Characterization of Lipschitz spaces via commutators of the Hardy-Littlewood maximal function
aa r X i v : . [ m a t h . C A ] J a n Characterization of Lipschitz spaces via commutatorsof the Hardy-Littlewood maximal function ∗ Pu Zhang
Department of Mathematics, Mudanjiang Normal University, Mudanjiang 157011, P. R. ChinaE-mail: [email protected]
Abstract
Let M be the Hardy-Littlewood maximal function and b be a locally inte-grable function. Denote by M b and [ b, M ] the maximal commutator and the (nonlinear)commutator of M with b . In this paper, the author consider the boundedness of M b and[ b, M ] on Lebesgue spaces and Morrey spaces when b belongs to the Lipschitz space,by which some new characterizations of the Lipschitz spaces are given. Keywords
Hardy-Littlewood maximal function; commutator; Lipschitz space; Morreyspace.
MR(2010) Subject Classification
Let T be the classical singular integral operator, the commutator [ b, T ] generated by T anda suitable function b is given by [ b, T ] f = bT ( f ) − T ( bf ) . (1.1)A well known result due to Coifman, Rochberg and Weiss [6] (see also [13]) states that b ∈ BM O ( R n ) if and only if the commutator [ b, T ] is bounded on L p ( R n ) for 1 < p < ∞ . In 1978,Janson [13] gave some characterizations of the Lipschitz space ˙Λ β ( R n ) (see Definition 1.1 below)via commutator [ b, T ] and proved that b ∈ ˙Λ β ( R n )(0 < β <
1) if and only if [ b, T ] is boundedfrom L p ( R n ) to L q ( R n ) where 1 < p < n/β and 1 /p − /q = β/n (see also Paluszy´nski [18]).For a locally integrable function f , the Hardy-Littlewood maximal function M is given by M ( f )( x ) = sup Q ∋ x | Q | Z Q | f ( y ) | dy, ∗ Supported by the National Natural Science Foundation of China (Grant Nos. 11571160 and 11471176). Pu Zhang the maximal commutator of M with a locally integrable function b is defined by M b ( f )( x ) = sup Q ∋ x | Q | Z Q | b ( x ) − b ( y ) || f ( y ) | dy, where the supremum is taken over all cubes Q ⊂ R n containing x .The mapping properties of the maximal commutator M b have been studied intensively bymany authors. See [3], [9], [11], [12], [20], [21] and [26] for instance. The following result isproved by Garc´ıa-Cuerva et al. [9]. See also [20] and [21]. Theorem A ( [9])
Let b be a locally integrable function and < p < ∞ . Then the maximalcommutator M b is bounded from L p ( R n ) to L p ( R n ) if and only if b ∈ BM O ( R n ) . The first part of this paper is to study the boundedness of M b when the symbol b belongsto Lipschitz space. Some characterizations of Lipschitz space via such commutator are given. Definition 1.1
Let < β < , we say a function b belongs to the Lipschitz space ˙Λ β ( R n ) ifthere exists a constant C such that for all x, y ∈ R n , | b ( x ) − b ( y ) | ≤ C | x − y | β . The smallest such constant C is called the ˙Λ β norm of b and is denoted by k b k ˙Λ β . Our first result can be stated as follows.
Theorem 1.1
Let b be a locally integrable function and < β < , then the followingstatements are equivalent:(1) b ∈ ˙Λ β ( R n ) .(2) M b is bounded from L p ( R n ) to L q ( R n ) for all p, q with < p < n/β and /q = 1 /p − β/n .(3) M b is bounded from L p ( R n ) to L q ( R n ) for some p, q with < p < n/β and /q =1 /p − β/n .(4) M b satisfies the weak-type (1 , n/ ( n − β )) estimates, namely, there exists a positive con-stant C such that for all λ > , (cid:12)(cid:12) { x ∈ R n : M b ( f )( x ) > λ } (cid:12)(cid:12) ≤ C (cid:0) λ − k f k L ( R n ) (cid:1) n/ ( n − β ) . (1.2) (5) M b is bounded from L n/β ( R n ) to L ∞ ( R n ) . Morrey spaces were originally introduced by Morrey in [17] to study the local behaviorof solutions to second order elliptic partial differential equations. Many classical operators ofharmonic analysis were studied in Morrey type spaces during the last decades. We refer thereaders to Adams [2] and references therein. haracterization of the Lipschitz spaces via commutators of maximal function Definition 1.2
Let ≤ p < ∞ and ≤ λ ≤ n . The classical Morrey space is defined by L p,λ ( R n ) = (cid:8) f ∈ L p loc ( R n ) : k f k L p,λ < ∞ (cid:9) , where k f k L p,λ := sup Q (cid:18) | Q | λ/n Z Q | f ( x ) | p dx (cid:19) /p . It is well known that if 1 ≤ p < ∞ then L p, ( R n ) = L p ( R n ) and L p,n ( R n ) = L ∞ ( R n ). Theorem 1.2
Let b be a locally integrable function and < β < . Suppose that < p Let b be a locally integrable function and < β < . Suppose that < p 0. This operator can be used in studying the product of a function in H and a functionin BM O (see [5] for instance). In 2000, Bastero, Milman and Ruiz [4] studied the necessaryand sufficient conditions for the boundedness of [ b, M ] on L p spaces when 1 < p < ∞ . Zhangand Wu obtained similar results for the fractional maximal function in [25] and extended thementioned results to variable exponent Lebesgue spaces in [26] and [27]. Recently, Agcayazi etal. [3] gave the end-point estimates for the commutator [ b, M ]. Zhang [24] extended these resultsto the multilinear setting.We would like to remark that operators M b and [ b, M ] essentially differ from each other. Forexample, M b is positive and sublinear, but [ b, M ] is neither positive nor sublinear.The second part of this paper is to study the mapping properties of the (nonlinear) commu-tator [ b, M ] when b belongs to some Lipschitz space. To state our results, we recall the definitionof the maximal operator with respect to a cube. For a fixed cube Q , the Hardy-Littlewoodmaximal function with respect to Q of a function f is given by M Q ( f )( x ) = sup Q ⊇ Q ∋ x | Q | Z Q | f ( y ) | dy, where the supremum is taken over all the cubes Q with Q ⊆ Q and Q ∋ x . Pu Zhang Theorem 1.4 Let b be a locally integrable function and < β < . Suppose that < p Let b ≥ be a locally integrable function, < β < and b ∈ ˙Λ β ( R n ) . Thenthere is a positive constant C such that for all λ > , (cid:12)(cid:12) { x ∈ R n : | [ b, M ]( f )( x ) | > λ } (cid:12)(cid:12) ≤ C (cid:0) λ − k f k L ( R n ) (cid:1) n/ ( n − β ) . Theorem 1.6 Let b be a locally integrable function and < β < . Suppose that < p Let b be a locally integrable function and < β < . Suppose that < p For a measurable set E , we denote by | E | the Lebesgue measure and by χ E the characteristicfunction of E . For p ∈ [1 , ∞ ], we denote by p ′ the conjugate index of p , namely, p ′ = p/ ( p − f and a cube Q , we denote by f Q = ( f ) Q = | Q | R Q f ( x ) dx. To prove the theorems, we need some known results. It is known that the Lipschitz space˙Λ β ( R n ) coincides with some Morrey-Companato space (see [14] for example) and can be char-acterized by mean oscillation as the following lemma, which is due to DeVore and Sharpley [8]and Janson, Taibleson and Weiss [14] (see also Paluszy´nski [18]). haracterization of the Lipschitz spaces via commutators of maximal function Lemma 2.1 Let < β < and ≤ q < ∞ . Define ˙Λ β,q ( R n ) := (cid:26) f ∈ L ( R n ) : k f k ˙Λ β,q = sup Q | Q | β/n (cid:18) | Q | Z Q | f ( x ) − f Q | q dx (cid:19) /q < ∞ (cid:27) . Then, for all < β < and ≤ q < ∞ , ˙Λ β ( R n ) = ˙Λ β,q ( R n ) with equivalent norms. Let 0 < α < n and f be a locally integrable function, the fractional maximal function of f is given by M α ( f )( x ) = sup Q | Q | − α/n Z Q | f ( y ) | dy where the supremum is taken over all cubes Q ⊂ R n containing x .The following strong and weak-type boundedness of M α are well-known, see [10] and [7]. Lemma 2.2 Let < α < n , ≤ p ≤ n/α and /q = 1 /p − α/n .(1) If < p < n/α then there exists a positive constant C ( n, α, p ) such that k M α ( f ) k L q ( R n ) ≤ C ( n, α, p ) k f k L p ( R n ) . (2) If p = n/α then there exists a positive constant C ( n, α ) such that k M α ( f ) k L ∞ ( R n ) ≤ C ( n, α ) k f k L n/α ( R n ) . (3) If p = 1 then there exists a positive constant C ( n, α ) such that for all λ > (cid:12)(cid:12)(cid:8) x ∈ R n : M α ( f )( x ) > λ (cid:9)(cid:12)(cid:12) ≤ C ( n, α ) (cid:0) λ − k f k L ( R n ) (cid:1) n/ ( n − α ) . Spanne (see [19]) and Adams [1] studied the boundedness of the fractional integral I α inclassical Morrey spaces. We note that the fractional maximal function enjoys the same bound-edness as that of the fractional integral since the pointwise inequality M α ( f )( x ) ≤ I α ( | f | )( x ).These results can be summarized as follows (see also [22]): Lemma 2.3 Let < α < n , < p < n/α and < λ < n − αp . (1) If /q = 1 /p − α/ ( n − λ ) then there is a constant C > such that k M α ( f ) k L q,λ ( R n ) ≤ C k f k L p,λ ( R n ) for every f ∈ L p,λ ( R n ) . (2) If /q = 1 /p − α/n and λ/p = µ/q . Then there is a constant C > such that k M α ( f ) k L q,µ ( R n ) ≤ C k f k L p,λ ( R n ) for every f ∈ L p,λ ( R n ) . Lemma 2.4 ( [15]) Let ≤ p < ∞ and < λ < n , then there is a constant C > thatdepends only on n such that k χ Q k L p,λ ( R n ) ≤ C | Q | n − λnp . Pu Zhang Proof of Theorem 1.1 If b ∈ ˙Λ β ( R n ), then M b ( f )( x ) = sup Q ∋ x | Q | Z Q | b ( x ) − b ( y ) || f ( y ) | dy ≤ C k b k ˙Λ β sup Q ∋ x | Q | − β/n Z Q | f ( y ) | dy = C k b k ˙Λ β M β ( f )( x ) . (3.1)Obviously, (2), (3), (4) and (5) follow from Lemma 2.2, Lemma 2.3 and (3.1).(3) = ⇒ (1): Assume M b is bounded from L p ( R n ) to L q ( R n ) for some p, q with 1 < p < n/β and 1 /q = 1 /p − β/n . For any cube Q ⊂ R n , by H¨older’s inequality and noting that 1 /p + 1 /q ′ =1 + β/n , one gets1 | Q | β/n Z Q | b ( x ) − b Q | dx ≤ | Q | β/n Z Q (cid:18) | Q | Z Q | b ( x ) − b ( y ) | dy (cid:19) dx = 1 | Q | β/n Z Q (cid:18) | Q | Z Q | b ( x ) − b ( y ) | χ Q ( y ) dy (cid:19) dx ≤ | Q | β/n Z Q M b ( χ Q )( x ) dx ≤ | Q | β/n (cid:18) Z Q [ M b ( χ Q )( x )] q dx (cid:19) /q (cid:18) Z Q χ Q ( x ) dx (cid:19) /q ′ ≤ C | Q | β/n k M b k L p → L q k χ Q k L p ( R n ) k χ Q k L q ′ ( R n ) ≤ C k M b k L p → L q . This together with Lemma 2.1 gives b ∈ ˙Λ β ( R n ).(4) = ⇒ (1): We assume (1.2) is true and will verify b ∈ ˙Λ β ( R n ). For any fixed cube Q ⊂ R n ,since for any x ∈ Q , | b ( x ) − b Q | ≤ | Q | Z Q | b ( x ) − b ( y ) | dy, then, for all x ∈ Q , M b ( χ Q )( x ) = sup Q ∋ x | Q | Z Q | b ( x ) − b ( y ) | χ Q ( y ) dy ≥ | Q | Z Q | b ( x ) − b ( y ) | χ Q ( y ) dy = 1 | Q | Z Q | b ( x ) − b ( y ) | dy ≥ | b ( x ) − b Q | . haracterization of the Lipschitz spaces via commutators of maximal function (cid:12)(cid:12)(cid:8) x ∈ Q : | b ( x ) − b Q | > λ (cid:9)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:8) x ∈ Q : M b ( χ Q )( x ) > λ (cid:9)(cid:12)(cid:12) ≤ C (cid:0) λ − k χ Q k L ( R n ) (cid:1) n/ ( n − β ) = C (cid:0) λ − | Q | (cid:1) n/ ( n − β ) . Let t > Z Q | b ( x ) − b Q | dx = Z ∞ (cid:12)(cid:12)(cid:8) x ∈ Q : | b ( x ) − b Q | > λ (cid:9)(cid:12)(cid:12) dλ = Z t (cid:12)(cid:12)(cid:8) x ∈ Q : | b ( x ) − b Q | > λ (cid:9)(cid:12)(cid:12) dλ + Z ∞ t (cid:12)(cid:12)(cid:8) x ∈ Q : | b ( x ) − b Q | > λ (cid:9)(cid:12)(cid:12) dλ ≤ t | Q | + C Z ∞ t (cid:0) λ − | Q | (cid:1) n/ ( n − β ) dλ ≤ t | Q | + C | Q | n/ ( n − β ) Z ∞ t λ − n/ ( n − β ) dλ ≤ C ( n, β ) (cid:0) t | Q | + | Q | n/ ( n − β ) t − n/ ( n − β ) (cid:1) . Set t = | Q | β/n in the above estimate, we have Z Q | b ( x ) − b Q | dx ≤ C | Q | β/n . It follows from Lemma 2.1 that b ∈ ˙Λ β ( R n ) since Q is an arbitrary cube in R n .(5) = ⇒ (1): If M b is bounded from L n/β ( R n ) to L ∞ ( R n ), then for any cube Q ⊂ R n ,1 | Q | β/n Z Q | b ( x ) − b Q | dx ≤ | Q | β/n Z Q (cid:18) | Q | Z Q | b ( x ) − b ( y ) | χ Q ( y ) dy (cid:19) dx ≤ | Q | β/n Z Q M b ( χ Q )( x ) dx ≤ | Q | β/n k M b ( χ Q ) k L ∞ ( R n ) ≤ C | Q | β/n k M b k L n/β → L ∞ k χ Q k L n/β ( R n ) ≤ C k M b k L n/β → L ∞ . This together with Lemma 2.1 gives b ∈ ˙Λ β ( R n ).The proof of Theorem 1.1 is completed since (2) = ⇒ (1) follows from (3) = ⇒ (1). (cid:3) Proof of Theorem 1.2 Assume b ∈ ˙Λ β ( R n ). By (3.1) and Lemma 2.3 (1), we have k M b ( f ) k L q,λ ≤ k b k ˙Λ β k M β ( f ) k L q,λ ≤ C k b k ˙Λ β k f k L p,λ . Pu Zhang Conversely, if M b is bounded from L p,λ ( R n ) to L q,λ ( R n ), then for any cube Q ⊂ R n ,1 | Q | β/n (cid:18) | Q | Z Q | b ( x ) − b Q | q dx (cid:19) /q ≤ | Q | β/n (cid:18) | Q | Z Q (cid:20) | Q | Z Q | b ( x ) − b ( y ) | χ Q ( y ) dy (cid:21) q dx (cid:19) /q ≤ | Q | β/n (cid:18) | Q | Z Q [ M b ( χ Q )( x )] q dx (cid:19) /q = 1 | Q | β/n (cid:18) | Q | λ/n | Q | (cid:19) /q (cid:18) | Q | λ/n Z Q [ M b ( χ Q )( x )] q dx (cid:19) /q ≤ | Q | − β/n − /q + λ/ ( nq ) k M b ( χ Q ) k L q,λ ( R n ) ≤ C | Q | − β/n − /q + λ/ ( nq ) k M b k L p,λ → L q,λ k χ Q k L p,λ ( R n ) ≤ C k M b k L p,λ → L q,λ , where in the last step we have used 1 /q = 1 /p − β/ ( n − λ ) and Lemma 2.4.It follows from Lemma 2.1 that b ∈ ˙Λ β ( R n ). This completes the proof. (cid:3) Proof of Theorem 1.3 By a similar proof to the one of Theorem 1.2, we can obtainTheorem 1.3. (cid:3) Proof of Theorem 1.4 (1) = ⇒ (2): For any fixed x ∈ R n such that M ( f )( x ) < ∞ , since b ≥ | [ b, M ]( f )( x ) | = | b ( x ) M ( f )( x ) − M ( bf )( x ) | = (cid:12)(cid:12)(cid:12)(cid:12) sup Q ∋ x | Q | Z Q b ( x ) | f ( y ) | dy − sup Q ∋ x | Q | Z Q b ( y ) | f ( y ) | dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ sup Q ∋ x | Q | Z Q | b ( x ) − b ( y ) || f ( y ) | dy = M b ( f )( x ) . (4.1)It follows from Theorem 1.1 that [ b, M ] is bounded from L p ( R n ) to L q ( R n ) since b ∈ ˙Λ β ( R n ).(2) = ⇒ (3): For any fixed cube Q ⊂ R n and all x ∈ Q , we have (see the proof of Proposition4.1 in [4], see also (2.4) in [25]) M ( χ Q )( x ) = χ Q ( x ) and M ( bχ Q )( x ) = M Q ( b )( x ) . haracterization of the Lipschitz spaces via commutators of maximal function | Q | β/n (cid:18) | Q | Z Q (cid:12)(cid:12) b ( x ) − M Q ( b )( x ) (cid:12)(cid:12) q dx (cid:19) /q = 1 | Q | β/n (cid:18) | Q | Z Q (cid:12)(cid:12) b ( x ) M ( χ Q )( x ) − M Q ( bχ Q )( x ) (cid:12)(cid:12) q dx (cid:19) /q = 1 | Q | β/n (cid:18) | Q | Z Q (cid:12)(cid:12) [ b, M ]( χ Q )( x ) (cid:12)(cid:12) q dx (cid:19) /q ≤ | Q | β/n +1 /q (cid:13)(cid:13) [ b, M ]( χ Q ) (cid:13)(cid:13) L q ( R n ) ≤ C | Q | β/n +1 /q (cid:13)(cid:13) χ Q (cid:13)(cid:13) L p ( R n ) ≤ C, (4.2)which implies (3) since the cube Q ⊂ R n is arbitrary.(3) = ⇒ (1): To prove b ∈ ˙Λ β ( R n ), by Lemma 2.1, it suffices to verify that there is a constant C > Q , 1 | Q | β/n Z Q | b ( x ) − b Q | dx ≤ C. (4.3)For any fixed cube Q , let E = { x ∈ Q : b ( x ) ≤ b Q } and F = { x ∈ Q : b ( x ) > b Q } . Thefollowing equality is trivially true (see [4] page 3331): Z E | b ( x ) − b Q | dx = Z F | b ( x ) − b Q | dx. Since for any x ∈ E we have b ( x ) ≤ b Q ≤ M Q ( b )( x ), then for any x ∈ E , | b ( x ) − b Q | ≤ | b ( x ) − M Q ( b )( x ) | . Thus, 1 | Q | β/n Z Q | b ( x ) − b Q | dx = 1 | Q | β/n Z E ∪ F | b ( x ) − b Q | dx = 2 | Q | β/n Z E | b ( x ) − b Q | dx ≤ | Q | β/n Z E | b ( x ) − M Q ( b )( x ) | dx ≤ | Q | β/n Z Q | b ( x ) − M Q ( b )( x ) | dx. (4.4)On the other hand, it follows from H¨older’s inequality and (1.3) that1 | Q | β/n Z Q (cid:12)(cid:12) b ( x ) − M Q ( b )( x ) (cid:12)(cid:12) dx ≤ | Q | β/n (cid:18) Z Q (cid:12)(cid:12) b ( x ) − M Q ( b )( x ) (cid:12)(cid:12) q dx (cid:19) /q | Q | /q ′ ≤ | Q | β/n (cid:18) | Q | Z Q (cid:12)(cid:12) b ( x ) − M Q ( b )( x ) (cid:12)(cid:12) q dx (cid:19) /q ≤ C. Pu Zhang this together with (4.4) gives (4.3) and so we achieve b ∈ ˙Λ β ( R n ).In order to prove b ≥ 0, it suffices to show b − = 0, where b − = − min { b, } . Let b + = | b |− b − ,then b = b + − b − . For any fixed cube Q , observe that0 ≤ b + ( x ) ≤ | b ( x ) | ≤ M Q ( b )( x )for x ∈ Q and therefore we have that, for x ∈ Q ,0 ≤ b − ( x ) ≤ M Q ( b )( x ) − b + ( x ) + b − ( x ) = M Q ( b )( x ) − b ( x ) . Then, it follows from (1.3) that, for any cube Q ,1 | Q | Z Q b − ( x ) dx ≤ | Q | Z Q | M Q ( b )( x ) − b ( x ) |≤ (cid:18) | Q | Z Q | b ( x ) − M Q ( b )( x ) | q dx (cid:19) /q = | Q | β/n (cid:26) | Q | β/n (cid:18) | Q | Z Q | b ( x ) − M Q ( b )( x ) | q dx (cid:19) /q (cid:27) ≤ C | Q | β/n . Thus, b − = 0 follows from Lebesgue’s differentiation theorem.The proof of Theorem 1.4 is completed. (cid:3) Proof of Theorem 1.5 Obviously, Theorem 1.5 follows from (4.1) and Theorem 1.1. (cid:3) Proof of Theorem 1.6 (1) = ⇒ (2): Assume b ≥ b ∈ ˙Λ β ( R n ), then by (4.1) andTheorem 1.2 we see that [ b, M ] is bounded from L p,λ ( R n ) to L q,λ ( R n ).(2) = ⇒ (1): Assume that [ b, M ] is bounded from L p,λ ( R n ) to L q,λ ( R n ). Similarly to (4.2),we have, for any cube Q ⊂ R n ,1 | Q | β/n (cid:18) | Q | Z Q (cid:12)(cid:12) b ( x ) − M Q ( b )( x ) (cid:12)(cid:12) q dx (cid:19) /q = 1 | Q | β/n (cid:18) | Q | Z Q (cid:12)(cid:12) [ b, M ]( χ Q )( x ) (cid:12)(cid:12) q dx (cid:19) /q ≤ | Q | λ/ ( nq ) | Q | β/n +1 /q (cid:13)(cid:13) [ b, M ]( χ Q ) (cid:13)(cid:13) L q,λ ( R n ) ≤ C | Q | λ/ ( nq ) | Q | β/n +1 /q (cid:13)(cid:13) χ Q (cid:13)(cid:13) L p,λ ( R n ) ≤ C, where in the last step we have used 1 /q = 1 /p − β/ ( n − λ ) and Lemma 2.4.This shows by Theorem 1.4 that b ∈ ˙Λ β ( R n ) and b ≥ (cid:3) Proof of Theorem 1.7 By the same way of the proof of Theorem 1.6, Theorem 1.7 canbe proven. We omit the details. 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