TWO WEIGHT INEQUALITIES FOR HARDY OPERATOR AND COMMUTATORS

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1 Journal of Mathematical Inequalities Volume 9, Number 3 (215), doi:1.7153/jmi-9-55 TWO WEIGHT INEQUALITIES FOR HARDY OPERATOR AND COMMUTATORS WENMING LI, TINGTING ZHANG AND LIMEI XUE (Communicated by J. Pečarić) Abstract. For the maximal oerator N related to the Hardy oerator P and its adjoint Q, we give the characterizations for weights (u,v) such that N is bounded from L (v) to L, (u) and from L (v) to L (u) resectively. We also obtain some A tye conditions which are sufficient for the two-weight inequalities for the Hardy oerator P, the adjoint oerator Q and the commutators of these oerators with CMO functions. and 1. Introduction Let P and Q be the Hardy oerator and its adjoint on (, ), Pf(x)= 1 x x f (y)dy, Qf(x)= x Hardy [8, 9] established the Hardy integral inequalities f (y) y dy. Pf(y) dy f (y) dy, > 1, Qf(y) dy f (y) dy, > 1, where = /( 1). The two inequalities above go by the name of Hardy s integral inequalities. For the earlier develoment of this kind of inequality and many alications in analysis, see [1, 15]. The Calderón oerator S is defined as S = P + Q and lays a significant role in the theory of real interolation, see [1]. Duoandikoetxea, Martin-Reyes and Ombrosi in [7] introduced the maximal oerator N related to the Calderón oerator and obtained a Mathematics subject classification (21): 42B25, 4A3. Keywords and hrases: Hardy oerator, Calderón oerator, commutator, two-weight inequality, maximal function. Suorted by the Natural Science Foundation of Hebei Province (A , A215434) and Postdoctoral Science Foundation of Hebei Province (B21337). c D l,zagreb Paer JMI

2 654 W. LI, T.ZHANG AND L. XUE new characterization for the weighted inequalities on S. Given a measurable function f on (,), the maximal oerator N is defined as Nf(x)=su t>x 1 t f (y) dy. t Notice that Nf is a decreasing function, and that Pf Nf S( f ) for any f. Let 1 <, wesayb is a one-side dyadic CMO function, if su j Z ( 1 2 j 1/ 2 j b(y) b (,2 j ] dy) = b CMO q <, where b (,2 j ] = 1 2 j 2 j f (x)dx, we then say that b CMO. It is easy to see BMO(,) MO, where 1 <. CMO q MO for 1 < q <. Let b be a locally integrable function on (,),wedefine the commutators of the Calderón oerator S with b as S b = P b + Q b,where P b f (x)= 1 x x (b(x) b(y)) f (y)dy, Q b f (x)= x (b(x) b(y)) f (y) dy. y LongandWangin[12] established the Hardy s integral inequalities for commutators generated by P and Q with one-sided dyadic CMO functions. For oerator T such as N, P, Q, S and S b, it is natural to consider the roblem of characterizing the airs (u,v) of nonnegative measurable functions such that ( ) 1/q ( 1/ T f(y) q u(y)dy f (y) v(y)dy) (1.1) holds with a ositive constant C indeendent of f,where<,q <. For > 1, Muckenhout in [14] roved that P is bounded from L (v) to L (u) if and only if there exists C > such that for all t > it holds that ( t u(y) y dy ) 1/( t ) 1/ v 1 (y)dy. Bradley [2] and Maz ya [13] obtained the similar results for the case 1 < q <. For oerator Q, they also obtained the similar results. For 1 < <, we say a air of weights (u,v) satisfies the two-weight A, condition, denoted (u,v) A,,if ( 1 t )( 1 t ) / [u,v] = su u(y)dy v(y) / dy <. t> t t For = 1, we write (u,v) A 1,, for the class of nonnegative functions such that Nu(y) v(y), a.e. and [u,v] 1 denotes the constant for which the inequality holds.

3 TWO-WEIGHT INEQUALITIES FOR HARDY OPERATOR AND COMMUTATORS 655 When u = v, the classes of A, weights were introduced by Duoandikoetxea, Martin-Reyes and Ombrosi in [7]. They roved that the Calderón oerator S are bounded on L (w) if and only if w A, when > 1. For N, the result is same. In this aer, we obtain the characterizations for weights (u,v) such that N is bounded from L (v) to L, (u) and from L (v) to L (u) resectively. We also give some A tye conditions which are sufficient for the two-weight strong (, ) inequalities for the oerators P, P b, Q and Q b. Our conditions differ from the conditions in Muckenhout in [14], Bradley [2] and Maz ya [13]. THEOREM 1.1. For 1 <, N is bounded from L (v) to L, (u) if and only if (u,v) A,. More recisely, su λ u({x : Nf(x) > λ }) 1/ [u,v] 1/ f L (v). λ > THEOREM 1.2. For 1 < <, < q <, N is bounded from L (v) to L q (u) if and only if for any t >, (u,v) satisfies ( t ) 1/q ( t 1/ [N(v 1 χ (,b) )(y)] q u(y)dy v(y) dy) 1 <. (1.2) But for 1 < <, N is not bounded from L (v) to L (u) if (u,v) A,,the roof is same as the case for the Hardy-Littlewood maximal function on R n,see[6]. Notice that Pf Nf S( f ), by Theorem 1.1, we have that (u,v) A, is necessary but not sufficient for S is bounded from L (v) to L (u). THEOREM 1.3. Let 1 < <. (1) If (u,v) is a air of weights for which there exists r > 1 such that, for every t >, ( 1 t )( 1 t ) /r u(y)dy v(y) r / dy <. (1.3) t t Then Pf(x) f (x) v(x)dx. (1.4) (2) If (u,v) is a air of weights for which there exists r > 1 such that, for every t >, ( 1 t ) 1/r ( 1 t ) / u(y) r dy v(y) / dy <. (1.5) t t Then Qf(x) f (x) v(x)dx. (1.6) THEOREM 1.4. Let 1 < <, b CMO r max{, } and (u,v) be a air of weights for which there exists r > 1 such that for every t >, ( 1 t t ) 1/r ( 1 t ) /r u(y) r dy v(y) r / dy <, (1.7) t

4 656 W. LI, T.ZHANG AND L. XUE then and P b f (x) f (x) v(x)dx, (1.8) Q b f (x) f (x) v(x)dx. (1.9) 2. The Proofs of Theorem 1.1 and Theorem 1.2 In the roofs of Theorem 1.1 and Theorem 1.2, we need the maximal oerator N g associated to a fixed ositive measurable function g. Wedefined N g as t N g f (x)=su f (y) g(y)dy t t>x g(y)dy. THEOREM 2.1. [7] Let g be a nonnegative measurable function such that < g(,b)= b g(y)dy < for all b >. (i) N g is of weak tye (1,1) with resect to the measure g(t)dt. Actually, {x:n g f (x)>λ } g(y)dy 1 λ {x:n g f (x)>λ } f (y) g(y)dy for all λ > and all measurable functions f. (ii) N g is of strong tye (, ), 1 < <, with resect to the measure g(t)dt. More recisely, N g f (y) g(y)dy ( ) f (y) g(y)dy. Proof of Theorem 1.1. For 1 <, the roof for the necessity of A, weights is standard, we omitted here. For sufficiency, we observe that Nf is decreasing and continuous. Therefore, if {x : Nf(x) > λ } is not emty, then it is a bounded interval (,d), thus d dλ = f (y) dy. Then ( d ) 1/ λ u({x : Nf(x) > λ }) 1/ = λ u(y)dy 1 ( d ) 1/ ( d ) 1/ ( d ) 1/ f (y) v(y)dy u(y)dy v(y) / dy d [u,v] 1/ This ends the roof. ( d 1/. f (y) v(y)dy)

5 TWO-WEIGHT INEQUALITIES FOR HARDY OPERATOR AND COMMUTATORS 657 Proof of Theorem 1.2. Denote σ = v 1. The necessity of (1.2) follows by a standard argument if we substitute f = σχ (,b) into Nf L q (u) f L (v). To show that (1.2) is sufficient, fix a bounded nonnegative function f with comact suort. Since Nf is decreasing and continuous, for each k Z, if{x (,) : Nf(x) > } is not emty, then there exists d k such that {x (,) : Nf(x) > } = (,d k ). Thus < d k+1 d k, Ω k = {x (,) : < Nf(x) +1 } =[d k+1,d k ) and dk d k = f (y)dy. Fix a large integer K >, which will go to infinity later, and let Λ K = {k Z : k K}. Wehave I K = K k= K Ω k (Nf(y)) q u(y)dy = 2 q K dk u(y)dy k= K d k+1 dk = 2 q K u(y)dy k= K d k+1 = 2 q T K ( f σ 1 ) q dν, Z K k= K ( 1 dk f (y)dy d k ( 1 d k dk where ν is the measure on Z given by 2 (k+1)q dk ) q d k+1 u(y)dy ) q ( d k σ(y)dy ( f σ 1 )(y)σ(y)dy dk σ(y)dy dk ( 1 dk q, ν(k)= u(y)dy σ(y)dy) d k+1 d k and, for every measurable function h, the oerator T K is defined by dk T K h(k)= h(y)σ(y)dy dk σ(y)dy χ ΛK (k). If we rove that T K is uniformly bounded from L ((,),σ) to L q (Z,ν) indeendently of K, we shall obtain I K ( T K ( f σ 1 ) q dν Z ( q/. = C f (y) v(y)dy) ) q/ [( f σ 1 )(y)] σ(y)dy The uniformity in K of this estimate and the monotone convergence theorem will lead to the desired inequality. Now we rove that T K is a bounded oerator from L ((,),σ) to L q (Z,ν). It is clear that T K : L ((,),σ) L (Z,ν) with constant less than or equal 1. The Marcinkiewicz interolation theorem says that it is enough to rove the uniform boundedness of the oerators T K from L 1 ((,),σ) to L q/, (Z,ν). For this, fix h a ) q

6 658 W. LI, T.ZHANG AND L. XUE bounded function with comact suort and ut F λ = {k Z : T K h(k) > λ } = { k K : T K h(k) > λ }, and k = min{k : k F λ }.Using(1.2),wehave ν(f λ )= dk k F λ dk k F λ dk k F λ dk d k+1 ( 1 d k dk σ(y)dy) q d k+1 (N(σχ (,dk ))(x) q d k+1 (N(σχ (,dk ))(x) q (N(σχ (,dk ))(x) q ( dk ) q/ h(y)σ(y)dy ( 1 λ ( 1 λ dk ) q/ h(y)σ(y)σ(y)dy h(y)σ(y)dy) q/, where the constant C does not deend on K. This ends the roof. 2 j+1 1 x 2 j x 3. The Proofs of Theorem 1.3 and Theorem 1.4 LEMMA 3.1. [12] Let b CMO 1,j,k Z, then b(t) b (,2 j+1 ] b(t) b (,+1 ] + 2 j k b CMO 1. Proof of Theorem 1.3. We first rove (1.4). By Hölder inequality and condition (1.3), we have Pf(x) = f (y)dy 2 j+1 2 j 1 2 j ( 1 2 j j k= 1 2 j+1 2 j ( j k= j +1 k= ( +1 2 j f (y) dy f (y) v(y)dy ) 1/ ( +1 v(y) / dy ) 1/ ) 2 j+1 2 j ( +1 ) 1/ ( +1 ) 1/r ( +1 ) 1/r f (y) v(y)dy v(y) r / ) dy dy

7 TWO-WEIGHT INEQUALITIES FOR HARDY OPERATOR AND COMMUTATORS 659 ( j (k j) ( +1 2 r f (y) v(y)dy ) ) 1/ k= ( j (k j) ) / j (k j) r 2 2r f (y) v(y)dy k= k= f (y) v(y)dy. Now we rove (1.6). By Hölder inequality and condition (1.5), we have = Qf(x) 2 j+1 2 j x 2 j+1 2 j 2 j+1 2 j f (y) y dy f (y) dy ( ) 2k+1 1/ ( +1 ) 1/ ) f (y) v(y)dy v(y) / dy ( ( 2 j+1 ) 1/r2 u(x) r j dx r 2 j ( 1 ( ) 2k+1 1/ ( +1 ) 1/ ) f (y) v(y)dy v(y) / dy 2 ( j k ( +1 r f (y) v(y)dy ) ) 1/ ( 2 ( j k) ) / 2r 2 ( j k) +1 2r f (y) v(y)dy f (y) v(y)dy. This ends the roof. Proof of Theorem 1.4. We first rove (1.8). = P b f (x) (b(x) b(y)) f (y)dy 2 j+1 1 x 2 j x

8 66 W. LI, T.ZHANG AND L. XUE 2 j+1 j j 2 j (b(x) b(y)) f (y) dy k= 2 j / 2 j 2 j +2 / = I + II. 2 j+1 2 j 1 2 j j k= j k= (b(x) b (,2 j+1 ] ) f (y) dy +1 (b(y) b (,2 j+1 ] ) f (y) dy For I, by Hölder inequality and condition (1.7), we have I = 2 / 2 / 1 2 j+1 2 j 1 ( j ( +1 k= b CMO r ( j k= b CMO r b CMO r 2 j b(x) b (,2 j+1 ] ( j +1 k= ( 2 j+1 ) 1/r ( 2 j+1 2 j b(x) b (,2 j+1 ] r dx f (y) v(y)dy ) 1/ ( 2 ( j+1) ( 1 2 j 2 j+1 2 j+1 +1 f (y) dy v(y) r / dy ) 1/r ( u(x) r dx ) 1/r 2 ( j+1) r + (k+1) ( +1 r f (y) v(y)dy ) 1/ ( 1 2 j+1 ( j k= b f (y) v(y)dy. CMO r 2 j+1 (k j) ( +1 2 r f (y) v(y)dy ) ) 1/ ) ) 1/r u(x) r dx +1 ( j (k j) ) / j (k j) r 2 2r f (y) v(y)dy k= k= dy ) 1/r ) v(y) r/ dy ) 1/r ) For II, by Lemma 3.1, we have II = 2 / 1 2 j+1 j +1 2 j 2 j k= 2 j / = II 1 + II j 2 j j k= (b(y) b (,+1 ] ) f (y) dy 2( j k)c b CMO 1 f (y) dy

9 TWO-WEIGHT INEQUALITIES FOR HARDY OPERATOR AND COMMUTATORS 661 For II 1,byHölder inequality and condition (1.7), we have II 1 = 2 / 1 2 j+1 2 j 2 j ( j k= ( +1 b(y) b (,+1 ] r dy ) 1/r ( +1 ) 1/ ( +1 ) 1/r ) f (y) v(y)dy v(y) r / dy b 2 CMO r ( j+1) ( j 2 j 2 (k+1) r + ( j+1) ( +1 ) 1/ ) r f (y) v(y)dy k= b CMO r f (y) v(y)dy. For II 2,wehave II 2 = C b CMO j+1 2 j 2 j ( j k= ( +1 ( j k) ( +1 ) 1/r ( +1 ) 1/r v(y) r / ) dy dy b CMO r b CMO r f (y) v(y)dy. f (y) v(y)dy ( j (k j) ( +1 ( j k)2 r f (y) v(y)dy k= ) 1/ ) ) 1/ Now we rove (1.9). We have = Q b f (x) 2 j+1 2 j x 2 j+1 2 j 2 j+1 2 / +2 / = J + JJ. (b(x) b(y)) f (y) y j 2 j+1 2 j dy (b(x) b(y)) f (y) dy 1 +1 (b(x) b (,2 j+1 ] ) f (y) dy 1 +1 (b(y) b (,2 j+1 ] ) f (y) dy

10 662 W. LI, T.ZHANG AND L. XUE For J, by Hölder inequality and condition (1.7), we have J = 2 / 2 / 2 j+1 b(x) b (,2 2 j j+1 ] ( ( 2 j+1 ) 1/r ( 2 j+1 b(x) b (,2 j+1 ] r dx ( 1 ( 2k+1 f (y) v(y)dy ) 1/( +1 b CMO 2 j/r ( 1 +1 ) 1/r r +1 u(x) r dx ( 2 k+ r k + k ( +1 f (y) v(y)dy ) 1/( 1 +1 b CMO r b CMO r 1 +1 ) f (y) dy ) 1/r u(x) r dx v(y) / dy ) 1/ ) +1 2 ( j k ( +1 r f (y) v(y)dy ) ) 1/ ( b f (y) v(y)dy. CMO r For JJ, by Lemma 3.1, we have JJ = 2 / 2 j j +2 / = JJ 1 + JJ 2. 2 j+1 2 j 2 ( j k) ) / 2r 2 ( j k) +1 2r f (y) v(y)dy +1 v(y) / dy ) 1/ ) (b(y) b (,+1 ] ) f (y) dy 2(k j)c b CMO 1 f (y) dy 1 +1 For JJ 1,byHölder inequality and condition (1.7), we have 2 j+1 JJ 1 = 2 / 2 j ( +1 b CMO r 2 j/r ( 2 j+1 f (y) v(y)dy 1 +1 (b(y) b (,2 k+1 ] ) f (y) dy ) 1/r ( u(x) r 1 ( dx 2k+1 b(y) b (,2 k+1 ] r dy ) 1/ ( +1 ) 1/r ) v(y) r / dy 2 j/r ( (k+1) r k+ r + (k+1) ( +1 r f (y) v(y)dy ) 1/r ) 1/ )

11 TWO-WEIGHT INEQUALITIES FOR HARDY OPERATOR AND COMMUTATORS 663 b CMO r 2 ( j k ( +1 r f (y) v(y)dy ) ) 1/ b CMO r f (y) v(y)dy. For JJ 2,wehave JJ 2 = C b CMO 1 2 j+1 k j ( ) ( 2 2k+1 1/ j f (y) v(y)dy ( +1 ) 1/ ) v(y) / dy b CMO r This ends the roof. ( b CMO r f (y) v(y)dy. (k j)2 j k ( +1 r ) 1/ ) f (y) v(y)dy Acknowledgement. The authors are grateful to the referee for invaluable comments and suggestions. REFERENCES [1] J. BASTERO, M. MILMAN AND F. J. RUIZ, On the connection between weighted norm inequalities, commutators and real interolation, Mem.Amer.Math.Soc.,154, (21), no [2] J. BRADLEY, Hardy inequalities with mixed norms, Canad. Math. Bull., 21, (4) (1978), [3] M. I. A. CAÑESTRO, P. O. SALVADOR AND C. R. TORREBLANCA, Weighted bilinear Hardy inequalities, J. Math. Anal. Al., 387, (212), [4] D. CRUZ-URIBE SFO, New roofs of Two-weight norm inequalities for the maximal oerator, Georgian Math. J., 7, (2), [5] J. GARCÍA-CUERVA, AND J. M. MARTELL, Two-weight norm inequalities for maximal oerators and fractional integrals on non-homogeneous saces, Indiana Univ. Math. J., 5, (21), [6] J. GARCÍA-CUERVA AND J. L. RUBIO DE FRANCIA, Weighted Norm Inequalities and Related oics, North-Holland Math. Stud., 116, [7] J. DUOANDIKOETEXA,F.J.MARTTN-REYES AND S. OMBROSI, Calderón weights as Muckenhout weights, Indiana Univ. Math. J., 62, (213), [8] G. H. HARDY, Note on a theorem of Hilbert, Math. Z., 6, (192), [9] G. H. HARDY, Note on some oints in the integral calculus, Messenger Math., 57, (1928), [1] G. H. HARDY, J. E. LITTLEWOOD AND G. POLYA, Inequalities, Cambridge University Press, Cambridge, UK, [11] A. KUFNER AND L. E. PERSSON, Weighted Inequalities of Hardy Tye, World Scientific, 23. [12] S. LONG AND J. WANG, Commutator of Hardy oerators, J. Math. Anal. Al., 274, (22),

12 664 W. LI, T.ZHANG AND L. XUE [13] V. G. MAZ YA, Sobolev Saces, Sringer-Verlage, [14] B. MUCKENHOUPT, Hardy s inequality with weights, Studia Math., 44, (1972), [15] B. OPIC, A. KUFNER, Hardy-tye Inequalities, Longman, 199. (Received November 4, 213) Wenming Li College of Mathematics and Information Science Hebei Normal University Shijiazhang, 516, Hebei, P. R. China lwmingg@sina.com Tingting Zhang College of Mathematics and Information Science Hebei Normal University Shijiazhang, 516, Hebei, P. R. China, @163.com Limei Xue School of Mathematics and Science Shijiazhuang University of Economics Shijiazhuang 531, Hebei, P. R. China xuelimei@126.com Journal of Mathematical Inequalities jmi@ele-math.com