RUBIO DE FRANCIA S EXTRAPOLATION THEOREM FOR B p WEIGHTS

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 138, Numbe 2, Febuay 21, Pages S (9)14- Aticle electonically published on Septembe 1, 29 RUBIO DE FRANCIA S EXTRAPOLATION THEOREM FOR B p WEIGHTS MARÍA J. CARRO AND MARÍA LORENTE (Communicated by Nigel J. Kalton) Abstact. In this pape, we pove some of Rubio de Fancia s extapolation esults fo the class B p of weights fo which the Hady opeato is bounded on L p (w) esticted to deceasing functions. Applications to the boundedness of opeatos on L p dec (w) ae given. We also pesent an extension to the B case and some connections with classical A p theoy. 1. Intoduction In 1984, J.L. Rubio de Fancia [1] poved that if T is a sublinea opeato that is bounded on L (w) fo evey w in the Muckenhoupt class A (>1) with constant depending only on ( )( ) w A =sup w w 1/( 1) Q Q Q Q I whee the supemum is taken ove all cubes Q, then fo evey 1 <p<, T is bounded on L p (w) fo evey w A p with constant depending only on w Ap.Since then, many esults concening this topic have been published (see [8], [6], [7]). Fom these esults, it is now known that, in fact, the opeato T plays no ole; that is, if (f,g) ae a pai of functions such that fo some 1 p <, f p (x)w(x)dx C g p (x)w(x)dx R n R n fo evey w A p with C depending on w Ap, then fo evey 1 <p<, f p (x)w(x)dx C g p (x)w(x)dx R n R n fo evey w A p with C depending on w Ap. The theoy has also been genealized to the case of A weights and many inteesting consequences have been deived fom it. The pupose of this pape is to develop a completely paallel theoy in the setting of B p weights. The techniques ae diffeent as usually happens with these Received by the editos Mach 3, 29, and, in evised fom, May 8, Mathematics Subject Classification. Pimay 26D1, 46E2. Key wods and phases. Extapolation, monotone functions, B p weights, Caldeón and Riemann-Liouville opeato. This wok was suppoted by MTM27-65, by CURE 25SGR556 and MTM C3-2, and by Junta de Andalucía FQM354 and P6-FQM-159. c 29 Ameican Mathematical Society Revets to public domain 28 yeas fom publication 629

2 63 MARÍA J. CARRO AND MARÍA LORENTE two theoies and things ae, in some sense, cleae and moe natual. We think that the esults in this pape should help to claify what is happening in the A p context and we hope to solve that case in a fothcoming pape. Befoe pesenting the main esults of this pape, let us just ecall some impotant facts concening B p weights which will be fundamental fo ou puposes. Fist of all, let us ecall that a positive and locally integable function w on (, ) is called a B p weight if the following condition holds: w Bp =inf { C>; w(t)dt + p w(t) t p dt C } w(t)dt, > <. It is known ([1]) that w B p with p>ifand only if, fo evey deceasing function f, ( 1 t p f(s)ds) w(t)dt C f p (s)w(s)ds t with C depending on w Bp. Obseve also that w Bp > 1ifw is not identically zeo. An impotant popety that these classes of weights satisfy (see [4], Chapte 3, Section 3.3) is that, fo evey p> and evey w B p,theeexistsε> such that w B p ε ;moeove, C w Bp (1.1) w Bp ε 1 εα p, w Bp whee C and <α<1ae univesal constants and ε is such that 1 εα p w Bp >. Since B p B q fo evey B p. Letusalsodefine w B =inf{ w Bp ; w B p }. We shall denote by C a univesal constant depending possibly on p but independent of the weight w. Also C might not be the same in all instances. We wite A B if thee exists a univesal constant C such that A CB and A B if A B and B A. 2. Main esults Ou fist esult is the countepat in this setting of the new vesion of Rubio de Fancia s extapolation esult: Theoem 2.1. Let ϕ be an inceasing function on (, ), let(f,g) be a pai of positive deceasing functions defined on (, ) and let and w B p, f p w ϕ( w Bp ) g p w,

3 R. DE FRANCIA S EXTRAPOLATION THEOREM FOR B p WEIGHTS 631 whee with C as in (1.1). ϕ( w Bp ) = Similaly, in the B setting: inf p <ε< pα p w B p ϕ ( p ε ) p/p C w Bp 1 ε p p α p w Bp Theoem 2.2. Let ϕ be an inceasing function on (, ), let(f,g) be a pai of positive deceasing functions defined on (, ) and let and w B, f p w ϕ(1) p/p w B g p w In ode to pove these two esults, we shall use the following lemmas. Lemma 2.3. Let ϕ be an inceasing function on (, ), let(f,g) be a pai of positive deceasing functions defined on (, ) and let Then, fo evey <ε, t ( ) t f(s)s p 1 ε p ds ϕ g(s)s p 1 ε ds ε Poof. Let w(t) =v(t)t p 1 ε with v a deceasing function and let us assume that w L 1 loc.then w(t) dt + p w(t) t p dt = w(t) dt + p v(t) dt t1+ε w(t) dt + 1 ε v()p ε = w(t) dt + p ε v() t p ε 1 dt ε p w(t) dt, ε and hence w B p with constant less than o equal to p /ε. In paticula, taking v(t) =χ (,s) (t) and applying the hypothesis, we obtain that f(u)up 1 ε ( ) du g(u)up 1 ε du ϕ p < ε and the esult follows. Let φ be a positive deceasing locally integable function defined on (, ) and let Φ(x) = x φ(t)dt. The genealized Hady opeato associated to φ is defined, fo f deceasing, by S φ f(x) = 1 x f(t)φ(t)dt. Φ(x)

4 632 MARÍA J. CARRO AND MARÍA LORENTE Lemma 2.4. Let <p<. Then,S φ is bounded on L p dec (w) with constant A if and only if (2.1) w(x)dx +Φ() p w(x) dx Ap Φ(x) p w(x)dx, fo all >. Poof. This esult has been poved in [5] (Theoem 4.1) fo the case p>1. The poof also woks (and is easie) fo p =1. Let us now pove the case <p<1. The necessay condition follows as in [5] by taking f = χ (,). Convesely, let f be deceasing. Then, f(s) 1 f(t)φ(t)dt Φ(s) fo evey s> and theefoe Taking this into account, ( p 1 f(t)φ(t)dt) f(s) p 1 Φ(s) p 1. (2.2) ( 1 x p (S φ f(x)) p w(x)dx = f(s)φ(s)ds) w(x)dx Φ(x) x ( ) p 1 = p f(t)φ(t)dt f(s)φ(s)ds w(x) Φ(x) p dx x p f(s) p φ(s)φ(s) p 1 ds w(x) Φ(x) p dx. Since f is deceasing, Coollay 2.2 in [5] gives that the chain of inequalities in (2.2) can be continued as follows: p p = = = λf p (y) min{λf p (y),x} A p χ (,x) (s)φ(s)φ(s) p 1 ds dy w(x) Φ(x) p dx φ(s)φ(s) p 1 ds dy w(x) Φ(x) p dx Φ(min{λ f p(y),x}) p dy w(x) Φ(x) p dx Φ(min{λ f p(y),x}) p w(x) dx dy Φ(x) p ( λf p (y) w(x)dx +Φ(λ f p(y)) p λf p (y) λ f p (y) ) w(x) Φ(x) p dx dy w(x)dx dy = A p f(y) p w(y) dy, whee the last inequality is obtained fom the hypothesis.

5 R. DE FRANCIA S EXTRAPOLATION THEOREM FOR B p WEIGHTS 633 Poof of Theoem 2.1. Let p>, w B p and <ε<p. Using the fact that f is deceasing and Lemma 2.3, we get (2.3) ϕ = ϕ f(t) p w(t)dt ( p ( p ( p ε t p ε t t f(s) p s p 1 ε ds) p/p w(t) dt ) p/p ( p/p p ε ε t p g(s) p s ds) p 1 ε w(t) dt ε ) p/p (S φ g p (t)) p/p w(t) dt, ε whee φ(t) =t p 1 ε. The poof will be finished once we compute A such that (S φ g p (t)) p/p w(t) dt A and by Lemma 2.4, we only have to compute A such that w(x)dx + (p ε)p p w(x) x (p ε)p p dx A g(t) p w(t)dt, w(x)dx, which is equivalent to saying that w B (p ε)p with A = w B (p. ε)p p p Now, since w B p thee exists ε >sothatw B p ε. Then, it suffices to take ε small enough so that p ε = (p ε)p p to get the esult. Moeove, by (1.1), we have that C w Bp A = w B (p = w ε)p Bp ε p 1 ε p p α p. w Bp Consequently, fo evey <ε< f(t) p w(t)dt ϕ ( p p pα p w B, p ε ) p/p C w Bp 1 ε p p α p w Bp g(t) p w(t)dt, and the esult follows by taking the infimum of such ε s. Poof of Theoem 2.2. By hypothesis we have that f p w ϕ( w ) g p w, fo evey w B. Then, taking w(t) =χ (,s) (t)t β with s>andβ> 1, we have that w B and w B = 1. Hence (2.4) f p (t)t β dt ϕ(1) g p (t)t β dt, fo all t>,β > 1. Now let p> and let w B be abitay. Then, by definition of B,theeexists q> such that w B q. Using again that f is deceasing and inequality (2.4), we

6 634 MARÍA J. CARRO AND MARÍA LORENTE obtain that fo evey β> 1, (2.5) f(t) p w(t)dt ϕ(1) p/p = ϕ(1) p/p ( 1+β t t 1+β f(s) p s β ds) p/p w(t) dt ( 1+β t β+1 t (S φ g p (t)) p/p w(t) dt, g(s) p s β ds) p/p w(t) dt whee φ(t) =t β. To finish the poof we only have to check that S φ is bounded in L p/p dec (w) and this is equivalent to showing that w B (1+β)p. Theefoe, it suffices p to choose β> 1such that (1+β)p 1, to get that p = q, i.e., β = qp p f(t) p w(t)dt ϕ(1) p/p w Bq g(t) p w(t)dt. Taking the infimum of such q s we ae done. 3. Application and examples In this section, we shall pesent mainly two applications which have inteesting consequences. Both of them ae consequences of the following obsevation: Remak 3.1. It has been implicitly poved that, given <p< fixed and a pai of deceasing functions (f,g), f(t)w(t)dt C w g(t)w(t)dt holds fo evey w B p with constant C w depending only on w Bp if and only if, fo evey s> and evey 1 <β<p 1, f(t)t β dt C β g(t)t β dt, with C β independent of s. Application I. The above obsevation is especially useful fo chaacteizing the boundedness on L p dec (w) of cetain opeatos. Theoem 3.2. Let T be an opeato such that i) fo evey deceasing function f, Tf is also a deceasing function wheneve it is well defined; ii) fo evey deceasing function g, afunctiont g is well defined by T f(t)g(t)dt = f(t)t g(t)dt, f. Let <p< be fixed. Then, (3.1) T : L p dec (w) Lp (w) is bounded fo evey w B p with constant depending only on w Bp if and only if, fo evey, s > and evey 1 <α<, (3.2) with C α independent of and s. Tχ (,) (t)t α dt C α min(, s) α+1,

7 R. DE FRANCIA S EXTRAPOLATION THEOREM FOR B p WEIGHTS 635 Poof. If T satisfies (3.1), then taking f to be a deceasing function, we can apply Theoem 2.1 to the pai (Tf,f) to deduce that T : L 1 dec(w) L 1 (w) fo evey w B 1, and by the pevious emak this is equivalent to having that, fo evey s> and evey 1 <α<, f(t)t (u α χ (,s) (u))(t)dt = Tf(t)t α dt C α f(t)t α dt Now, it is known (see [5]) that the above inequality holds fo evey deceasing f if and only if, fo evey >, min(s,) Tχ (,) (t)t α dt = T (u α χ (,s) (u))(t)dt C α t α dt as we wanted to show. C α min(, s) α+1 In paticula, we can conside integal opeatos with positive kenel, which have been intensively studied in [9]. Coollay 3.3. Let Tf(x) = f(t)k(x, t)dt with k a positive kenel such that, fo evey deceasing function f, Tf is also a deceasing function wheneve it is well defined. Then, T : L p dec (w) Lp (w) is bounded fo evey w B p with constant C w depending only on w Bp if and only if, fo evey, s > and evey 1 <α<, (3.3) with C α independent of and s. k(x, t)x α dt dx C α min(, s) α+1, Similaly, in the case of two linea opeatos: Coollay 3.4. If T 1 and T 2 ae two linea opeatos satisfying the hypothesis of Theoem 3.2, we have a) (3.4) (T 1 f) p (t)w(t)dt C w (T 2 f) p (t)w(t)dt fo evey w B p and evey deceasing function f with C w depending only on w Bp if and only if, fo evey, s > and evey 1 <α<, T 1 χ (,) (t)t α dt C α T 2 χ (,) (t)t α dt, with C α independent of and s.

8 636 MARÍA J. CARRO AND MARÍA LORENTE b) If T j ae integal opeatos with positive kenels k j satisfying the hypothesis of Coollay 3.3, then(3.4) holds fo evey w B p if and only if, fo evey, s > and evey 1 <α<, (3.5) k 1 (x, t)x α dt dx C α k 2 (x, t)x α dt dx, with C α independent of and s. Examples Let us now give some examples of well known opeatos fo which boundedness on L p dec (w) is tue fo evey w B p and examples in which this condition fails. Example I. The Caldeón opeato. Let λ, β, γ > with λ βγ and let us conside the opeato x β Tf(x) =x λ t γ 1 f(t)dt. Then, T is an integal opeato with kenel k(x, t) =x λ χ (,xβ )(t)t γ 1 and hence using Coollay 3.3 it is immediate to see the following esult: Theoem 3.5. Let T be the Caldeón opeato defined above. Then, the following conditions ae equivalent: (i) Thee exists <p< such that T : L p dec (w) Lp (w) is bounded fo evey w B p. (ii) Fo evey <p<, T : L p dec (w) Lp (w) is bounded fo evey w B p. (iii) β =1and γ = λ 1. Example II. The Riemann-Liouville factional opeato is defined by x R λ f(x) =x λ (x t) λ 1 f(t)dt, with <λ 1. Theoem 3.6. Fo evey <p<, the opeato R λ : L p dec (w) Lp (w) is bounded fo evey w B p. Poof. In this case k(x, t) =x λ χ (,x) (t)(x t) λ 1. We aleady know that, in ode to pove the esult, it is enough to show that fo all 1 <α<andall, s > we have (3.6) k(x, t)x α dt dx C α min(, s) α+1. To see this, suppose fist that s. Then, fo x (,s), k(x, t)dt = x x λ (x t) λ 1 dt = 1 λ.

9 R. DE FRANCIA S EXTRAPOLATION THEOREM FOR B p WEIGHTS 637 Theefoe, k(x, t)x α dt dx = 1 x α dx = Cs α+1 = C min(, s) α+1. λ Suppose now that <s. Then thee ae two possible cases: s 2 and 2 <s.in the case whee s 2 we have If 2 <s,then k(x, t)x α dt dx 2 2 k(x, t)x α dt dx = 2 k(x, t)x α dt dx C(2) α+1 = C min(, s) α+1. k(x, t)x α dt dx + Fo the fist summand we poceed as in the pevious case: 2 k(x, t)x α dt dx k(x, t)x α dt dx. k(x, t)x α dt dx C(2) α+1 = C min(, s) α+1. Let us estimate the second one. By the mean value theoem applied to the function f(u) =(x u) λ on the inteval [,], we have that thee exists c (,) such that (x ) λ x λ = λ(x c) λ 1.Then k(x, t)dt = x λ ( x λ (x ) λ λ ) = x λ (x c) λ 1 x λ xλ x = x. Theefoe, k(x, t)x α dt dx x α s 2 2 x dx = x α 1 x 2 x dx. Since the function g :[2, s] R given by g(x) = x x is deceasing and α<, we have that ( s k(x, t)x α α (2) α ) ( (2) α s α ) dt dx 2 =2 α α 2 and (3.6) is poved. C(2) α+1 = C min(, s) α+1, Remak 3.7. With the same technique, we can also pove that neithe the adjoint Caldeón opeato defined by 1 Tf(x) =x λ t γ 1 f(t)dt x β with λ, β, γ > no the Laplace opeato Lf(x) = e xt f(t)dt satisfy the condition of boundedness on L p dec (w) fo evey w B p. Inthefistcasethekenelis k(x, t) =x λ χ (xβ,1)(t)t γ 1 anditisenoughtoshowthatitisnottuethatfoeach 1 <α<and, s >, k(x, t)x α dt dx min(, s) α+1.

10 638 MARÍA J. CARRO AND MARÍA LORENTE Let <s<1 <.Then Hence, k(x, t)dt = x λ χ (xβ,1)(t)t γ 1 dt = x λ 1 x β t γ 1 dt = 1 γ x λ (1 x βγ ). k(x, t)x α dt dx = 1 x α λ (1 x βγ )dx 1 x α λ (1 s βγ )dx γ γ = 1 sβγ x α λ dx =, γ fo any α such that 1 <α< 1+λ. In the second case the kenel is k(x, t) =e xt.letustake<s<and obseve that e xt dt = Hf(x), whee H denotes the Hady opeato and f(t) =e t. Then, making the substitution x = u, weget x α k(x, t)dt dx = x α Hf(x)dx = 1 u α Hf(u)du = 1 α u α 1 e u du. u If we keep s =1andlet tend to infinity, then 1 e u uα u du = 1 1 e u uα u du is 1 a positive constant and, as 1 <α<, while min(, s) α+1 = s α+1. α Application II. Let g (t) = inf {s >:λ g (s) t} be the deceasing eaangement of g, wheeλ g (y) = {x R n : g(x) >y} is the distibution function of g with espect to Lebesgue measue, and let f (t) = 1 t t f (s)ds. In [3] the space S p (w) defined by ( 1/p f Sp (w) = (f (t) f (t)) w(t)dt) p < was studied and it was poved that it coincides with the Loentz space Γ p (w) defined by ( 1/p f Γp (w) = (f (t)) w(t)dt) p < if w RB p ;thatis,foevey>, w(s)ds p α w(s) s p ds. To see this, it was poved that if w RB p, the following inequality holds: (f (t)) p w(t)dt (f (t) f (t)) p w(t)dt. Now, making the change of vaiable u =1/t the pevious inequality is the same as (3.7) ( 1 u ) p ( 1 f (s)ds) u p 2 w du u) ( 1 ( u f ( 1 u f ( 1 u )) ) p ( 1 u p 2 w du. u)

11 R. DE FRANCIA S EXTRAPOLATION THEOREM FOR B p WEIGHTS 639 On the othe hand, the following( hold: ) i) w RB p if and only if u p 2 w B p. 1 u ii) g(u) = 1 u f (s)ds is clealy ) a deceasing function. iii) h(u) = 1 u (f ( 1u ) f ( 1u ) is also a deceasing function (see [3]). Theefoe, inequality (3.7) can be ead as g(u) p v(u)du h(u) p v(u)du fo evey v B p with g and h being deceasing functions, and thus it is equivalent to poving that fo evey s>, g(u)u α du h(u)u α du fo evey 1 <α<, which can be seen with an easy computation. 4. Final comments 1) In the context of A p weights developed in [1], [6], [7] and [3], we have a pai of positive functions (f,g) not necessaily deceasing such that, fo some 1 depending only on w Ap satisfying f p w C g p w. Then, it is natual to ask whethe it is tue that thee exists an opeato T satisfying f Tf, Tf Tg, and T : L p (w) L p (w) fo evey w A p. Obseve that if this wee the case, then fo evey w A p, f p w (Tf) p w (Tg) p w C g p w, and we get the extapolation esult in the afoementioned papes. Also obseve that this is what happens in the B p context since upon taking ( 1 t ) 1/p Tf(t) = t p f p (s)s p 1 ε ds ε we have that T satisfies the thee conditions mentioned above fo f a deceasing function. 2) In the context of the intepolation theoy of Banach spaces, we also have a simila esult to the ones developed in [2] (Theoems 3.8 and 5.2): Given two compatible Banach spaces Ā and B and a linea opeato T such that, fo some <p<, (4.1) T : Āp,w;K B p,w;k is bounded fo evey w B p with constant depending only on w Bp,wehavethat

12 64 MARÍA J. CARRO AND MARÍA LORENTE fo evey w B q and evey <q<, T : Āq,w;K B q,w;k is bounded with constant depending only on w Bq. To see this, obseve that by hypothesis, ( ) p K(t, T f; B) ( ) p K(t, f; Ā) w(t)dt C w w(t)dt, t t and since K(t,T f; B) t is a deceasing function, we can apply ou esults diectly. Moeove, we have that (4.1) holds fo some p and evey w B p (o equivalently, fo evey <p< and evey w B p ) if and only if, fo evey > and evey 1 <α<, K(t, T f; B)t α 1 dt C α K(t, f; Ā)tα 1 dt, with C α independent of >. Refeences [1] M.A. Aiño and B. Muckenhoupt, Maximal functions on classical Loentz spaces and Hady s inequality with weights fo noninceasing functions, Tans.Ame.Math.Soc.32 (199), no. 2, MR98957 (9k:4234) [2] J. Basteo, M. Milman and F. Ruiz, On the connection between weighted nom inequalities, commutatos and eal intepolation, Mem. Ame. Math. Soc. 154 (21), no MR (23c:4633) [3] M.J. Cao, A. Gogatishvili, J. Matín and L. Pick, Functional popeties of eaangement invaiant spaces defined in tems of oscillations, J. Funct. Anal. 229 (25), no. 2, MR (26g:4651) [4] M.J. Cao, J.A. Raposo and J. Soia, Recent developments in the theoy of Loentz spaces and weighted inequalities, Mem. Ame. Math. Soc. 187 (27), no MR23859 (28b:4234) [5] M.J. Cao and J. Soia, Boundedness of some integal opeatos, Can.J.Math.45 (1993), MR (95d:4764) [6] D. Cuz-Uibe, J.M. Matell and C. Péez, Extensions of Rubio de Fancia s extapolation theoem, Collect.Math.Exta (26), no. 6, MR (28a:4214) [7] D. Cuz-Uibe, J.M. Matell and C. Péez, Extapolation fom A weights and applications, J. Funct. Anal. 213 (24), no. 2, MR (25g:4229) [8] G. Cubea, J. Gacía-Cueva, J.M. Matell and C. Péez, Extapolation with weights, eaangement-invaiant function spaces, modula inequalities and applications to singula integals, Adv. Math. 23 (26), no. 1, MR (27d:424) [9] Shanzhong Lai, Weighted nom inequalities fo geneal opeatos on monotone functions, Tans. Ame. Math. Soc. 34 (1993), no. 2, MR (94b:425) [1] J.L Rubio de Fancia, Factoization theoy and A p weights, Ame. J. Math. 16 (1984), no. 3, MR74514 (86a:4728a) Depatament de Matemàtica Aplicada i Anàlisi, Univesitat de Bacelona, 871 Bacelona, Spain addess: cao@ub.edu Depatamento de Análisis Matemático, Univesidad de Málaga, 2971 Málaga, Spain addess: loente@anamat.cie.uma.es

ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION. 1. Introduction. 1 r r. r k for every set E A, E \ {0},

$ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION. 1. Introduction. 1 r r. r k for every set E A, E \ {0},$ ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION E. J. IONASCU and A. A. STANCU Abstact. We ae inteested in constucting concete independent events in puely atomic pobability