WEIGHTED NORM INEQUALITIES FOR FRACTIONAL MAXIMAL OPERATORS - A BELLMAN FUNCTION APPROACH

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1 WEIGHTED NORM INEUALITIES FOR FRACTIONAL MAXIMAL OPERATORS - A BELLMAN FUNCTION APPROACH RODRIGO BAÑUELOS AND ADAM OSȨKOWSKI ABSTRACT. We tudy claical weighted L p L q inequalitie for the fractional maximal operator on R d, proved originally by Muckenhoupt and Wheeden in the 70. We etablih a lightly tronger verion of thi inequality with the ue of a novel extenion of Bellman function method. More preciely, the etimate i deduced from the exitence of a certain pecial function which enjoy appropriate majorization and concavity. From thi reult and an explicit verion of the A p ε theorem, derived alo with Bellman function, we obtain the harp inequality of Lacey, Moen, Pérez and Torre.. INTRODUCTION The motivation for the reult of thi paper come from the quetion about weighted L p L q -norm inequalitie for fractional maximal operator on R d, with the contant of optimal order. To introduce the neceary background and notation, let 0 α < d be a fixed contant. The fractional maximal operator M α i given by M α ϕ(x) = up { αd ϕ(u) du : R d i a cube containing x, where ϕ i a locally integrable function on R d, denote the Lebegue meaure of and the cube we conider above have ide parallel to the axe. In particular, if we put α = 0, then M α reduce to the claical Hardy-Littlewood maximal operator. The above fractional operator play an important role in analyi and PDE, and form a convenient tool to tudy differentiability propertie of function. In particular, it i often of interet to obtain optimal, or at leat good bound for norm of thee operator. We will be motly intereted in the weighted etting. In what follow, the word weight refer to a locally integrable, poitive function on R d, which will uually be denoted by w. Given p (, ), we ay that w belong to the Muckenhoupt A p cla (or, in hort, that w i an A p weight), if the A p characteritic [w] Ap, given by [w] Ap := up ( ) ( ) p w w /(p ), i finite. One can alo define the appropriate verion of thi condition for p = and p =, by paing above with p to the appropriate limit (ee e.g. [6], [9]). However, we omit the detail, a in thi paper we will be mainly concerned with the cae < p <. A hown by Muckenhoupt [3], the A p condition arie naturally during the tudy of weighted L p bound for the Hardy-Littlewood maximal operator. To be more precie, for 200 Mathematic Subject Claification. Primary: 42B25. Secondary: 46E30. Key word and phrae. Maximal, dyadic, Bellman function, bet contant. R. Bañuelo i upported in part by NSF Grant # DMS. A. Oȩkowki i upported in part by the NCN grant DEC-202/05/B/ST/0042.

2 2 RODRIGO BAÑUELOS AND ADAM OSȨKOWSKI a given < p <, the inequality M 0 ϕ L p (w) C p,d,w ϕ L p (w) hold true with ome contant C p,d,w depending only on the parameter indicated, if and only if w i an A p weight. Here, of coure, ϕ L p (w) = ( ϕ p wdu ) /p R i the uual d norm in the weighted L p pace. Thi reult wa extended to the fractional etting by Muckenhoupt and Wheeden [4]. Let p, q be poitive exponent atifying the relation q = p α d. Then the inequality ( ( M α ϕ(x) ) ) /q ( ) /p q w(x) q dx C p,α,d,w ϕ(x) p w(x) p dx R d R d if and only if ( ) ( ) q/p up w q w p <, where p = p/(p ) i the harmonic conjugate to p. In other word, paing to w q, we ee that M α Lp (w p/q ) L q (w) C p,α,d,w if and only if w A q/p +. Actually, one can chooe the above contant C p,d,w and C p,α,d,w o that the dependence on w i through the appropriate characteritic of the weight only. Then there arie a very intereting quetion, concerning the harp decription of thi dependence. The firt reult in thi direction, going back to early 90, i that of Buckley []. Specifically, he proved that Hardy-Littlewood operator atifie (.) M 0 ϕ L p (w) C p,d [w] /(p ) A p ϕ L p (w) and howed that the power /(p ) cannot be decreaed in general. By now, there are everal different proof of thi inequality (which produce variou upper bound for the involved contant C). For intance, we refer the intereted reader to work of Coifman and Fefferman [3], Lerner [2], Nazarov and Treil [5] and, for a lightly tronger tatement, to the recent paper of Hytönen and Perez [7]. In the fractional etting, Lacey et al. [] proved that (.2) M α ϕ Lq (w) C p,α,d [w] ( α/d)p /q A q/p + ϕ L p (w p/q ), and the exponent ( α/d)p /q cannot be improved. See alo the recent paper of Cruz- Uribe and Moen [4] for certain generalization of the above reult. The purpoe of thi paper i to provide yet another extenion of (.) and (.2). Here i the precie tatement. Theorem.. Suppoe that 0 α < d i fixed and let p, q be exponent atifying q = p α d. Then for any A q/p + weight w and any locally integrable function ϕ on R d we have (.3) M α ϕ L q (w) C d inf <r<q/p + { [w] /(q ) A r where C i an abolute contant (for intance, C = 2 work fine) and (.4) = (r, p, α) = q rq 2 (r )(q p) + pq. ) /(q ) ϕ Lp (w p/q ),

3 WEIGHTED INEUALITIES 3 To ee that thi reult implie (.) and (.2), we will need to etablih the following verion of the A p ε theorem which i harper than what the claical reult give, ee Coifman and Fefferman [3]. Theorem.2. Suppoe that w i an weight ( < β < ) and put (.5) r = β( + 2βd/(β ) β β/(β ) [w] /(β ) ) (, β). β + 2 βd/(β ) β β/(β ) [w] /(β ) Then w i an A r weight and [w] Ar 2 rd+r [w] (r )/(β ). Now (.) and (.2) follow eaily: apply Theorem.2 with β = q/p + and plug r given by (.5) into (.3). We have ( ) q (r ) q = p + q β r and o q/ c p,α,d [w] /(β ). Therefore, β(β ) β r =, β + 2 βd/(β ) β β/(β ) [w] /(β ) ( [w] /(q ) q ) /(q ) A r 2 (rd+r)/(q ) [w] (r )/(β )(q ) A β and it remain to note that C p,α,d [w] r/(β )(q ), r (r )(q p) + pq (β )(q p) + pq = (β )(q ) (β )q 2 (β )q 2 = c /(q ) p,α,d [w] /(β )(q ) ( α ) p d q. A few word about our approach are in order. Actually, we will work with the dyadic verion of the fractional maximal operator, given by M α d ϕ(x) = up { αd ϕ(u) du : R d i a dyadic cube containing x. Here the dyadic cube are thoe formed by the grid 2 N Z d, N Z. By a tandard comparion of the ize of M α ϕ and M α d ϕ (ee e.g. Grafako [6] or Stein [6]), we will be done if we etablih the following verion of Theorem.. Theorem.3. Suppoe that 0 α < d i fixed and let p, q be exponent atifying q = p α d. Then for any A q/p + weight w and any locally integrable function ϕ on R d we have (.6) M α d ϕ Lq (w) inf <r<q/p + { [w] /(q ) A r ) /(q ) ϕ L p (w p/q ). The paper i organized a follow. In the next ection we etablih Theorem.2. Section 3 i devoted to the proof of Theorem.3. Both thee tatement are hown with the ue of the o-called Bellman function method, a powerful tool exploited widely in analyi and probability, and it certain enhancement. We end with ome remark concerning martingale verion of the above weighted norm inequalitie.

4 4 RODRIGO BAÑUELOS AND ADAM OSȨKOWSKI 2. PROOF OF THEOREM.2 Throughout thi ection, β (, ), d and c are fixed parameter. Let u define the hyperbolic domain We tart with the following geometric fact. Ω c = {(w, v) IR + IR + : wv β c. Lemma 2.. Let α [2 d, 2 d ] and uppoe that point P, and R = αp + ( α) lie in Ω c. Then the whole line egment P i contained within Ω 2 βd c. Proof. Uing a imple geometrical argument, it i enough to conider the cae when the point P and R lie on the curve wv β = c (the upper boundary of Ω c ) and lie on the curve wv β = (the lower boundary of Ω c ). Then the line egment R i contained within Ω c, and hence alo within Ω 2 βd c, o it i enough to enure that the egment PR i contained in Ω 2 βd c. Let P = (P x, P y ), = ( x, y ) and R = (R x, R y ). We conider two cae. If P x < R x, then P y = α R y (α ) y < 2 d R y, o the egment PR i contained in the quadrant {(x, y) : x R x, y 2 d R y. Conequently, PR lie below the hyperbola xy β = γ paing through (R x, 2 d R y ), that i, correponding to γ = 2 d(β ) R x Ry β 2 βd c. Thi prove the aertion in the cae P x < R x. In the cae P x R x the reaoning i imilar. Indeed, we check eaily that the line egment PR lie below the hyperbola xy β = γ paing through (2 d R x, R y ). That i, the one with γ = 2 d R x Ry β 2 βd c. The proof of Theorem.2 will ret on a Bellman function which wa invented by Vayunin in [8] during the tudy of power etimate for weight on the real line. To recall thi object, we need ome more notation. Let be the unique negative number atifying the equation ( )( /β) β = 2 βd /c (the exitence and uniquene of i clear: the function F (u) = ( u)( u/β) β i trictly increaing on (, 0] and atifie lim u F (u) = 0, F (0) = ). For r ( β( )/(β ), β ), define [ C = C β,d,r,c = (2 βd c) r/(β(r )) ( ) (β r)/(β( r)) + β r ] β(r ). Thi i preciely the contant C max (β, (r ), 2 βd c) appearing on p. 50 in [8]. A function defined on a ubet of R 2 i aid to be locally concave if it i concave along any line egment contained in it domain. We will need the following lemma from [8]. Lemma 2.2. There i a locally concave function B on Ω 2 pd c which atifie (2.) B(w, v) = w /(r ) if wv β = and uch that (2.2) B(w, v) C β,d,r,c w /(r ) for all (w, v) Ω 2 βd c.

5 WEIGHTED INEUALITIES 5 Combining thi with Lemma 2. and a traightforward induction argument, we ee that the above function B ha the following property. If P, P 2,..., P 2 d and P = 2 d (P +P P 2 d) lie in Ω c, then d 2 (2.3) B(P) 2 d B(P k ). We turn to the proof of Theorem.2. Let u firt etablih two auxiliary fact. k= Lemma 2.3. The number atifie the double inequality (2.4) 2 βd/(β ) β β/(β ) c /(β ) β β/(β ) c /(β ). Proof. Let u firt focu on the right-hand ide inequality. By the definition of, we ee that we mut prove that or ( + β β/(β ) c /(β ) )( + β /(β ) c /(β ) ) β 2 βd /c, ( + β /(β ) c /(β ) ) β 2 βd c( + β β/(β ) c /(β ) ). But thi i imple and follow immediately from the etimate ( + β /(β ) c /(β ) ) β (2β /(β ) c /(β ) ) β 2 βd β β/(β ) c β/(β ). We turn our attention to the left-hand ide inequality in (2.4). Thi time we mut how that ( + 2 βd/(β ) β /(β ) c /(β ) ) β 2 βd c( + 2 βd/(β ) β β/(β ) c /(β ) ). Uing the bound ( + a) β a β + a β, we may write ( + 2 βd/(β ) β /(β ) c /(β ) ) β 2 β2 d/(β ) β β/(β ) c β/(β ) + 2 βd βc and the proof i finihed. = 2 βd c(β + 2 βd/(β ) β β/(β ) c /(β ) ), Lemma 2.4. Suppoe that w i an weight with [w] Aβ = c. Then for any r ( β( )/(β ), β ) we have [w] Ar C r β,d,r,c. Proof. Fix a cube R d and conider the family of it dyadic ubcube. That i, for a given n 0, let n be the collection of 2 nd pairwie dijoint cube contained in, each of which ha meaure 2 nd. Let (w n ) n 0 and (v n ) n 0 be the conditional expectation of w and w /(β ) with repect to ( n ) n 0. That i, for any x and any nonnegative integer n, define w n (x) = w and v n (x) = w /(β ), where i the unique element of n which contain x. Directly from thi definition, we ee that w n, v n are contant on each n and we have w n = 2 d w = R R, R n+ R 2 d w n+ R R, R n+ and v n = 2 d w /(p ) = R R, R n+ R 2 d v n+ R. R, R n+

6 6 RODRIGO BAÑUELOS AND ADAM OSȨKOWSKI Therefore, by (2.3), we ee that B(w n+, v n+ )du B(w n, v n )du for all n = 0,, 2,..., and hence, umming over all n, we get B(w n+, v n+ )du B(w n, v n )du. Conequently, by induction, we obtain ( ) B(w n, v n )du B(w 0, v 0 )du = B wdu, w /(β ) du ( /(r ) C β,d,r,c wdu), where in the lat paage we have exploited (2.2). Now if we let n, then w n w and v n w /(β ) almot everywhere, in view of Lebegue differentiation theorem. Therefore the above etimate, combined with Fatou lemma and (2.), give ( /(r ) w /(r ) du = B(w, w /(β ) )du C β,d,r,c wdu). ( ) /(r ) Multiplying both ide by wdu and taking the upremum over all, we obtain the deired upper bound for [w] Ar. Proof of Theorem.2. Put c = [w] Aβ. Let u firt note that the number r defined in (.5) belong to the interval ( β( )/(β ), β ). Indeed, the inequality r < β i evident, o all we need i the lower bound r > β( )/(β ). After ome eay manipulation, it can be tranformed into > 2 βd/(β )+ β β/(β ) c /(β ), which follow from the left inequality in (2.4). Thu we are allowed to apply Lemma 2.4 with the above choice of r and therefore we will be done if we give the appropriate upper bound for C r β,d,r,c = (2βd c) r/β ( ) (r β)/β [ + β r ] r β(r ). Let u analyze the three factor appearing in thi expreion. We have (2 βd c) r/β = 2 rd c r/β, and, ince (r β)/β < 0, the right inequality in (2.4) give ( ) (r β)/β ( ) (r β)/β (β β/(β ) c /(β ) ) (r β)/β c (r β)/(β(β )). Finally, by the left inequality in (2.4), we get and hence + β r β(r ) β r β(r ) 2βd/(β ) β β/(β ) c /(β ) = 2, [ + β r ] r β(r ) < 2 r. Putting all the above fact together, we obtain [w] Ar 2 rd+r c (r )/(β ), which i the deired claim.

7 WEIGHTED INEUALITIES 7 3. WEIGHTED INEUALITIES FOR FRACTIONAL MAXIMAL OPERATOR Let 0 < α < d be fixed, and let p, q be given two parameter atifying q = p α d. Next, put u = p p q + and let r be an arbitrary number belonging to the interval (, u). In what follow, we will alo need the parameter = (p, q, α) given by (.4) and the number rpq t = t(r, p, α) = (r )(q p) + pq. Suppoe that w i a weight atifying the condition A r and let ϕ : R d R be a function belonging to L p (w p/q ). Of coure, we may aume that ϕ 0: in (.6), the paage ϕ ϕ doe not affect the L p -norm of ϕ, and doe not decreae the L q norm of M α ϕ. Furthermore, it i enough to deal with bounded ϕ only, by a tandard truncation argument. Next, let be a fixed dyadic cube and, for each n 0, let n be the collection of all dyadic cube of meaure /2 nd, contained within. Conider the equence (ϕ n ) n 0, (w n ) n 0 and (v n ) n 0 of conditional expectation of ϕ, w and w /(r ) with repect to ( n ) n 0. That i, in analogy with the previou ection, for any x and any nonnegative integer n, define (3.) ϕ n (x) = ϕ, w n (x) = w and v n (x) = w /(r ), where i the element of n which contain x. We will alo ue the notation ψ n (x) = α/d max { ϕ 0, 2 α ϕ, 2 2α ϕ 2,..., 2 nα ϕ n. Before we proceed, let u comment on the reaoning we are going to preent. A in the preceding ection, the proof will be baed on the propertie of a certain pecial function B. Thi time we have four parameter involved, correponding to the equence (ϕ n ) n 0, (ψ n ) n 0, (w n ) n 0 and (v n ) n 0, and thu it i natural to define B on the four-dimenional domain D = {(x, y, w, v) : x 0, y 0, w > 0, wv r c. Here and below, we ue the ame letter w to denote the weight and the third coordinate; a we hope, thi hould not caue any confuion and it hould be clear from the context which object we are uing at the moment. A natural idea to follow i to find B for which the equence ( B(ϕ n, ψ n, w n, v n )dz ) i nonincreaing, nonpoitive and atifie the n 0 majorization of the form lim inf n B(ϕ n, ψ n, w n, v n )dz M α d ϕ q L q (;w) ) q/(q ) c q/(q ) ϕ q L p (;w p/q ) or, after ome tandard limiting argument (Lebegue differentiation theorem, Fatou lemma, etc.), (3.2) B(ϕ, M α d ϕ, w, w /(r ) )dz M α d ϕ q L q (;w) ( cq ) q/(q ) ϕ q L p (;w p/q ). A typical approach in the tudy of uch majorization i to prove the correponding pointwie bound. The problem i that the right hand ide above i a mixture of L p and L q norm and i not of integral form. That i, there eem to be no pointwie inequality which after

8 8 RODRIGO BAÑUELOS AND ADAM OSȨKOWSKI integration would yield the above majorization. In addition, no manipulation with the inequality (3.2) (for intance, replacing the right hand ide by ) /(q ) M α d ϕ Lq (;w) c /(q ) ϕ L p (;w p/q ) or other expreion of thi type) eem to lead to the convenient majorization in the integral form. To overcome thi difficulty, we will make ue of the following novel argument which, to the bet of our knowledge, ha not been ued before. Namely, to etablih the inequality (.6), we will work with a pecial function B which itelf depend on ϕ. Specifically, put B(x, y, w, v) = B ϕ (x, y, w, v) = y q w q ϕ q t L p (;w p/q ) cxt y v t. In the two lemma below, we tudy certain crucial propertie of thi object. We tart with the following monotonicity condition. Lemma 3.. Suppoe that [w] Ar = c. Then for any nonnegative integer n we have (3.3) B(ϕ n, ψ n, w n, v n )dx B(ϕ 0, ψ 0, w 0, v 0 )dx. Proof. Note that (ϕ n, ψ n, w n, v n ) D (o the compoition B(ϕ n, ψ n, w n, v n ) make ene) due to aumption [w] Ar = c. Clearly, it uffice to how the inequality B(ϕ n, ψ n, w n, v n )dz B(ϕ n, ψ n, w n, v n )dz. Thi will be done in the everal eparate tep below. Step. Firt we will how the pointwie etimate B ( ϕ n (z), ψ n (z), w n (z), v n (z) ) (3.4) B ( ϕ n (z), ψ n (z), w n (z), v n (z) ), z. Let n (z) be the element of n which contain z. The above bound i trivial when ψ n (z) > n (z) α/d ϕ n (z) = α/d 2 nα ϕ n (z), ince then ψ n (z) = ψ n (z). Suppoe that ψ n (z) n (z) α/d ϕ n (z) (o actually we have equality: ee the definition of the equence ψ). Since ψ n (z) ψ n (z), we will be done if we how that (3.5) y B( ϕ n (z), y, w n (z), v n (z) ) 0 for y ( 0, n (z) α/d ϕ n (z) ), n = 0,, 2,.... The partial derivative equal [ qy w n (z) y q ϕ q t L p (;w p/q ) cϕ n(z) t w n (z) v n (z) t]. However, we have w n (z)v n (z) r c, directly from the aumption [w] Ar = c. Furthermore, we have y < n (z) α/d ϕ n (z), which ha been jut impoed above. Conequently, we ee that the partial derivative in (3.5) i not larger than [ qy w n (z)ϕ n (z) n (z) α(q )/d ϕ n (z) q t ϕ q t L p (;w p/q ) v n(z) r t].

9 WEIGHTED INEUALITIES 9 However, by Hölder inequality, we may write n (z) ϕ n (z) = ϕ n (z) ( p/q)/p ( ) (r )/q ϕ p w w /(r ) n (z) /p (r )/q n (z) n (z) ϕ L p (;w p/q )v n (z) (r )/q n (z) /p (r )/q, which i equivalent to n (z) α(q )/d ϕ n (z) q t ϕ q t L p (;w p/q ) v n(z) r t 0. Thi how that (3.5), and hence alo (3.4), hold true. Step 2. Now, oberve that the C function G : [0, ) (0, ) [0, ) given by G(x, v) = x t v t i convex. Thi i traightforward: for x, v > 0, the Heian matrix [ ] D 2 t(t )x G(x, v) = t 2 v t t( t)x t v t r( t)x t v t t(t )x t v t i nonnegative-definite. Step 3. We are ready to etablih the deired bound (3.3). Pick an arbitrary element of n and apply (3.4) to get B(ϕ n, ψ n, w n, v n )dz B(ϕ n, ψ n, w n, v n )dz. Directly from the definition of the equence (ϕ n ) n 0, (w n ) n 0 and (v n ) n 0, we ee that ϕ n, w n and v n are contant on and equal to ϕ n dz, w n dz and v n dz there, repectively. Now, we ue Step 2 and the formula for B to get B(ϕ n, ψ n, w n, v n )dz B(ϕ n, ψ n, w n, v n )dz and thu B(ϕ n, ψ n, w n, v n )dz It remain to um over all n to get the claim. We will alo require the following propertie. B(ϕ n, ψ n, w n, v n )dz. Lemma 3.2. (i) If (x, y, w, v) D atifie y = α/d x, then we have the pointwie inequality B(x, y, w, v) 0. (ii) For any (x, y, w, v) D we have the majorization B(x, y, w, v) q [ ) ] q/(q ) y p q(q t)/(q ) w ϕ c q/(q ) (xw /q ) p. q L p (;w p/q )

10 0 RODRIGO BAÑUELOS AND ADAM OSȨKOWSKI Proof. (i) Thi follow from (3.5) with n = 0. Indeed, for any w, v atifying wv c there i a weight w with [w] Ar c atifying w 0 = w, v 0 = v (ee e.g. [8]). Thu, if we put ϕ x, then ϕ 0 = x, ψ 0 = y and hence (3.5) give B(x, y, w, v) B(ϕ 0 (z), 0, w 0 (z), v 0 (z)) = 0. (ii) Of coure, it uffice to how the bound for y > 0. By the mean value property, for any β 0 we have β q/(q ) q (β ). q Plugging β = q ϕ q t L p (w p/q ) cxt y q w (p q)(q )/q2 + and multiplying both ide by y q w, we obtain an inequality which i equivalent to B(x, y, w, w /( r) ) [ q ) ] q/(q ) y p q(q t)/(q ) w ϕ c q/(q ) (xw /q ) p. q L p (;w p/q ) It remain to oberve that B(x, y, w, v) increae when v increae, and therefore we have B(x, y, w, v) B(x, y, w, w /( r) ). We are ready to etablih our main reult. Proof of (.6). Combining the previou two lemma, we obtain the bound ψ n (z) q w n (z)dz ) q/(q ) q(q t)/(q ) ϕ c q/(q ) ϕ L p (;w p/q ) n (z) p w n (z) p/q dz. All that i left i to carry out appropriate limiting procedure. Firt, let n go to infinity. Then ψ n increae to { ψ = up α ϕ : x, k for ome k. In addition, we have ϕ n ϕ almot everywhere (by Lebegue differentiation theorem) and w n w in L (), by the tandard fact concerning conditional expectation (ee e.g. Doob [5]). Putting thee fact together, and combining them with the boundedne of ϕ aumed at the beginning, we get ψ w q lim inf n ψ q nw n ) q/(q ) ϕ q(q t)/(q ) L p (w p/q ) c q/(q ) lim inf ϕ p nw p/q n n ) q/(q ) q(q t)/(q ) ϕ c q/(q ) ϕ L p (w p/q ) L p (w p/q ) ) q/(q ) = c q/(q ) ϕ q L p (w p/q ). Next, aume that 2... i a trictly increaing equence of dyadic cube, and apply the above etimate with repect to = n. Then, a n, we have ψ M α d ϕ and hence (.6) follow from Lebegue monotone convergence theorem.

11 WEIGHTED INEUALITIES 4. A WEIGHTED VERSION OF DOOB S INEUALITY In the particular cae α = 0, Theorem.3 give the following tatement for Hardy- Littlewood maximal operator. Theorem 4.. Let w be an A p weight, < p <. Then for any locally integrable function ϕ on R d we have { ( ) /r (4.) M d ϕ Lp (w) inf [w] /r p <r<p A r ϕ Lp (w). p r Thi theorem can be given a probabilitic meaning, in term of martingale, and uch a reult wa proved in [0]. Let u recall the general etting of A p martingale. Suppoe that (Ω, F, P) i a probability pace with a non-decreaing right continuou family (F t ) t 0 of ub σ-field of F uch that F 0 contain all P-null et. Fix a random variable Z uch that Z > 0 a.. and uch that E[Z] =. With p >, we ay that the random variable Z i an A p -weight if ( Z t (4.2) [Z] Ap := up t E [ ( ) /(p ) ] ) p Ft Z L <, where Z t = E [Z F t ]. The following reult can be derived by keeping track of the contant in the proof of [0, Theorem 2]. Theorem 4.2. Let X t = E [X F t ] be the martingale generated by the P integrable random variable X. Suppoe Z A r, for ome r >. Then for all p > r, ( ) /r p (4.3) X L p (ˆP) [Z]/r A r X p r, Lp(ˆP) where a uual X = up t X t and dˆp denote the the meaure ZdP. We give the proof of thi reult ince it i imple and hort. We aume, a we may, that X Lp (ˆP) <. A imple calculation (jut the definition of conditional expectation) give that Ê[X F t ] = E[ZX F t] Z t and thi hold almot urely with repect to both probability meaure P and ˆP. Applying thi with (/Z)X in place of X we ee that X t = Z t Ê[(/Z)X F t ]. If we let r 0 be the conjugate exponent of r, Hölder inequality give { [ [( ) r0 ]] r X t r Zt r Ê F t Ê[ X r F t ] Z [ ( ) ]] r0 r = Z t [E F t Ê[ X r F t ] Z [Z] Ar Ê[ X r F t ] Now, applying Doob inequity with p/r > to the ˆP martingale in the econd term give the inequality.

12 2 RODRIGO BAÑUELOS AND ADAM OSȨKOWSKI A Theorem.3, thi propoition can alo be obtained uing the Bellman function technique a above, bypaing Doob inequality. We now addre the martingale verion of Theorem.2. While Theorem 4.2 hold for arbitrary martingale, it i proved in Uchiyama [7] that the A p ε reult doe not hold in general for arbitrary martingale but it doe o if the martingale have continuou trajectorie (Brownian martingale) or if they are regular. Recall that the nondecreaing filtration (F n ) n 0 on the probability pace (Ω, F, P) with n= F n = F i aid to be regular if F n i atomic for each n and there i a univeral contant C 0 uch that P(A)/P(B) < C 0 for any two atom A F n and B F n uch that B A. Dyatic filtration on R d are for example regular with C 0 = 2 d. The proof of Theorem.2 applie to A p weight on a regular filtration and we obtain the following theorem which give a martingale verion of Buckley inequality. Theorem 4.3. Let X n = E [X F n ] be the martingale generated by the P integrable random variable X and aume that (F n ) n 0 i regular with contant C 0. Suppoe Z A p, < p <. There i a contant C depending on C 0 uch that (4.4) X L p (ˆP) Cp p [Z]/(p ) A p X L p (ˆP). Except for the contant C, thi inequality i harp. The ame reult hold for martingale with continuou trajectorie for ome univeral contant C. ACKNOWLEDGMENT The reult of thi paper were obtained during the fall emeter of 203 when the econd-named author viited Purdue Univerity. REFERENCES [] S. M. Buckley, Etimate for operator norm on weighted pace and revere Jenen inequalitie, Tran. Amer. Math. Soc. 340 (993), [2] D. L. Burkholder, Boundary value problem and harp inequalitie for martingale tranform, Ann. Probab. 2 (984), [3] R. R. Coifman and C. Fefferman, Weighted norm inequalitie for maximal function and ingular integral, Studia Math. 5 (974), [4] D. Cruz-Uribe and K. Moen, A Fractional Muckenhoupt-Wheeden Theorem and it Conequence, Integral Equation Operator Theory 76 (203), pp [5] J. L. Doob, Stochatic procee, John Wiley & Son, Inc., New York; Chapman & Hall, Limited, London, 953. [6] L. Grafako, Claical Fourier analyi, Second Edition, Springer, New York, [7] T. Hytönen and C. Perez, Sharp weighted bound involving A, available at [8] T. Hytönen, The A 2 theorem: Remark and Complement, available at [9] S. V. Hručev, A decription of weight atifying the A condition of Muckenhoupt, Proc. Amer. Math. Soc. 90 (984), [0] M. Izumiawa and N. Kazamaki, Weighted norm inequalitie for martingale, Tôhoku Math. Journ. 29 (977), [] M. T. Lacey, K. Moen, C. Pérez and R. H. Torre, Sharp weighted bound for fractional integral operator, J. Funct. Anal. 259 (200), pp [2] A. K. Lerner, An elementary approach to everal reult on the Hardy-Littlewood maximal operator, Proc. Amer. Math. Soc. 36 (2008), [3] B. Muckenhoupt, Weighted norm inequalitie for the Hardy maximal function, Tran. Amer. Math. Soc. 65 (972), [4] B. Muckenhoupt and R. Wheeden, Weighted norm inequalitie for fractional integral, Tran. Amer. Math. Soc. 92 (974), pp

13 WEIGHTED INEUALITIES 3 [5] F. L. Nazarov and S. R. Treil, The hunt for a Bellman function: application to etimate for ingular integral operator and to other claical problem of harmonic analyi, St. Peterburg Math. J. 8 (997), pp [6] E. M. Stein, Singular integral and Differentiability Propertie of Function, Princeton Univerity Pre, Princeton, 970. [7] A. Uchiyama, Weight function on probability pace, Tôhoku Math. Journ. 30 (978), [8] V. Vayunin, The exact contant in the invere Hölder inequality for Muckenhoupt weight (Ruian), Algebra i Analiz 5 (2003), 73 7; tranlation in St. Peterburg Math. J. 5 (2004), pp DEPARTMENT OF MATHEMATICS, PURDUE UNIVERSITY, WEST LAFAYETTE, IN 47907, USA addre: banuelo@math.purdue.edu DEPARTMENT OF MATHEMATICS, INFORMATICS AND MECHANICS, UNIVERSITY OF WARSAW, BA- NACHA 2, WARSAW, POLAND addre: ado@mimuw.edu.pl