Characterizations of weighted BMO space and its application
aa r X i v : . [ m a t h . F A ] J u l CHARACTERIZATIONS OF WEIGHTED BMO SPACE AND ITSAPPLICATION
DINGHUAI WANG, JIANG ZHOU ∗ AND ZHIDONG TENG
Abstract.
In this paper, we prove that the weighted BMO space as followsBMO p ( ω ) = n f ∈ L : sup Q k χ Q k − L p ( ω ) (cid:13)(cid:13) ( f − f Q ) ω − χ Q (cid:13)(cid:13) L p ( ω ) < ∞ o is independent of the scale p ∈ (0 , ∞ ) in sense of norm when ω ∈ A . Moreover, we canreplace L p ( ω ) by L p, ∞ ( ω ). As an application, we characterize this space by the boundednessof the bilinear commutators [ b, T ] j ( j = 1 , b , from L p ( ω ) × L p ( ω ) to L p ( ω − p ) with1 < p , p < ∞ , 1 /p = 1 /p + 1 /p and ω ∈ A . Thus we answer the open problem proposedin [2] affirmatively. Introduction
A locally integrable function f is said to belong to BMO space if there exists a constant C > Q ⊂ R n ,1 | Q | Z Q | f ( x ) − f Q | dx ≤ C, where f Q = | Q | R Q f ( x ) dx and the minimal constant C is defined by k f k ∗ .There are a number of classical results that demonstrate BMO functions are the rightcollections to do harmonic analysis on the boundedness of commutators. A well knownresult of Coifman, Rochberg and Weiss [4] states that the commutator[ b, T ]( f ) = bT ( f ) − T ( bf )is bounded on some L p , 1 < p < ∞ , if and only if b ∈ BMO, where T is the Hilbert transform.Janson extended the result in [8] via the commutators of Calder´on-Zygmund operators withsmooth homogeneous kernels; Chanillo in [3] did the same for commutators of the fractional Primary 42B20, 47B07; Secondary: 42B25,47G99.
Key words and phrases.
Boundedness; Calder´on-Zygmund operators; Characterization; Commutators;Weighted BMO space.The research was supported by National Natural Science Foundation of China (Grant No.11661075 andNo. 11271312).* Corresponding author, [email protected]. integral operator with the restriction that n − α be an even integer. The theory was thenextended and generalized to several directions. For instance, Bloom [1] investigated thesame result in the weighted setting; Uchiyama [16] extended the boundednss results on thecommutator to compactness; Krantz and Li in [10] and [11] have applied commutator theoryto give a compactness characterization of Hankel operators on holomorphic Hardy spaces H ( D ), where D is a bounded, strictly pseudoconvex domain in C n . It is perhaps for thisimportant reason that the boundedness of [ b, T ] attracted ones attention among researchersin PDEs.Recently, Chaffee [2] considered the multilinear setting and proved that for 0 ≤ α < n ,1 < p , p < ∞ and 1 p + 1 p − αn = 1 q , if q >
1, then(1.1) [ b, T ] j : L p × L p → L q ⇔ b ∈ BMOfor j = 1 ,
2, where T is a bilinear operator of convolution type with a homogeneous kernelof degree − n + α . In his proof required the use of H¨older inequality with q and q ′ , theexponent q must be larger than 1. Thus, he asked Problem 1. If < q < b, T ] j is a bounded operator from L p × L p to L q , is b inBMO space?At the same time, Wang, Jiang and Pan [18] obtain the similar result as (1.1) for bilinearfractional integral operator, they also asked Problem 2. If ~b = ( b , b ) and [Π ~b, I α ] is a bounded operator from L p × L p to L q , is ~b ∈ BMO × BMO?In this paper, we will give an answer of Problem 1 and show that the answer of Problem2 is affirmative for the case ~b = ( b, b ), using the following Theorem 1.1 and Theorem 1.2.We focus on proving the case α = 0. For 0 < α < n , a similar arguments are applied withnecessary modifications, one can obtain the desired result. Moreover, we extend the resultto weighted case. To state our result, we first give the following denotations.We recall the definition of A p weight introduced by Muckenhoupt [13]. For 1 < p < ∞ and a nonnegative locally integrable function ω on R n , ω is in the Muckenhoupt A p class ifit satisfies the condition[ ω ] A p := sup Q (cid:18) | Q | Z Q ω ( x ) dx (cid:19)(cid:18) | Q | Z Q ω ( x ) − p − dx (cid:19) p − < ∞ . And a weight function ω belongs to the class A if[ ω ] A := 1 | Q | Z Q ω ( x ) dx (cid:16) ess sup x ∈ Q ω ( x ) − (cid:17) < ∞ . We write A ∞ = S ≤ p< ∞ A p .Let ω ∈ A ∞ and p ∈ (0 , ∞ ). We let L p ( ω ) be the space of all measurable functions f suchthat k f k L p ( ω ) := (cid:18) Z R n | f ( x ) | p ω ( x ) dx (cid:19) /p < ∞ . Let 0 < p < ∞ . Given a nonnegative locally integrable function ω , the weighted BMOspace BMO p ( ω ) is defined by the set of all functions f ∈ L ( R n ) such that k f k BMO p ( ω ) = sup Q (cid:18) ω ( Q ) Z Q | f ( y ) − f Q | p ω ( y ) − p dy (cid:19) /p = sup Q k χ Q k L p ( ω ) (cid:13)(cid:13) ( f − f Q ) χ Q ω (cid:13)(cid:13) L p ( ω ) < ∞ , where ω ( Q ) = R Q ω ( x ) dx . We write BMO ( ω ) = BMO( ω ) simple. In [6], Garc´ıa-Cuervaproved that if ω ∈ A , BMO( ω ) = BMO p ( ω ) for 1 < p < ∞ with equivalence of thecorresponding norms. Problem 3.
Let X be a quasi-Banach function space and ω ∈ A . Is the norm k f k BMO( ω ) equivalent to k f k BMO X ( ω ) = sup Q k χ Q k X (cid:13)(cid:13) ( f − f Q ) χ Q ω (cid:13)(cid:13) X < ∞ ?The aim of this paper is to show that the answer of Problem 3 is affirmative for X = L p, ∞ ( ω )(1 < p < ∞ ) and X = L r ( ω )(0 < r <
1) .
Theorem 1.1.
Let ω ∈ A and X = L p, ∞ ( ω ) with < p < ∞ . Then BMO( ω ) = BMO X ( ω ) with equivalence of the corresponding norms. Theorem 1.2.
Let ω ∈ A and < r < . Then BMO( ω ) = BMO r ( ω ) with equivalence of the corresponding norms. Remark 1.1.
In the unweighted setting, Str¨omberg in [15] showed that for < s ≤ , p > ,there exists a constant C such that s /p k f k BMO ∗ s ≤ k f k BMO p ≤ C k f k BMO ∗ s , where k f k BMO ∗ s = sup Q inf c inf (cid:8) t ≥ (cid:12)(cid:12) { x ∈ Q : | f ( x ) − c | > t } (cid:12)(cid:12) < s | Q | (cid:9) . Recall that bilinear singular integral operator T is a bounded operator which satisfies k T ( f , f ) k L p ≤ C k f k L p k f k L p , for some 1 < p , p < ∞ with 1 /p = 1 /p + 1 /p and the function K , defined off the diagonal y = y = y in ( R n ) , satisfies the conditions as follow:(1) The function K satisfies the size condition. | K ( x, y , y ) | ≤ C (cid:0) | x − y | + | x − y | (cid:1) n ;(2) The function K satisfies the regularity condition. For some γ >
0, if | y − y ′ | ≤ max {| x − y | , | x − y |}| K ( x, y , y ) − K ( x ′ , y , y ) | ≤ C | y − y ′ | γ (cid:0) | x − y | + | x − y | (cid:1) n + γ ;if | y − y ′ | ≤ max {| x − y | , | x − y |}| K ( x, y , y ) − K ( x ′ , y , y ) | ≤ C | y − y ′ | γ (cid:0) | x − y | + | x − y | (cid:1) n + γ . Then we say K is a bilinear Calder´on-Zygmund kernel. If x / ∈ supp f T supp f , then T ( f , f )( x ) = Z R n Z R n K ( x, y , y ) f ( y ) f ( y ) dy dy . The linear commutators are defined by[ b, T ] ( f , f )( x ) := b ( x ) T ( f , f )( x ) − T ( bf , f )( x ) , and [ b, T ] ( f , f )( x ) := b ( x ) T ( f , f )( x ) − T ( f , bf )( x ) . The iterated commutator is defined by[Π ~b, T ]( f , f )( x ) := [ b , [ b , T ] ] ( f , f )( x ) . In this paper, we say that an operator is of ’convolution type’ if the kernel K ( x, y , y ) isactually of the form K ( x − y , x − y ). The applications of Theorem 1.1 and Theorem 1.2as follows. Theorem 1.3.
Let ω ∈ A , ~b = ( b, b ) and T be a bilinear convolution type operator definedby T ( f , f )( x ) = Z R n Z R n K ( x − y , x − y ) f ( y ) f ( y ) dy dy for all x / ∈ supp f T supp f , where K is a bilinear Calder´on-Zygmund kernel and such thatfor any cube Q ⊂ R n with / ∈ Q , the Fourier series of K is absolutely convergent. For < p , p < ∞ with /p = 1 /p + 1 /p , the following statements are equivalent: (a1) b ∈ BMO( ω ) ; (a2) There exists a positive constant C such that for j = 1 , , k [ b, T ] j ( f , f ) · ω − k L p ( ω ) ≤ C k f k L p ( ω ) k f k L p ( ω ) . (a3) There exists a positive constant C such that for j = 1 , , k [ b, T ] j ( f , f ) · ω − k L p, ∞ ( ω ) ≤ C k f k L p ( ω ) k f k L p ( ω ) . (a4) There exists a positive constant C such that k [Π ~b, T ]( f , f ) · ω − k L p ( ω ) ≤ C k f k L p ( ω ) k f k L p ( ω ) . (a5) There exists a positive constant C such that k [Π ~b, T ]( f , f ) · ω − k L p, ∞ ( ω ) ≤ C k f k L p ( ω ) k f k L p ( ω ) . Specially, if ω ( x ) ≡
1, we have
Crolorrary 1.4.
Let ~b = ( b, b ) and T be a bilinear convolution type operator defined by T ( f , f )( x ) = Z R n Z R n K ( x − y , x − y ) f ( y ) f ( y ) dy dy for all x / ∈ supp f T supp f , where K is a bilinear Calder´on-Zygmund kernel and such thatfor any cube Q ⊂ R n with / ∈ Q , the Fourier series of K is absolutely convergent. For < p , p < ∞ with /p = 1 /p + 1 /p , the following statements are equivalent: (b1) b ∈ BMO ; (b2) [ b, T ] j is a bounded operator from L p × L p to L p for j = 1 , ; (b3) [ b, T ] j is a bounded operator from L p × L p to L p, ∞ for j = 1 , ; (b4) [Π ~b, T ] is a bounded operator from L p × L p to L p ; (b5) [Π ~b, T ] is a bounded operator from L p × L p to L p, ∞ . A same argument we also have the following result.
Crolorrary 1.5.
Let ~b = ( b, b ) and I α be a bilinear fractional integral operator defined by I α ( f , f )( x ) = Z R n Z R n f ( y ) f ( y ) (cid:0) | x − y | + | x − y | (cid:1) n − α dy dy . For < α < n , < p , p < ∞ with /q = 1 /p + 1 /p − α/n , the following statements areequivalent: (c1) b ∈ BMO ; (c2) [ b, I α ] j is a bounded operator from L p × L p to L q for j = 1 , ; (c3) [ b, I α ] j is a bounded operator from L p × L p to L q, ∞ for j = 1 , ; (c4) [Π ~b, I α ] is a bounded operator from L p × L p to L q ; (c5) [Π ~b, I α ] is a bounded operator from L p × L p to L q, ∞ . Finally, two open problems will be given.
Problem A.
Let ~b = ( b , b ) with b = b and [Σ ~b, T ] := [ b , T ] + [ b , T ] . If [Σ ~b, T ] is abounded operator from L p × L p to L q , is ~b in BMO × BMO?
Problem B.
Let ~b = ( b , b ) with b = b . If [Π ~b, T ] is a bounded operator from L p × L p to L q , is ~b in BMO × BMO? 2.
Main Lemmas
Throughout this paper, the letter C denotes constants which are independent of mainvariables and may change from one occurrence to another. Q ( x, r ) denotes a cube centeredat x , with side length r , sides parallel to the axes.For X = L q , ∞ ( ω ), it is clear that BMO q ( ω ) is contained in BMO X ( ω ) and k · k BMO X ( ω ) ≤k · k BMO q ( ω ) ≤ k · k BMO q ( ω ) if 1 < q ≤ q < ∞ . However, for 1 < q < q < ∞ , one has thereverse inequality as follows. Lemma 2.1.
Let < q < q < ∞ , ω ∈ A ∞ and X = L q , ∞ ( ω ) . Then BMO X ( ω ) iscontained in BMO q ( ω ) and k · k BMO q ( ω ) ≤ C k · k BMO X ( ω ) .Proof. Let f ∈ BMO X ( ω ). Given a fixed cube Q ⊂ R n , it is easy to see that k χ Q k L q, ∞ ( ω ) = ω ( Q ) /q , then for any λ > ω ( Q ) /q (cid:16) λ q ω (cid:8) x ∈ Q : | f ( x ) − f Q | > λω ( x ) (cid:9)(cid:17) /q ≤ k f k BMO X ( ω ) ;that is, ω (cid:8) x ∈ Q : | f ( x ) − f Q | > λω ( x ) (cid:9) ≤ k f k q BMO X ( ω ) ω ( Q ) λ − q . Choose N = k f k BMO X ( ω ) (cid:16) q q − q (cid:17) /q . Thus, Z Q | f ( x ) − f Q | q ω ( x ) − q dx = Z Q (cid:16) | f ( x ) − f Q | ω ( x ) (cid:17) q ω ( x ) dx = q Z ∞ λ q − ω (cid:8) x ∈ Q : | f ( x ) − f Q | > λω ( x ) (cid:9) dλ ≤ q Z N λ q − ω ( Q ) dλ + q Z ∞ N λ q − k f k q BMO q X ( ω ) | Q | λ − q dλ = ω ( Q ) N q + q q − q k f k q BMO X ( ω ) ω ( Q ) N q − q , which gives (cid:18) ω ( Q ) Z Q | f ( y ) − f Q | q ω ( x ) − q dy (cid:19) /q ≤ (cid:16) q q − q (cid:17) /q k f k BMO X ( ω ) . Then k f k BMO q ( ω ) ≤ (cid:16) q q − q (cid:17) /q k f k BMO X ( ω ) and the lemma follows. (cid:3) Let ω ∈ A and dµ ( x ) = ω ( x ) dx . For 0 < r < ∞ , we set k f k BMO r ( ω ) = sup Q inf c n µ ( Q ) Z Q (cid:16) | f ( x ) − c | ω ( x ) (cid:17) r dµ ( x ) o /r , and BMO r ( ω ) = { f ∈ L loc : k f k BMO r ( ω ) < ∞} . Lemma 2.2.
Let < r < , ω ∈ A and dµ ( x ) = ω ( x ) dx . Suppose k f k BMO r ( ω ) = 1 and foreach cube Q let c Q be the value which minimizes Z Q (cid:16) | f ( x ) − c | ω ( x ) (cid:17) r dµ ( x ) . Then µ (cid:16)(cid:8) x ∈ Q : | f ( x ) − c Q | ω ( x ) > t (cid:9)(cid:17) ≤ c e − c t µ ( Q ) , where c and c are positive constants.Proof. Take any cube Q , write E Q = { x ∈ Q : | f ( x ) − c Q | ω ( x ) > t } . Then µ ( E Q ) ≤ Z E Q | f ( x ) − c Q | r t r ω ( x ) r dµ ( x ) ≤ t r µ ( Q ) µ ( Q ) Z Q | f ( x ) − c Q | r ω ( x ) r dµ ( x ) ≤ t r µ ( Q ) . Write F ( t ) = t r , then µ ( E Q ) ≤ F ( t ) µ ( Q ) . Let s > t ∈ (0 , ∞ ) such that 2 n +1 r s [ ω ] /rA ≤ t . Fix a cube Q , there is a Calderon-Zygmund decomposition of disjoint cubes { Q j } such that Q j ⊂ Q and( i ) s r < µ ( Q j ) Z Q j (cid:16) | f ( x ) − c Q | ω ( x ) (cid:17) r dµ ( x ) ≤ n s r , ( ii ) | f ( x ) − c Q | ω ( x ) ≤ s for x ∈ (cid:0) S j Q j (cid:1) c .Since ω ∈ A and x ∈ Q , then ω ( x ) ≤ [ ω ] A | Q | µ ( Q ) . By ( i ) and 0 < r < Z Q j (cid:16) | f ( y ) − c Q | ω ( y ) (cid:17) r ω ( y ) r dy = Z Q j (cid:16) | f ( y ) − c Q | ω ( y ) (cid:17) r ω ( y ) r − dµ ( y ) ≤ (cid:16) [ ω ] A | Q j | µ ( Q j ) (cid:17) − r Z Q j (cid:16) | f ( y ) − c Q | ω ( y ) (cid:17) r dµ ( y ) ≤ n s r [ ω ] − rA | Q j | − r µ ( Q j ) r . Notice that Z Q j (cid:16) | f ( x ) − c Q j | ω ( x ) (cid:17) r dµ ( x ) ≤ Z Q j (cid:16) | f ( x ) − c Q | ω ( x ) (cid:17) r dµ ( x ) , which implies that Z Q j (cid:16) | f ( y ) − c Q j | ω ( y ) (cid:17) r ω ( y ) r dy ≤ n s r [ ω ] − rA | Q j | − r µ ( Q j ) r . Therefore, (cid:16) | c Q j − c Q | ω ( x ) (cid:17) r = ω ( x ) − r | Q j | Z Q j | c Q j − c Q | r dy ≤ | Q j | ω ( x ) r Z Q j (cid:16) | f ( y ) − c Q j | ω ( y ) (cid:17) r ω ( y ) r dy + 1 | Q j | ω ( x ) r Z Q j (cid:16) | f ( y ) − c Q | ω ( y ) (cid:17) r ω ( y ) r dy ≤ n +1 s r [ ω ] − rA (cid:16) µ ( Q j ) | Q j | ω ( x ) (cid:17) r ≤ n +1 s r [ ω ] A . From the fact that [ ω ] A ≥
1, we have t > s , by ( i ) and ( ii ), we have µ ( E Q ) = X j µ (cid:0) { x ∈ Q j : | f ( x ) − c Q | ω ( x ) > t } (cid:1) ≤ X j µ (cid:0) { x ∈ Q j : | f ( x ) − c Q j | ω ( x ) + | c Q j − c Q | ω ( x ) > t } (cid:1) ≤ X j µ (cid:0) { x ∈ Q j : | f ( x ) − c Q j | ω ( x ) > t − n +1 r s [ ω ] /rA } (cid:1) ≤ X j F ( t − n +1 r s [ ω ] /rA ) · µ ( Q j ) ≤ F ( t − n +1 r s [ ω ] /rA ) X j s r Z Q j (cid:16) | f ( x ) − c Q | ω ( x ) (cid:17) r dµ ( x ) ≤ F ( t − n +1 r s [ ω ] /rA ) s r Z Q (cid:16) | f ( x ) − c Q | ω ( x ) (cid:17) r dµ ( x ) ≤ F ( t − n +1 r s [ ω ] /rA ) s r µ ( Q ) . Let F ( t ) = F ( t − n +1 r s [ ω ] /rA ) s r . Continue this process indefinitely, we obtain for any k ≥ F k ( t ) = F k − ( t − n +1 r s [ ω ] /rA ) s r , and µ ( E Q ) ≤ F k ( t ) µ ( Q ) . We fix a constant t >
0. If k · n +1 r [ ω ] /rA s < t ≤ ( k + 1) · n +1 r [ ω ] /rA s. for some k ≥
1, thus µ ( E Q ) ≤ µ (cid:0) { x ∈ Q : | f ( x ) − c Q | ω ( x ) > t } (cid:1) ≤ µ (cid:0) { x ∈ Q : | f ( x ) − c Q | ω ( x ) > k · n +1 r [ ω ] /rA s } (cid:1) ≤ F k ( k · n +1 r [ ω ] /rA s ) µ ( Q )= F (2 n +1 r [ ω ] /rA s ) s ( k − r µ ( Q ) = 12 n +1 [ ω ] A s kr µ ( Q ) ≤ e − kr log s n +1 [ ω ] A µ ( Q ) ≤ e r log s n +1 [ ω ] A exp (cid:16) − tr log s n +1 r [ ω ] /rA s (cid:17) µ ( Q )Since − k ≤ − t n +1 r [ ω ] /rA s . If t ≤ n +1 r [ ω ] /rA s , then use the trivial estimate µ ( E Q ) ≤ µ ( Q ) ≤ e − t e n +1 r [ ω ] /rA s µ ( Q ) . Recall that s is any real number larger than 1. Choosing s = e , this yields µ (cid:16)(cid:8) x ∈ Q : | f ( x ) − c Q | ω ( x ) > t (cid:9)(cid:17) ≤ c e − c t µ ( Q ) , for some positive constants c and c , which proves the inequality of the lemma 2.2. (cid:3) Lemma 2.3.
Let ω ∈ A and < r < . Then BMO r ( ω ) = BMO r ( ω ) . The norms are mutually equivalent.Proof.
By Lemma 2.2 and the homogeneity of k · k
BMO r ( ω ) , we obtain that for any f ∈ BMO r ( ω ), ω (cid:0) { x ∈ Q : | f ( x ) − c Q | ω ( x ) > t } (cid:1) ≤ c exp ( − c t/ k f k BMO r ( ω ) ) ω ( Q ) . This gives us1 ω ( Q ) Z Q | f ( x ) − c Q | dx = 1 ω ( Q ) Z ∞ ω (cid:0) { x ∈ Q : | f ( x ) − c Q | ω ( x ) > t } (cid:1) dt ≤ ω ( Q ) Z ∞ c exp ( − c t/ k f k BMO r ( ω ) ) ω ( Q ) dt ≤ C k f k BMO r ( ω ) . Therefore,1 ω ( Q ) Z Q (cid:16) | f ( x ) − f Q | ω ( x ) (cid:17) r ω ( x ) dx ≤ ω ( Q ) Z Q (cid:16) | f ( x ) − c Q | ω ( x ) (cid:17) r ω ( x ) dx + 1 ω ( Q ) Z Q (cid:16) | c Q − f Q | ω ( x ) (cid:17) r ω ( x ) dx ≤ k f k r BMO r ( ω ) + (cid:18) ω ( Q ) Z Q | f ( x ) − c Q | dx (cid:19) r ≤ C k f k r BMO r ( ω ) . Conversely, k · k
BMO r ( ω ) ≤ k · k BMO r ( ω ) is obvious. Thus, the equivalence of k · k BMO r ( ω ) and k · k BMO r ( ω ) is shown. (cid:3) Standard real analysis tools as the maximal function M ( f ), the weighted maximal function M ω ( f ) and the sharp maximal function M ♯ ( f ) carries over to this context, namely, M ( f )( x ) = sup Q ∋ x | Q | Z Q | f ( y ) | dy ; M ω ( f )( x ) = sup Q ∋ x ω ( Q ) Z Q | f ( y ) | ω ( y ) dy ; M ♯ ( f )( x ) = sup Q ∋ x inf c | Q | Z Q | f ( y ) − c | dy ≈ sup Q ∋ x | Q | Z Q | f ( y ) − f Q | dy. A variant of weighted maximal function and sharp maximal operator M ω,s ( f )( x ) = (cid:0) M ω ( f s ) (cid:1) /s and M ♯δ ( f )( x ) = (cid:0) M ♯ ( f δ )( x ) (cid:1) /δ , which will become the main tool in our scheme.The following relationships between M δ and M ♯δ to be used is a version of the classicalones due to Fefferman and Stein [5]. Lemma 2.4.
Let < p, δ < ∞ and ω ∈ A ∞ . There exists a positive C such that Z R n ( M δ f ( x )) p ω ( x ) dx ≤ C Z R n ( M ♯δ f ( x )) p ω ( x ) dx, for any smooth function f for which the left-hand side is finite. Lemma 2.5.
Let ω ∈ A and b ∈ BMO( ω ) . Then, there exists a constant C such that M ♯ (cid:0) [ b, T ] ( f , f ) (cid:1) ( x ) ≤ C k b k BMO( ω ) ω ( x ) M ( T ( f , f )( x ))+ C k b k BMO( ω ) ω ( x ) M ω,s ( f )( x ) M ( f )( x ) , (2.1) M ♯ (cid:0) [ b, T ] ( f , f ) (cid:1) ( x ) ≤ C k b k BMO( ω ) ω ( x ) M ( T ( f , f )( x ))+ C k b k BMO( ω ) ω ( x ) M ( f )( x ) M ω,s ( f )( x ) , (2.2) and M ♯ / (cid:0) [Π ~b, T ]( f , f ) (cid:1) ( x ) ≤ Cω ( x ) k b k ω ) M (cid:0) T ( f , f ) (cid:1) ( x )+ Cω ( x ) k b k BMO( ω ) M / (cid:0) [ b, T ] ( f , f ) (cid:1) ( x )+ Cω ( x ) k b k BMO( ω ) M / (cid:0) [ b, T ] ( f , f ) (cid:1) ( x )+ Cω ( x ) k b k ω ) M ω,s ( f )( x ) M ω,s ( f )( x ) , (2.3) for any < s < ∞ and bounded compact supported functions f , f .Proof. we only prove (2.1) and the proof of (2.2) and (2.3) are very similar to that of (2.1).Let Q := Q ( x , r ) be a cube and x ∈ Q . Then, (cid:18) | Q | Z Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ b, T ] ( f , f )( z ) (cid:12)(cid:12) / − | c | / (cid:12)(cid:12)(cid:12) dz (cid:19) ≤ C (cid:18) | Q | Z Q (cid:12)(cid:12) [ b, T ] ( f , f )( z ) − c (cid:12)(cid:12) / dz (cid:19) ≤ C (cid:18) | Q | Z Q (cid:12)(cid:12) ( b ( z ) − λ ) T ( f , f )( z ) (cid:12)(cid:12) / dz (cid:19) + (cid:18) | Q | Z Q (cid:12)(cid:12) T (( b − λ ) f , f )( z ) − c (cid:12)(cid:12) / dz (cid:19) =: A + A , where λ = b Q .We first consider the term A . By H¨older inequality, we obtain that A = (cid:18) | Q | Z Q (cid:12)(cid:12)(cid:12) ( b ( z ) − λ ) T ( f , f )( z ) (cid:12)(cid:12)(cid:12) / dz (cid:19) ≤ C k b k BMO( ω ) ω ( Q ) | Q | · | Q | Z Q (cid:12)(cid:12) T ( f , f )( z ) (cid:12)(cid:12) dz ≤ Cω ( x ) k b k BMO( ω ) M (cid:0) T ( f , f ) (cid:1) ( x ) . Let us consider next the term A . LetΩ = { ( y , y ) ∈ R n × R n : | x − y | + | x − y | ≤ √ nr } and for k ≥ k = { ( y , y ) ∈ R n × R n : 2 k +1 √ n ≥ | x − y | + | x − y | > k √ nr } . We write A ≤ (cid:18) | Q | Z Q (cid:12)(cid:12)(cid:12)(cid:12) Z Z Ω ( b ( y ) − λ ) K ( z − y , z − y ) f ( y ) f ( y ) dy dy (cid:12)(cid:12)(cid:12)(cid:12) / dz (cid:19) + (cid:18) | Q | Z Q (cid:12)(cid:12)(cid:12)(cid:12) Z Z R n × R n \ Ω ( b ( y ) − λ ) K ( z − y , z − y ) f ( y ) f ( y ) dy dy − c (cid:12)(cid:12)(cid:12)(cid:12) / dz (cid:19) =: A + A . It is obvious that Ω ⊂ √ nQ × √ nQ , we write f i = f i χ √ nQ . By Kolmogorov inequalityand the fact that T is bounded from L × L to L / , ∞ , we get A ≤ | Q | k T (( b − b Q ) f , f ) k L / , ∞ ≤ C | Q | Z √ nQ | b ( y ) − b Q || f ( y ) | dy Z √ nQ | f ( y ) | dy ≤ Cω ( x ) k b k BMO s ′ ( ω ) M ω,s ( f )( x ) M ( f )( x ) . Let c = 1 | Q | Z Q Z Z R n × R n \ Ω ( b ( y ) − λ ) K ( z ′ − y , z ′ − y ) f ( y ) f ( y ) dz ′ For any z, z ′ ∈ Q and y , y such that | x − y | + | x − y | > √ nr , then | z − z ′ | ≤ √ nr ≤
12 max {| x − y | , | x − y |} , which gives us that (cid:12)(cid:12)(cid:12) K ( z − y , z − y ) − K ( z ′ − y , z ′ − y ) (cid:12)(cid:12)(cid:12) ≤ C | z − z ′ | γ (cid:0) | z − y | + | z − y | (cid:1) n + γ . Therefore, A ≤ C ∞ X k =1 r γ | Q | Z Q Z Z Ω k | b ( y ) − λ || f ( y ) || f ( y ) | (cid:0) | z − y | + | z − y | (cid:1) n + γ dy dy dz ≤ C ∞ X k =1 (cid:16) kn (cid:17) γ · | k Q | Z k +1 √ nQ Z k +1 √ nQ | b ( y ) − λ || f ( y ) || f ( y ) | dy dy ≤ Cω ( x ) k b k BMO s ′ ( ω ) M ω,s ( f )( x ) M ( f )( x ) . Collecting our estimates, we have shown that M ♯ (cid:0) [ b, T ] ( f , f ) (cid:1) ( x ) ≤ C k b k BMO( ω ) ω ( x ) M ( T ( f , f ))( x )+ C k b k BMO( ω ) ω ( x ) M ω,s ( f )( x ) M ( f )( x )for any 1 < s < ∞ and bounded compact supported functions f , f . (cid:3) Proof of Theorem 1.1 ∼ Theorem 1.3
Proof of Theorem 1.1.
Let 1 < p < ∞ , ω ∈ A and X = L p, ∞ ( ω ). By Lemma 2.1, we have k · k BMO( ω ) ≤ C k · k BMO X ( ω ) . From the fact that BMO( ω ) = BMO p ( ω ) and k · k BMO X ( ω ) ≤ k · k BMO p ( ω ) , it follows that k · k BMO X ( ω ) ≤ C k · k BMO( ω ) . Thus we complete the proof of Theorem 1.1. (cid:3)
Proof of Theorem 1.2.
Let f ∈ BMO r ( ω ). In the proof of lemma 2.3, we have shown that1 ω ( Q ) Z Q | f ( x ) − c Q | dx ≤ C k f k BMO r ( ω ) . Therefore, 1 ω ( Q ) Z Q | f ( x ) − f Q | dx ≤ ω ( Q ) Z Q | f ( x ) − c Q | dx ≤ C k f k BMO r ( ω ) ≤ C k f k BMO r ( ω ) . As a result, k f k BMO( ω ) ≤ C k f k BMO r ( ω ) . The opposite inequality is a consequence of H¨olderinequality, then the equivalence of k f k BMO( ω ) and k f k BMO r ( ω ) is shown. (cid:3) Proof of Theorem 1.3. ( a ⇒ ( a ω ∈ A , then ω − p ∈ A ∞ . By Lemma 2.4 andLemma 2.5 with 1 < s < min { p , p } , from a standard argument that we can obtain k [Σ ~b, T ]( f , f ) ω − k L p ( ω ) = k [Σ ~b, T ]( f , f ) k L p ( ω − p ) ≤ k M (cid:0) [Σ ~b, T ]( f , f ) (cid:1) k L p ( ω − p ) ≤ C k M ♯ (cid:0) [Σ ~b, T ]( f , f ) (cid:1) k L p ( ω − p ) ≤ C k b k BMO( ω ) (cid:13)(cid:13) M (cid:0) T ( f , f ) (cid:1)(cid:13)(cid:13) L p ( ω ) + C k b k BMO( ω ) k M ( f )( x ) M ω,s ( f ) k L p ( ω ) ≤ C k b k BMO( ω ) 2 Y i =1 k f i k L pi ( ω ) . We observe that to use the Fefferman-Stein inequality, one needs to verify that certainterms in the left-hand side of the inequalities are finite. We can assume that f , f arebounded functions with compact support, applying a similar argument as in [7, pp.32-33]and Fatou’s lemma, one gets the desired result.( a ⇒ ( a
3) is obvious. ( a ⇒ ( a z ∈ R n such that | ( z , z ) | > √ n and let δ ∈ (0 ,
1) small enough.Take B = B (cid:0) ( z , z ) , δ √ n (cid:1) ⊂ R n be the ball for which we can express K as an absolutelyconvergent Fourier series of the form1 K ( y , y ) = X j a j e iv j · ( y ,y ) , ( y , y ) ∈ B, with P j | a j | < ∞ and we do not care about the vectors v j ∈ R n , but we will at timesexpress them as v j = ( v j , v j ) ∈ R n × R n . Set z = δ − z and note that (cid:0) | y − z | + | y − z | (cid:1) / < √ n ⇒ (cid:0) | δy − z | + | δy − z | (cid:1) / < δ √ n. Then for any ( y , y ) satisfying the inequality on the left, we have1 K ( y , y ) = δ − n K ( δy , δy ) = δ − n X j a j e iδv j · ( y ,y ) . Let Q = Q ( x , r ) be any arbitrary cube in R n . Set ˜ z = x + rz and take Q ′ = Q (˜ z, r ) ⊂ R n .So for any x ∈ Q and y , y ∈ Q ′ , we have (cid:12)(cid:12)(cid:12) x − y r − z (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) x − x r (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) y − ˜ zr (cid:12)(cid:12)(cid:12) ≤ √ n, (cid:12)(cid:12)(cid:12) x − y r − z (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) x − x r (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) y − ˜ zr (cid:12)(cid:12)(cid:12) ≤ √ n, which implies that (cid:18)(cid:12)(cid:12)(cid:12) x − y r − z (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) x − y r − z (cid:12)(cid:12)(cid:12) (cid:19) / ≤ √ n. Let s ( x ) = sgn( R Q ′ ( b ( x ) − b ( y )) dy ). Then | b ( x ) − b Q ′ | = s ( x ) (cid:0) b ( x ) − b Q ′ (cid:1) = s ( x ) | Q ′ | Z Q ′ Z Q ′ (cid:0) b ( x ) − b ( y ) (cid:1)(cid:1) dy dy (3.1)Setting g j ( y ) = e − i δr v j · y χ Q ′ ( y ) ,h j ( y ) = e − i δr v j · y χ Q ′ ( y ) ,m j ( x ) = e i δr v j · ( x,x ) χ Q ( x ) s ( x ) , which shows that | b ( x ) − b Q ′ | = s ( x ) r n δ − n | Q ′ | Z Q ′ Z Q ′ b ( x ) − b ( y ) (cid:0) | x − y | + | x − y | (cid:1) n − α/ × X j a j e i δr v j · ( x − y ,x − y ) dy dy = X j a j [ b, T ] ( g j , h j )( x ) m j ( x ) . If p >
1, we have the following estimate λω ( Q ) /p ω (cid:0) x ∈ Q : | b ( x ) − b Q ′ | ω ( x ) > λ (cid:1) /p = λω ( Q ) /p ω (cid:0) x ∈ Q : | b ( x ) − b Q ′ | ω ( x ) > λ (cid:1) /p ≤ λω ( Q ) /p ω (cid:0) x ∈ Q : P j | a j | (cid:12)(cid:12) [ b, T ] ( g j , h j )( x ) (cid:12)(cid:12) ω ( x ) > λ (cid:1) /p ≤ Cω ( Q ) /p X j | a j |k [ b, T ] ( g j , h j ) k L p, ∞ ( ω ) ≤ C X j | a j | . We write k b k BMO ∗ ( ω ) := sup Q sup λ> λω ( Q ) /p ω (cid:0) x ∈ Q : | b ( x ) − b Q ′ | ω ( x ) > λ (cid:1) /p , then k b k BMO ∗ ( ω ) ≤ C P j | a j | . The same estimate as lemma 2.1, we conclude that | b Q − b Q ′ | ≤ | Q | Z Q | b ( x ) − b Q ′ | dx ≤ ω ( Q ) | Q | k b k BMO ∗ ( ω ) ≤ C ω ( Q ) | Q | X j | a j | . By the definition of A weights, we concluded that ω ( Q ) ≤ | Q | ω ( x ), which implies that forany cube Q and λ > λω ( Q ) /p ω (cid:0) x ∈ Q : | b ( x ) − b Q | ω ( x ) > λ (cid:1) /p ≤ λω ( Q ) /p ω (cid:0) x ∈ Q : | b ( x ) − b Q ′ | ω ( x ) > λ (cid:1) /p + λω ( Q ) /p ω (cid:0) x ∈ Q : | b Q ′ − b Q | ω ( x ) > λ (cid:1) /p ≤ C X j | a j | + λω ( Q ) /p ω (cid:0) x ∈ Q : C P j | a j | ω ( Q ) | Q | ω ( x ) > λ (cid:1) /p ≤ C X j | a j | . This shows that b ∈ BMO X ( ω ) with X = L p, ∞ ; that is, the symbol b belongs to BMO( ω ).If p ≤
1, choose q ∈ (0 , p ). By the fact that L p, ∞ ( ω ) ⊂ M pq ( ω ) in [17, Corollary 2.3](see also [12, Lemma 1.7] for the unweighted case), M pq ( ω ) stands for the weighted Morreyspaces; that is, for 0 < q < p < ∞ , M pq ( ω ) = (cid:26) f ∈ L q loc : k f k M pq = sup Q ω ( Q ) /q − /p (cid:16) Z Q | f ( y ) | q ω ( y ) dy (cid:17) /q < ∞ (cid:27) . Therefore, inf c (cid:18) ω ( Q ) Z Q (cid:16) | b ( x ) − c | ω ( x ) (cid:17) q ω ( x ) dx (cid:19) /q ≤ (cid:18) ω ( Q ) Z Q (cid:16) | b ( x ) − b Q ′ | ω ( x ) (cid:17) q ω ( x ) dx (cid:19) /q ≤ (cid:18) Cω ( Q ) Z Q (cid:12)(cid:12)(cid:12) X j | a j | [ b, T ] ( g j , h j )( x ) ω ( x ) − (cid:12)(cid:12)(cid:12) q ω ( x ) dx (cid:19) /q ≤ Cω ( Q ) − /p X j | a j |k [ b, T ] ( g j , h j ) ω − k M pq ( ω ) ≤ Cω ( Q ) − /p X j | a j |k [ b, T ] ( g j , h j ) ω − k L p, ∞ ( ω ) ≤ C X j | a j | . Thus showing that b ∈ BMO q ( ω ). The desired result follows from here.By the inequality (2.3) in lemma 2.4 and the same argument as ( a ⇒ ( a a ⇒ ( a a ⇒ ( a a ⇒ ( a
1) follows themethod that of ( a ⇒ ( a
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