Let x ∈ Q ∩ supp ( µ ) and Q ( x ) the largest doubling cube centered at x andhaving sidelength k − j ℓ ( Q ) for some j ∈ N . Existence of Q ( x ) can be ensured for µ -almost all x ∈ R d . Set Q ( j ) := { Q ( x ) : ℓ ( Q ( x )) = k − j ℓ ( Q ) } , j ∈ N . By Besicovitch’s covering lemma we can take Q ( j ) ⊂ Q ( j ) so that X R ∈Q ( j ) χ R ≤ d χ Q and that x ∈ [ R ∈Q ( j ) R for µ -almost all x ∈ Q with ℓ ( Q ( x )) = k − j ℓ ( Q ).Volume argument gives us that ♯ ( Q ( j )) ≤ d k jd . Since (cid:18)Z Q | f ( y ) | q dµ ( y ) (cid:19) q ≤ ∞ X j =1 X R ∈Q ( j ) (cid:18)Z R | f ( y ) | q dµ ( y ) (cid:19) q and µ ( R ) ≤ β − j µ (2 Q ) for all R ∈ Q ( j ), we have µ (2 Q ) p − q (cid:18)Z Q | f ( y ) | q dµ ( y ) (cid:19) q ≤ ∞ X j =1 β j ( p − q ) X R ∈Q ( j ) µ ( R ) p − q (cid:18)Z R | f ( y ) | q dµ ( y ) (cid:19) q ≤ ∞ X j =1 d k jd β j ( p − q ) k f : M pq ( µ ) k d = ∞ X j =1 d exp (cid:26) j (cid:18) d log k + (cid:18) p − q (cid:19) log β (cid:19)(cid:27) k f : M pq ( µ ) k d ≤ C k f : M pq ( µ ) k d , where the constant C is finite, provided β > k dpqp − q . Throughout the rest of this paper we assume that µ satisfy the growth condi-tion (1). We do not assume that µ is doubling. Before we formulate our theorems,let us recall the definition of the RBMO spaces due to Tolsa [10]. Given two cubes Q ⊂ R with Q ∈ Q ( µ ), we denote δ ( Q, R ) := Z ℓ ( Q R ) ℓ ( Q ) µ ( Q ( z Q , l )) l n dll , K Q,R = 1 + δ ( Q, R ) , Q R denotes the smallest cube concentric to Q containing R . Here and belowwe abbreviate the (2 , d +1 )-doubling cube to the doubling cube and Q ( µ ; 2 , d +1 )to Q ( µ, Q ∈ Q ( µ ), we set Q ∗ as the smallest doubling cube R of theform R = 2 j Q with j = 0 , , . . . . Tolsa defined a new BMO for the growth measures, which is suitable for theCalder´on-Zygmund theory. We say that f ∈ L loc ( µ ) is an element of RBMO if itsatisfies k f k ∗ := sup Q ∈Q ( µ ) µ (cid:0) Q (cid:1) Z Q | f ( x ) − m Q ∗ ( f ) | dµ ( x )+ sup Q ⊂ RQ,R ∈Q ( µ, | m Q ( f ) − m R ( f ) | K Q,R < ∞ , where m Q ( f ) := 1 µ ( Q ) Z Q f ( y ) dµ ( y ). Further details may be found in [10, Section2]. The following lemma is due to Tolsa. Lemma 3.1. [10, Corollary 3.5]
Let f ∈ RBMO.1. There exist positive constants C and C ′ independent of f so that, for every λ > and every cube Q ∈ Q ( µ ) , µ { x ∈ Q : | f ( x ) − m Q ∗ ( f ) | > λ } ≤ C µ (cid:18) Q (cid:19) exp (cid:18) − C ′ λ k f k ∗ (cid:19) .
2. Let ≤ q < ∞ . Then there exists a constant C independent of f , so that,for every cube Q ∈ Q ( µ ) , µ (cid:0) Q (cid:1) Z Q | f ( x ) − m Q ∗ ( f ) | q dµ ( x ) ! q ≤ C k f k ∗ . Elementary property of δ ( · , · ) Below we list elementary properties of δ ( · , · )used in this paper. Lemma 3.2.
Let Q ∈ Q ( µ ) . Then the following properties hold :(1) For ρ > , we have δ ( Q, ρQ ) ≤ C log ρ. (2) δ ( Q, Q ∗ ) ≤ C n +1 log 2 . (3) Let k ∈ N and α > . Assume, for some θ > , α ≤ µ ( Q ) ≤ µ (2 k Q ) ≤ θ α. Then δ ( Q, k Q ) ≤ n log 2 · θ C c n , where c n := ∞ X k =0 − nk . By the growth condition (1) there are a lot of big doubling cubes. Precisely speaking,given a cube Q ∈ Q ( µ ), we can find j ∈ N with 2 j Q ∈ Q ( µ,
2) (see [10]).
Given the cubes P ⊂ Q ⊂ R with P ∈ Q ( µ ) , then | δ ( P, R ) − ( δ ( P, Q ) + δ ( Q, R )) | ≤ C, where C is a constant depending only on C , n, d . (5) Let
Q, R ∈ Q ( µ ) . Suppose, for some constant c > , Q ⊂ R and ℓ ( R ) ≤ c ℓ ( Q ) . Then there exists a doubling cube S ∈ Q ( µ, such that Q ∗ , R ∗ ⊂ S and δ ( Q ∗ , S ) , δ ( R ∗ , S ) ≤ C , where C is a constant depending only on c , C , n, d .Proof. In [9], we have proved (1)–(4). For reader’s convenience the full proof isgiven here. (1) is obvious. To prove (2) we set Q ∗ = 2 k Q . We may assume that k ≥
1. The dyadic argument yields that δ ( Q, k Q ) = Z ℓ (2 k Q ) ℓ ( Q ) µ ( Q ( z Q , l )) l n dll ≤ n log 2 k X k =1 µ (2 k Q ) ℓ (2 k Q ) n . Note that 2 d +1 µ (2 k − Q ) ≤ µ (2 k Q ) for k = 1 , , . . . , k ,since 2 k − Q is not doubling, which yields, together with the fact that d ≥ n , δ ( Q, k Q ) ≤ n log 2 µ (2 k Q ) ℓ (2 k Q ) n k X k =1 (2 n − d − ) k − k ≤ C n +1 log 2 . We prove (3). It follows by the dyadic argument and the assumption that δ ( Q, k Q ) ≤ n log 2 k X k =1 µ (2 k Q ) ℓ (2 k Q ) n ≤ n log 2 · θαℓ ( Q ) n k X k =0 − nk ≤ n log 2 · θ C c n . Now we prove (4). It suffices to prove that A := | δ ( P Q , R ) − δ ( Q, R ) | ≤ C. (2)We decompose A as A = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ℓ ( P R ) ℓ ( P Q ) µ ( Q ( z P , l )) l n dll − Z ℓ ( Q R ) ℓ ( Q ) µ ( Q ( z Q , l )) l n dll (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z ℓ ( P Q ) ℓ ( Q ) µ ( Q ( z Q , l )) l n dll + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z min { ℓ ( P R ) ,ℓ ( Q R ) } ℓ ( P Q ) ( µ ( Q ( z P , l )) − µ ( Q ( z Q , l ))) dll n +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + Z max { ℓ ( P R ) ,ℓ ( Q R ) } min { ℓ ( P R ) ,ℓ ( Q R ) } (cid:18) µ ( Q ( z P , l )) l n + µ ( Q ( z Q , l )) l n (cid:19) dll =: A + A + A .
6y (1) the integrals A and A are easily estimated above by some constant C .So we estimate A . Bound A from above by A ≤ Z ∞ ℓ ( P Q ) µ ( Q ( z P , l )∆ Q ( z Q , l )) dll n +1 = Z ∞ ℓ ( P Q ) Z R d χ Q ( z P ,l )∆ Q ( z Q ,l ) ( y ) dµ ( y ) dll n +1 . A simple geometric observation tells us that χ Q ( z P ,l )∆ Q ( z Q ,l ) ( y ) = 0 if l / ∈ [min {| y − z P | ∞ , | y − z Q | ∞ } , max {| y − z P | ∞ , | y − z Q | ∞ } ] , where | y | ∞ := max {| y | , . . . , | y d |} . This observation and Fubini’s theorem yield A ≤ C Z R d \ P Q (cid:12)(cid:12)(cid:12)(cid:12) | y − z P | ∞ n − | y − z Q | ∞ n (cid:12)(cid:12)(cid:12)(cid:12) dµ ( y ) ≤ C Z | y − z P | ∞ ≥ ℓ ( P Q ) / | z P − z Q | ∞ | y − z P | ∞ n +1 dµ ( y ) ≤ C | z P − z Q | ∞ ℓ ( P Q ) ≤ C. This proves (2).Finally we establish (4). Let Q ∗ = 2 j Q . Then we claim δ ( R, j R ) ≤ C .Indeed, by virtue of the fact that Q ⊂ R we see that if l ≥ ℓ ( R ) then Q ( z R , l ) ⊂ Q ( z q , l ). As a consequence we obtain δ ( R, j R ) = Z j ℓ ( R ) ℓ ( R ) µ ( Q ( z R , l )) l n dll ≤ Z j ℓ ( R ) ℓ ( R ) µ ( Q ( z Q , l )) l n dll ≤ Z c j +1 ℓ ( Q ) ℓ ( Q ) µ ( Q ( z Q , l )) l n dll ≤ C. If we put S := (2 j +1 R ) ∗ , then δ ( R ∗ , S ) ≤ C . (1) and (4) finally give us δ ( Q ∗ , S ) ≤ δ ( Q ∗ , j +1 R ) + δ (2 j +1 R, S ) + C ≤ C. This is the desired result.
Scalar-valued Campanato space
Having cleared up the definition of RBMO,we will find a relationship between RBMO and the Morrey spaces. With the def-inition of RBMO in mind, we shall define the Campanato spaces.Let f ∈ L loc ( µ ). We define the Campanato spaces C pq ( k, µ ) normed by k f : C pq ( k, µ ) k := sup Q ∈Q ( µ ) µ ( kQ ) p − q (cid:18)Z Q | f ( x ) − m Q ∗ ( f ) | q dµ ( x ) (cid:19) q + sup Q ⊂ RQ,R ∈Q ( µ, µ ( Q ) p | m Q ( f ) − m R ( f ) | K Q,R , ≤ q ≤ p ≤ ∞ , k > . k , k >
1. Then C pq ( k , µ ) and C pq ( k , µ ) coincide as a set and their normsare mutually equivalent. Speaking more precisely, we have the norm equivalence k f : C pq ( k , µ ) k ∼ k f : C pq ( k , µ ) k . (3)To prove (3) we may assume that k = 2 k − C pq ( k, µ ) with respect to k . Then all we have to prove is µ ( k Q ) p − q (cid:18)Z Q | f ( x ) − m Q ∗ ( f ) | q dµ ( x ) (cid:19) q ≤ C k f : C pq ( k , µ ) k for fixed cube Q ∈ Q ( µ ). Divide equally Q into 2 d cubes and collect those in Q ( µ ).Let us name them Q , Q , . . . , Q N , N ≤ d . The triangle inequality reduces thematter to showing µ ( k Q ) p − q (cid:18)Z Q l | f ( x ) − m Q ∗ ( f ) | q dµ ( x ) (cid:19) q ≤ C k f : C pq ( k , µ ) k , ≤ l ≤ N. Note that k Q l ⊂ k Q . We apply Lemma 3.2 (5) to obtain an auxiliary doublingcube R which contains ( Q l ) ∗ , Q ∗ and satisfies K ( Q l ) ∗ ,R , K Q ∗ ,R ≤ C . Thus, weobtain µ ( k Q ) p − q (cid:18)Z Q l | f ( x ) − m Q ∗ ( f ) | q dµ ( x ) (cid:19) q ≤ µ ( k Q ) p − q (cid:18)Z Q l | f ( x ) − m ( Q l ) ∗ ( f ) | q dµ ( x ) (cid:19) q + µ ( Q l ) p | m ( Q l ) ∗ ( f ) − m R ( f ) | + µ ( Q l ) p | m R ( f ) − m Q ∗ ( f ) |≤ C k f : C pq ( k , µ ) k . As a result (3) is proved.Since C pq ( k , µ ) and C pq ( k , µ ) are isomorphic to each other as Banach spaces,no confusion can occur if we denote C pq ( µ ) = C pq (2 , µ ).Note that C ∞ q ( µ ) = RBM O , if 1 ≤ q < ∞ . This is an immediate consequenceof Lemma 3.1. Thus we can say that RBMO is a limit function space of C pq ( µ ) as p → ∞ with q ∈ [1 , ∞ ) fixed.Next, we observe Q ( µ,
2) can be seen as a net whose order is induced bynatural inclusion. With the aid of the following proposition, we shall cope withthe ambiguity of constant functions in the semi-norm of the Campanato spaces.
Proposition 3.3.
Let ≤ q ≤ p < ∞ . Then the limit M ( f ) := lim Q ∈Q ( µ, m Q ( f ) exists for every f ∈ C pq ( µ ) . That is, given ε > , we can find a doubling cube Q ∈ Q ( µ, such that | m R ( f ) − m Q ( f ) | ≤ ε or all R ∈ Q ( µ, engulfing Q . In particular there exists an increasing sequenceof concentric doubling cubes I ⊂ I ⊂ . . . ⊂ I k ⊂ . . . such that { m I k ( f ) } k ∈ N is Cauchy and [ k I k = R d . (4)We remark that the condition like (4) appears in [4]. We are mainly interestedin the function f ∈ C pq ( µ ) such that M ( f ) = 0. Proof.
Before we come to the proof of Proposition 3.3, we note that k | f | : C p ( µ ) k ≤ C k f : C p ( µ ) k . (5)Indeed, we have µ (cid:18) Q (cid:19) p − Z Q || f ( x ) | − m Q ∗ ( | f | ) | dµ ( x )= µ (cid:18) Q (cid:19) p − µ ( Q ∗ ) Z Q (cid:12)(cid:12)(cid:12)(cid:12)Z Q ∗ | f ( x ) | − | f ( y ) | dµ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) dµ ( x ) ≤ µ (cid:18) Q (cid:19) p − µ ( Q ∗ ) Z Q int Q ∗ || f ( x ) | − | f ( y ) || dµ ( y ) dµ ( x ) ≤ µ (cid:18) Q (cid:19) p − µ ( Q ∗ ) Z Q Z Q ∗ | f ( x ) − f ( y ) | dµ ( y ) dµ ( x ) ≤ µ (cid:18) Q (cid:19) p − Z Q | f ( x ) − m Q ∗ ( f ) | dµ ( x ) + µ ( Q ∗ ) p − Z Q ∗ | m Q ∗ ( f ) − f ( y ) | dµ ( y ) ≤ C k f : C p ( µ ) k . In the same way we can provesup Q ⊂ RQ,R ∈Q ( µ, µ ( Q ) p | m Q ( | f | ) − m R ( | f | ) | K Q,R ≤ C k f : C p ( µ ) k . As a consequence (5) is justified.We now turn to the proof of Proposition 3.3. By the monotonicity of C pq ( µ )with respect to q , we may assume q = 1. Case 1 µ is infinite. Take a sequence of concentric doubling cubes { Q j } j ∈ N such that for all j ∈ N µ ( Q ) ≥ , µ ( Q j +1 ) ≥ µ ( Q j ) , δ ( Q j , Q j +1 ) ≤ C C > C . Then by the definition of C p ( µ ) it holdsthat | m Q j ( f ) − m Q j +1 ( f ) | ≤ C − jp k f : C p ( µ ) k , j ∈ N . Thus we establish at least the existence of M ( f ) := lim j →∞ m Q j ( f ). Let Q ∈Q ( µ )( µ,
2) which contains Q j and does not contain Q j +1 . Set Q ′ = ( Q j Q ) ∗ . Thenby using Lemma 3.2 it is easy to see that δ ( Q, Q ′ ) ≤ C for some absolute constant C >
0. Then we have | m Q ′ ( f ) − m Q ( f ) | , | m Q ′ ( f ) − m Q j ( f ) | ≤ C, − jp k f : C p ( µ ) k , which implies | m Q ( f ) − M ( f ) | ≤ C − jp k f : C p ( µ ) k . Thus we finally establish M ( f ) = lim Q ∈Q ( µ, m Q ( f ). Case 2 µ is finite. In this case, we have only to prove
Claim 3.4. If µ is finite and k f : C p ( µ ) k < ∞ , then f ∈ L ( µ ) . In proving Claim 3.4, (5) allows us to assume f is positive.We take an increasing sequence of concentric doubling cubes { Q j } j ∈ N suchthat δ ( Q , Q k ) ≤ C for all k ∈ N . Then we have m Q k ( f ) ≤ m Q ( f ) + µ ( Q ) − p (1 + C ) k f : C pq ( µ ) k . Passage to the limit then gives Z R d f dµ ≤ µ ( R d ) (cid:16) m Q ( f ) + µ ( Q ) − p (1 + C ) k f : C pq ( µ ) k (cid:17) . This establishes f ∈ L ( µ ).The main theorem in this section is the following. Theorem 3.5.
Let ≤ q ≤ p < ∞ . Assume f ∈ C pq ( µ ) satisfies M ( f ) =lim Q ∈Q ( µ, m Q ( f ) = 0 . Then C − k f : C pq ( µ ) k ≤ k f : M pq ( µ ) k ≤ C k f : C pq ( µ ) k for some constant C > . The left inequality is obvious. To prove the right inequality we need a lemma.
Lemma 3.6.
Under the assumption of Theorem 3.5, given R ∈ Q ( µ, , thereexists a sequence of increasing doubling cubes { R k } Kk =1 such that . R k is concentric and R = R .2. If µ is finite, then so is K and R K = R d .3. For large K ∈ N , there exists R k so that R k ⊂ I K ⊂ R k +1 .4. µ ( R k ) ≥ k − µ ( R ) , k < K .5. δ ( R k , R k +1 ) ≤ C, k < K .Proof.
Suppose we have defined R k . If µ ( R d ) ≤ k µ ( R ), then we set R k +1 = R d and we stop. Suppose otherwise. We define R k +1 as the smallest doubling cube ofthe form 2 l R k with l ≥ µ -measure exceeds 2 k µ ( R ). By virtue of Lemma3.2 (3) it is easy to verify that { R k } Kk =1 obtained in this way satisfies the propertyof the lemma.Let us return to the proof of Theorem 3.5. Let R ∈ Q ( µ ). We shall estimate µ (2 R ) p − q (cid:18)Z R | f ( x ) | q dµ ( x ) (cid:19) q . The triangle inequality enables us to majorize the above integral by µ (cid:18) R (cid:19) p − q (cid:18)Z R | f ( x ) − m R ∗ ( f ) | q dµ ( x ) (cid:19) q + µ ( R ) p | m R ∗ ( f ) | . Consequently we can reduce the matters to the estimate of µ ( R ∗ ) p | m R ∗ ( f ) | . Now we invoke Lemma 3.6 for K taken so that µ ( R ) p | m I K ( f ) | ≤ k f : C pq ( µ ) k . Using the sequence { R k } Kk =1 , we obtain µ ( R ∗ ) p | m R k ( f ) − m R k +1 ( f ) |≤ C − kp µ ( R k ) p | m R k ( f ) − m R k +1 ( f ) | δ ( R k , R k +1 ) ≤ C − kp k f j : C pq ( l r , µ ) k . We also have µ ( R ∗ ) p | m R k ( f ) − m I K ( f ) | ≤ C − k p k f : C pq ( µ ) k , since by theproperties and of Lemma 3.6 we see that δ ( R k , R k +1 ) , δ ( I K , R k +1 ) aremajorized by some constants dependent only on C . The triangle inequalitygives us µ ( R ∗ ) p | m R ∗ ( f ) |≤ µ ( R ) p k − X k =1 | m R k ( f ) − m R k +1 ( f ) | + µ ( R ∗ ) p (cid:16) | m R k ( f ) − m I K ( f ) | + | m I K ( f ) | (cid:17) ≤ C ∞ X k =1 − kp ! k f : C pq ( µ ) k + µ ( R ∗ ) p | m I K ( f ) | ≤ C k f : C pq ( µ ) k . The proof of Theorem 3.5 is therefore complete.11 ector-valued extension
Finally we consider the vector-valued extensionsof Theorem 3.5. Let k a j : l r k denote the l r -norm of a = { a j } j ∈ N . If possibleconfusion can occur, then we write k{ a j } j ∈ N : l r k . For f ∈ L loc ( µ ), we definethe sharp maximal operator due to Tolsa by M ♯ f ( x ) := sup x ∈ Q ∈Q ( µ ) µ (cid:0) Q (cid:1) Z Q | f ( y ) − m Q ∗ ( f ) | dµ ( y )+ sup x ∈ Q ⊂ RQ,R ∈Q ( µ, | m Q ( f ) − m R ( f ) | K Q,R . Lemma 3.1 can be extended to the following vector-valued version.
Lemma 3.7. [9]
Let f j ∈ RBMO for j = 1 , , . . . . For any cube Q ∈ Q ( µ ) and q, r ∈ (1 , ∞ ) , there exists a constant C independent of f j such that µ (cid:0) Q (cid:1) Z Q k f j ( x ) − m Q ∗ ( f j ) : l r k q dµ ( x ) ! q ≤ C sup x ∈ R d (cid:13)(cid:13) M ♯ f j ( x ) : l r (cid:13)(cid:13) . (6)We now define the vector-valued Campanato spaces. Let 1 ≤ q ≤ p ≤ ∞ and r ∈ (1 , ∞ ). We say that { f j } j ∈ N belongs to the vector-valued Campanato spaces C pq ( l r , µ ) if each f j is µ -measurable and k f j : C pq ( l r , µ ) k := sup Q ∈Q ( µ ) µ (2 Q ) p − q (cid:18)Z Q k f j ( x ) − m Q ∗ ( f j ) : l r k q dµ ( x ) (cid:19) q + sup Q ⊂ RQ,R ∈Q ( µ, µ ( Q ) p k m Q ( f j ) − m R ( f j ) : l r k K Q,R < ∞ . As for the vector-valued spaces, the norm equivalence of the Campanato typestill holds.
Theorem 3.8.
Let ≤ q ≤ p < ∞ and let { f j } j ∈ N be a sequence in C pq ( µ ) .Assume that there exists an increasing sequence of concentric doubling cubes I ⊂ I ⊂ . . . ⊂ I k ⊂ . . . such that lim k →∞ m I k ( f j ) = 0 for all j and [ k I k = R d . Then there exists a constant
C > independent of { f j } j ∈ N such that C − k f j : C pq ( l r , µ ) k ≤ k f j : M pq ( l r , µ ) k ≤ C k f j : C pq ( l r , µ ) k . Using Lemma 3.7, we can say more about C ∞ q ( l r , µ ), which gives us a partialclue to the definition of the vector-valued RBMO spaces. Speaking precisely, weobtain the following proposition. 12 roposition 3.9. Let { f j } j ∈ N be a sequence of L loc ( µ ) functions. Then sup Q ⊂ RQ,R ∈Q ( µ, k m Q ( f j ) − m R ( f j ) : l r k K Q,R ≤ c sup x ∈ R d k M ♯ f j ( x ) : l r k . (7) In particular, we have k f j : C ∞ q ( l r , µ ) k ≤ c sup x ∈ R d k M ♯ f j ( x ) : l r k . (8) Proof.
Fix Q ⊂ R such that Q ∈ Q ( µ ). Then | m Q ( f j ) − m R ( f j ) | K Q,R ≤ c M ♯ f j ( x )for all x ∈ Q . By taking the l r -norm of both sides we obtain k m Q ( f j ) − m R ( f j ) : l r k K Q,R ≤ c sup x ∈ Q k M ♯ f j ( x ) : l r k ≤ c sup x ∈ R d k M ♯ f j ( x ) : l r k . Now since Q and R are taken arbitrarily, (7) is proved. (8) can be obtained withthe help of (6) and (7).Before we conclude this section, a remark may be in order. Remark 3.10.
Let 0 < α < n . For
Q, R ∈ Q ( µ ) with Q ⊂ R we define K ( α ) Q,R = 1 + N Q,R X k =1 (cid:18) µ (2 k Q ) ℓ (2 k Q ) n (cid:19) n − αn , where N Q,R is the least integer j with 2 j Q ⊃ R . For the definition of this constantwe refer to [3]. Theorems in this section still hold, if we replace K Q,R by K ( α ) Q,R whenever 1 ≤ q ≤ p < ∞ . Acknowledgement
The authors thank Prof. E. Nakai for discussing this paper with them.
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