Maximal functions and related weight classes

Publicacions Matemàtiques, Vol 38 (1994), 127-155 . MAXIMAL FUNCTIONS AN D RELATED WEIGHT CLASSE S CARLO SBORDONE AND INGEMAR WI K Abstract The famous result of Muckenhoupt on the connection betwee n weights w in Apclasses and the boundedness of the maximal operator in Lp (w) is extended to the case p = oo by the introductio n of the geometrical maximal operator . Estimates of the norm o f the maximal operators are given in terms of the A p - constants . The equality of two differently defined A . - constants is proved . Thereby an answer is given to a question posed by R . Johnson . For non -increasing functions on the positive real line a paralle l theory to the Ap-theory is established far the connection betwee n weights in Bpclasses and maximal functions, thereby extendin g and developing the recent results of Ariño and Muckenhoupt . 1 . Introductio n Let f be a nonnegative, locally integrable function defined on (o, oo) . The well-known Carleman inequality (see [4, p . 250 D 00  X  0 0 f exp(J1nf(t)dt)dx < e f f (x)dx , 0  0  0 in which e is the best possible constant, can be considered as the limi t case, as p tends to infinity, of the Hardy inequality for f 00  X  0 0 f (ff(t)dt) P dx (  xp P 1 )Pff() dx . o  o  o Work supported in part by the National Science Council of Sweden and the Italia n Ministero dell ' Università e della Ricerca Scientificae Tecnologica . 128  C . SBORDONE, I . WI K In fact, the geometrical mean of f, exp (1 f o In f (t)dt), satisfies (see [4 , p . 139] } x  x (1 .1)  lim ( 1 J f (t) P dt) P = exp(— flnf(t)dt) . P- , oc x  x o  o We recall that ( p~ 1 ) P is the best constant 1n Hardy's inequality and s o we deduce 00  X  DO (1 .2) lim sup J ( 1 J g(t)dt) p dx = sup J exp( 1 11911P=1 o  x o  IIf111=1 o  x pom °° f in f (t)dt)dx . 0 In the first part of the paper we study analogues of these results in n dimensions for maximal functions and corresponding weights . To b e more precise we need some notations . We let Q stand for a cube with axes parallell to the coordinate axe s and its Lebesgue measure . It is convenient to use a special sign fo r the mean value over Q of a function f ff(x)dx = QI J f (x)dx . Q  Q First we define, for g ELL(Ir) , q > o, the q -maximal function of g b y (1 .3)  M q g ( x ) = sup( ig(t)I q dt) q , QDx ~ where the supremum extends over all cubes Q c R' . For q = 1 we ge t the familiar Hardy - Littlewood maximal function Mg = As a limit case as q tends to zero, we introduce the geometrical maxima l function, Mog, by definin g Mog(x) = sup exp( f Q g(t) idt) . For non-increasing, non-negative functions f on (O, oo) it is easy to sho w that M~ f (x) _ (ff(t)dt) P and Mo f (x) = exp f ln f (t)dt . MAXIMAL FUNCTIONS AND RELATED WEIGHT CLASSES  12 9 The left and right hand sides of (1 .1) therefore are lim M ~ f(x) and p- .->0 0  p Mo f (x) respectively . We prove in Theorem 2 tha t p li m M P f(x) = Mof(x) for x E W . The corresponding limit relation to (1 .2) will be proved as a corollary t o this theorem, but in a much more general situation, where the Lebesgu e measure is replaced by a measure w(x)dx, with w a weight in the A oo - class of Muckenhoup t . In section 2 .2 we study the limit case as p tend s to infinity of the Apconstant of a weight function w . (1 .4)  A P (w) = s up + w(x)dx(-} W — ; 1 ' T (x)dx) ' J  J and defin e A~(w) = sup fw(x)dxexp(1l n Q  Q dx) , A 00 = lim A P (w) . 1 ( x Jensen's inequality implies A oo ( w ) < 71,, ( w ) Johnson in [6] left it as an open problem whether there exists a constan t c such that A,,(w) Ç cA,,(w) . We settle that problem by showing i n Theorem 1, that the two quantities are actually equal . In Theorem 3 we prove that the geometrical maximal function M o gives a bounded mapping of L 1 (w ) into L 1 (w ) if and only if the weigh t function belongs to thereby extrapolating from A p the classica l result of Muckenhoupt [7] on the Hardy-Littlewood maximal function . In the secondpart of the paper we restrict our concern to the cas e of non-increasing,non -negative functions on (o, oo) . Following a recen t paper of Ariño and Muckenhoupt [1], we continue to study the classe s of weights for which the maximal operator is bounded on non-increasin g functions in L p (w) . It turns out that we have here a more or less complet e analogy with the A p - classes . Also in this case we study the limit cas e as p tends to infinity . Our final specialization is to the case when wi s nondecreasing . 130  C . SBORDONE, I . WI K 2 . The general cas e 2 .1 . Notations and definitions . For a non-negative, locally integrable (weight)-function w we defin e L P (w) as the class of all mensurable functions f such tha t f f(x)w(x)dx < oo, with f ILP(w) = ( f I j~gn  pg n A p is the class of all weight functions w with finite A p (w) . As p tend s to infinity in (1 .4) the secondfactor on the right hand side tends t o exp(f ln  See [4, p . 71j . It is therefore natural to define th e Q A oo - constant of w a s (2 .1)  A~(w)=s~ f w(x)dx• exp(+1n W ~ x ) dx) . Q  Q Usually A~ is not defined as the class of functions for which the righ t hand side of (2 .1 ) is finite . However, it has been proved by S . V . Hrusce v [5], and J . Garcia-Cuerva- R . de Francia [3, p . 405] that this is a n alternative definition of A,, . When studying the boundedness of the maximal operators it is convenient tohave the following notations for weight functions w mp(w) = sup JM 1 f(x)w(x)dx . IIfIILm =1 R " It is easy to see tha t (2 .2)  sup fM 1 f(x)w(x)dx = sup f(Mf(x))7'w(x)dx) . IIfIILcm ) = P  IIfIILPc, ) - 1 We therefore pu t (2 .3)  m,,,(w) = sup fMof(x)w(x)dx . IIIIIL( .)=1 R n The non -increasing and non -decreasing rearrangements of a function f will be denoted by f* and f * respectively and are defined b y f* (t) = sup ecs inf f (x) , f * (t) = mf ecs cup f (x ) I E I= t E  I E I= t  E and then become continuous to the right and left respectively . MAXIMAL FUNCTIONS AND RELATED WEIGHT CLASSES  13 1 2 .2 . A~ as a limit case of A p . It is well known that A,,,, can be defined as U A p . Let w be a functio n p> 1 in A~ . Then there exists a number p 1 , 1 Ç p 1 C oo such that w E A p for p ~ p 1 . Since, by Hdlder's inequality, A p (w) is a decreasing functio n of p, we have two candidates for the A oo - constant of w, namel y .4(w) = lim A P (w) and A,,, 3 (w) as defined by (2 .1) . By Jensen ' s inequalit y exp (1 Ing(x)dx)  g(x)dx . Q  Q We apply this with g =  , raise both sides to the power p and obtai n exp  In ~~ x) dx) Ç  w(x x ~ Q  42  ) which means that A,,(w) < A P + 1 (w) and thus A oo (w) < A 00 (w) . It ha s been an open question, [6, p . 98], whether there exists a constant c suc h that fl c . ° (cv) < cA co (w) . Here is the answer . Theorem 1 . If w E A,„ Q , then A,,(w) = A co ( w ) . In the proof of the theorem we will use the following lemma . Lemma 1 . Let f be a non -negative integrable function on (0,1 ) an d p a real number, p ~ 1 . Pu t 9(x) _ (f (x)' zf f (x) < e { e,  elsewhere . 1 (2 .4) (f Í ( x ) dx ) P — (fg(x)dx) P P( f .f ( x ) dx ) ( f f ( x ) dx ) p— i 0  0  E  0 where E = {x E (0, 1) ; f(x) > e} . Then 132  C . SBORDONE, 1 . WI K Proof : This is an immediate consequence of the inequalit y b p - a p Ç p{b - a }b p- 1 , for O Ça Cb, p_ ~ 1 , f b = J f(x)dx and a = J g(x)dx . ■ o  a Proof of Theorem 1 : Suppose w is a function in A px (R n ) with A 1 - constant A . We will show that, for every p ~ p i - 1, we hav e d x (2 .5)  su w(x)dx • (  )p < (1 + (p,p i ,A)) . Aoo(w) , w ( x ) P where lim 5(p, p 1 , A) = O . This implies A . c . () (w) Ç A oo (w ) which prove s p -+ o o the theorem . Except for the supremum the left hand side of (2 .5) is invariant unde r changes of scale in Rn and also under multiplication of w by a positiv e constant . Without loss of generality we may therefore assume that tha t IQ1 =1 andw(Q) = f w(x)dx = 1 . ~ We denote here by w * (t), 0 < t Ç 1 the non -decreasing rearrangement of the restriction of w to Q . Then, from [10, p . 250] e .g ., an d the definition of A 00 ( w ) , we conclude tha t (2 .6)  w * (t) > A -i t pl-1 and exp (fin w ~ t ~dt) < A~(c~) . o Since es <1+x+x 2 , for x<1 , we hav e  dt P - J exp (-in  )dt< (1 + j  -  p w(t)  - 1  p ln  ~~ ~-p 2 (ln  ~~ )2)dt , c .~ * t  w * t 0 W* ` tJ  o  o if w * (t) > e –p . This means tha t 1  1  1 1 dt  ~p exppin(1 + fin  1 dt + 1 (ln(t))2dt) .  1 p o  W ( t )  o with MAxIMAL FUNCTIONS AND RELATED WEIGHT CLASSES  13 3 The inequality : ln(1 + x) Ç x, for x > -1, implie s 1  1  1 (2 .7)  (%  dt 1 )p exp (fin  1  dt) - exp (1 J [(in  1  ) Z dt) . w * (t)T ,  w .(t)  p  w .(t ) We now use (2 .6) to fin d [(in  1 o J  w . (t) ) 2 dt < f (lnA+ (p i – 1) In ~ ) Z dt = c ( p l, A ) . 0 The second factor on the right hand side of {2 .7} therefore converges t o 1 as p tends to infinity . This proves the theorem for functions w that ar e bounded below by a positive constant a . (We just choose p so large tha t e -p [ Q .} If that is not the case we construct a new functio n { w(x), if w(x) > e - P e- P ,  if w(x) < e -P . It is easy to check that (2 .7) is valid with w * replaced by (w) . After th e replacement we increase the right hand side of (2 .7) and use the secon d inequality of (2 .G) to fin d 1  1  1 (2 .8) (1  dt  )~ exp (fin  1 w .(t) -dt) • exp (1 J [(in w,, I (t) ) 2 dt) < (wp) .(t) P  P c A oo (w) • exp(c(p~ A ) ) . We now take a closer look at the left hand side of this inequality . W e want to replace (w p ) . by w * and estimate the difference in a way tha t is independent of our particular choice of cube Q . For this we will us e Lemma 1, applied with f (x) = w * (t) - t , . From (2 .6) we conclude tha t (t)" 1 >e  ~ A --1 t pI - 1 [ e - p ---> t[  t p . Hence E C (4, AP~x ~ e- P1~ 1 ) in the lemma and we have the estimat e t P dt  [ A l  d t w ¡t p  (1 — p l --1 . )  .~ -- p 1 .~ . ~ E  *l)  a  p c( p , p i , A) . 134  C . SBORDONE, 1 . WII { This estimate is used in (2 .8) and, combined with Lemma 1, the conclu - sion is 1  1 ( J w*( ~ > 1 ) P < A~(w) • exp c(p p A ~ + d ( p , p i, A )( f w*~t>p)P—1 ~ P where d(p, p 1i A) = p c(p, p 1 , A) . This quantity obviously tends to zer o as p tends to infinity . We now take an arbitrary E, o < E < 1, and choose p so large tha t exp e(p l' A) C(1+ E) and d(p,p i , A) Ç E . Put v = ( f -- d -~- } p . The n p  o w(t) P v 3 1 by Hólder ' s inequality an d v < (1 + E)A .(w) + evo c (1 + E )A,,(w) + ev , v < 1 +  E' A,,(w) < (1 + 3e)A oo (w) . Now we take supremum over all cubes Q and ge t A p + i (w) = ( 1 + 6 (P, PI , A)} . A .( w ) ç(1+ 3 e) A,,,(w) , where 6( p, p i , A) tends to zero as p tends to infinity . This means tha t lim Ap(w) ç A,,(w) , P-- >0 0 which concludes the proof of the theorem . ■ It is also possible to have an estimate of the rate of convergence . A simple analysis of the various inequalities will give us the estimat e c ( p i , A ) 6 (P, , A) ~  P where C ~p 1 , A) is a constant depending only on p i and A . MAXIMAL FUNCTIONS AND RELATED WEIGHT CLASSES  13 5 2 .3 . M o as a limit case af 1V1 I . p Corresponding to the preceding paragraph, we present here a resul t on the geometrical maximal function, Mo f, the precise importance o f which is demonstrated in Theorem 3 at the end of this paragraph . Theorem 2 . Suppose that f lies in Lzoc(Rn ) , for some a > O . The n we have P i , m M P f(x) _ M o .f ( x ), Vx . Proof : By Jensen's inequalit y exp In f (x)dx (ff(x)dx)P . We take the supremum over all Q that contain x and obtai n Mo f(x) ~~I x f(x) , Vx , and letting p tend to infinity this give s (2 .9)  Mof(x) Ç Iim N .fY f (x), Vx . p-+oo p It remains to prove the opposite inequality of (2 .9) . We assume first tha t the number a in the theorem equals one . Then we use Lemma 2 below , according to which we have, far every E E (0,1) and cube Q : 2 .10) i  1 ( f (x)dx) P < exp In f E (x)dx • exp (ln€)2 +  p  p  + ep —  1 f f(x)dx , Q  Q  Q where f (x),  if f> e f f (x)d x J-f( x ) dx elsewhere . QQ From this we conclud e su  t dt P Csu exP fin  (lnE) 2 ~-1  p P~ f~~ ~  P~  f£(t)dt•exp  + ep ij [ f(t)dt) . Q  Q f E is independent of p . Letting p tend to infinity therefore gives u s lim M I . f (x) sup exp ( 1n f E (t)dt) . p—}oo p  Q 3x Q 142  C . SBORDONE, 1 . WI K In analogy with the n - dimensional case we define, for p ~ 0 0 0 m' (w) = sup  f (x)w(x)dx , P 0 but now the supremum is taken over all non-increasing f on (0, oo) wit h II f II L(w) = 1 . We note that m ' P (w) is the infimum of all C such that (3 .1 ) holds . Correspondingly we define 0 0 m' oo (w) = sup I-Mof (x)w(x)dx , f o where the supremum is taken over the same class . 3 .2 . The analogy between A p and B p . In Lemma (2 .1) of [1] there is a proof, of the fact that w E B p implie s that w E B p _ E for some E > 0 (a similar result is in Strdmberg-Torchinsk y [S, p . 12D .. We give here a short and sharp proof of that lemma . Lemma 3 . Suppose that 0 C p < oo and w is a function in B p suc h tha t (3 .4)  J w(t ) tPdt < B f (t)dt,  Vx > 0 . x  0 Then w E Bp_ E f or E C - B -'+— 1 i . e . w E B pl for p i C B +~ p and B p (w) Ç  Bp  The upper bound of E is best possible . p -E(B+1) • Proof : Choose E < B+1, multiply (3 .4) by x E -1 and integrate fro m r to infinity . A change of the order of integration on both sides the n results in ~  r  7w(t ) 1 ~ W ~ t ~ (t E r E )dt <  B (f W  (t) dt + W  fdt) , e  t P  p — E  r P-E  t P- E T  0  r which gives us, after once more using (3 .4 ) 00  00  r ( 1 — B ) w(t) dt < rE w(t) dt +  B  w() d p - E ' 1 tP-1  E f tP  ( p_ € ) r p_Ef t t r  r 0 r 1  1  B < ( + p  )  f w(t)dt . 0 MAxIMAL FUNCTIONS AND RELATED WEIGHT CLASSES  14 3 This is to say that w E B p _, for ~ C ~--- and B p (w) C p— B  B+~  — Eus p +i} • To show that the limit is best possible we just take w(x) = x a , a > -1 , and p > a+ 1 . Then the Bp - constant of w is p  ' á % . By the result aboy e we see that w E B p , for  p  =a+1 . a+ i p -cx—i + 1 Of course no smaller p's are possible, if the left side of (3 .G) is to con - verge . ■ We will extend the results of [1] to the geometrical maximal functio n M q f (and also in some cases to 0 < p Ç 1 . To make apparent th e parallellity with the ordinary A p - classes, we introduce a class B oc . I t will soon become evident that the corresponding to the definition of A o o would be to define B~ as the class of weight functions, for which ther e exist two constants r < 1 and k > 0 such tha t t f w(u)d u 1> t r  °  > k . x f w(u)d u o This is equivalent to the following definition, which is more easy to grasp . Definition . B~ is the class of non -negative, locally integrable functions w on (O, oo) with the property that there exist two constant s r,OCrC 1 andC> Osuchtha t f f (3 .5)  C J w(t)dt > J w(t)dt, dx > O . 0  0 Remark . We could equally well have made the definition with r = instead of being arbitrary . This would seemingly be more restrictiv e for r >  However if w satisfies our definition with an r > we ca n iterate the inequality approximately (— log 2 r) —1 of times to see that i t is satisfied for r = but with a larger C . Definition . The doubling constant, d(w) is the minimum of all C such that (3 .5) is valid with r = If d(w) is finite we will say that w has the doubling property . It is immediately evident from the definition that B p , c B p and als o that B p (w) Ç Bp l (w) if p i C p . C . SBORDONE, I . WI K A function in B p obviously has the doubling property . (Just relax i n the definition (3 .2) by reducing the interval of integration on the left i n (3 .2) to become (x, 2x) . However, we can do much better and obtain a n estimate of C in (3 .5), an estimate that dependson r and also can b e used as an alternative characterization of B p . (Compare corollary 1, p . 250 of [10} . } Theorem 4 . A weight function w is in B p , if and only if there exis t constants p i , 0 <p i C p, and C such that t  x (3 .6)  fw(u)dú > C(~) pl fw(n)du,  for x > t . o  o If Cpl (w) is the maximal C for which (3 .6) holds, the n Cp 1 (w) >  1  for p i > 2B + and B p (w) <  p  2B P (W) -}- 1  2B ~ -  21p  CPl (w)(p —  pi ) Proof .• Suppose first that w E B p and put B p (cv) = B . By the precedin g lemma we know that, for p i = 2s+ a zs+i p < p , w E B p, with constan t 2B . Thu s 2 B f w( u ) du > 91 Iw(tt) du >  2 r(k+1)p1 fw(u)d u up  ~ _  1 _~  _ o  x  k =o  x2 k N–1 x2 k + 1 ~ . E 2 –(k+1)P1 fw(u)du .  k =o  x2 k This gives, for every x~ 0 , N–1 x2 k  x2 N (1 — 2--p1) E 2 -kp1  w(u)du + 2 –Np1  w(u)du Ç k=l  o  0 x C (2B+ 2–P1 ) weu, }du . 0 Therefore, taking only the last term on the left into account and replacin g x by x2 -N , we fin d x2 -N  x 1 J w(u)du > 2 N P1(2B + 1) f w(u)du . o  o 144 x  o0  0o  x2 k+1 MAXIMAL FUNCTIoNS AND RELATED WEIGHT CLASSES  14 5 For x2 -(N+1) c t C x2 - N we hav e t  x2 - ( N}1 )  x 1 fw(u)du > J w(u)du > 2 Np 1(2B + 1) f w(u)du > o  o x  >   1  ( t ) P f()du . 2B+1 x o Thereby we have proved the necessity of the condition and the firs t inequality between the constants . To prove the suffiency we assume p i < p and t  x fw(u)du~C(y '1 fw(u)du , for 0 < t < x . x o Multiply this inequality by t rpx x px - 1-p . We ge t t  x 1 Ctpl . x1+ p -Pl f w(u)du > xp+l fw () du . o  o This inequality is valid for o Ç t Ç x . We integrate with respect to x over the interval (t, o()) and changethe order of integration in the righ t member . The result is t  t  1  p f w(u)du > 1 f w(u) du + 1 J w ( u )du . C' (p pi) t  P  t  p  u o  o  t Hence w(u)du C  p  . 1 w(u)du . _ C' . (p _ p i ) t P u t  o This completes the proof of the necessity and the second inequality between the constants . ■ We complete the analogy b y Theorem 5 . B~ =U B p o o p>o C . SBORDONE, 1 . WI K .Proof• Suppose w E B p for some p > O . It is immediate from Theore m 4 that w satisfies the requirements for being in B oa . Thu s p> 0 Suppose on the other hand that w E B~ with d(w) = C, i .e . z x  x fw(t)dt C J w(t)dt . o  o 14 6 This means that 2 x f w(t)dt (C — x 1) f w(t)dt . o Thus 00 J W p ) dt = t x 2 k+i ac  2 k+i x w (t)O °  1 dt ç tp  E 2k px p  w(t)dt Ç (C —1 ) 00 k=0 2kx  k =o  2 k x 2 k x  x  x  2kPxP J w(t)dt < (C — 1) E 00  2k C p xp J t= (2 P  G , ) x ) p fw(t)d t k—0  o  k—0  o  o for p > log 2 C . So w E B p for p > log 2 C an d B oo C  Bp . p> o and the proof is complete . ■ 3 .3 . m ' 0 0 as limit case of mp . In Chis section we will for convenience use a special notation, L d (w) , for the set of all non-negative, non-increasing functions in L (w) . Theorem 6 . Ilrlo is a bounded operator on L d (w), (i .e . m' o ,, C oo), i f and only ifw E B oa an d d(w) < (2m(w) — 1 ) 2 Ç c o( d ( W )) s .a MAXIMAL FUNCTIONS AND RELATED WEIGHT CLASSES  14 7 where Co is an absolute constant . Proof of part 1 : We give first a short proof of the first part of th e theorem without estimates of the constants . Suppose therefore that M o is a bounded operator on Ld (w) . Since  f(x) tends monotonicall y p to Mo f(x) as p tends to infinity, it is an immediate consequence of th e monotone convergence theorem tha t lim m p ' (w) = m cc ( w ) C cc . p—to o Thus mp (w) C oo for p large enough . By [11 this implies that w E B p fo r p large enough and then, by Theorem 5, w E B oo . If on the other hand w E B o o , then, by Theorem 5 again, w E B p for p large enough and the result in [1] implies m p ' (w) < oa . Henc e m oo (w ) C oa, which means that Mo is bounded on L d (w) . ■ We will now present a complete proof of Theorem 6 that does no t rely on the results of Arino and Muckenhoupt, but is based on anothe r technique . It has the advantage that it gives estimates of m' .(w) i n terms of d(w) . To complete the proof we need the following lemma . Lemma 4 . Suppose E a k is a positive series with sum A . Foren a -D o new series with the convoluted terres a m E 2Ef k - m 1 ' m=—oa 2E+ 1 b k _ . ~ ak,  2 -E < bk+1 C 2 € and  b k Ç  A . — —  ~ bk  2~ — 1 Proof : bk = . . . + ak—22—2E + a k-1 2-E + a k + a k + 1 2 —2 ' + ak+22 --2E . . . Now the two first properties are trivial and the third follows from a change of order of summation . ■ Proof of Theorem 6 : Suppose first that m' oo (w) = K < oo . The n 00  D o (3 .7)  JMof(x)w(x)dx < K J f (x)w(x)dx, Vf E Ld(W) . 0  0 b k = Then k= —oo C . SBORDONE, 1 . WI K Choose a in o< a < 1 and put 1, oCxÇ r f(x) = a, r<x<2 r o,' x > 2r . Then f E L d (w) and { Í1a, l-z 7 , r o C C x x C r O,  x > 2r . 2 r Mof~x) = a  r We apply formula (3 .7) and obtai n 3 2r r 2 r f w(x)dx + J a l- =w(x)dx < K( J w(x)dx+ faw(x)dx) . 3 0 r Thus 2r  r a f (a= — K)w(x)dx < (K — 1)J w(x)dx . r  0 We choose a = (2K) -2 . Since > we obtai n 2r  f f w(x)dx < 4K(K — 1) J w(x)dx , r  O which means that w E .B 0 , 0 with doubling constant at most(2 K — 1) 2 . I t also follows from this inequality that K has to be strictly greater tha n 1, otherwise w has to beidentically zer o Suppose on the other hand thatw E B oo with doubling constant C . Choose the sequence {a k } "2 o such tha t a k f w(x)dx C- k . o Using the doubling property we se e ale  ak + 1  2a k + i C- k = w(x)dx = C  w(x)dx ~  w(x)dx . 0  0  0 148 o 111IAXIMAL FUNCTIONS AND RELATED WEIGHT CLASSES  14 9 Therefor e (3 .8)  a k ~ 2a k + 1 . Take an arbitrary f E L d (w) and pu t o a J f (x)w(x)dx = K . 0 Since f is nonincreasing this means tha t a CO  k 1 °°  ¡ E f( Cxk )  w(x)dx = c ---- E f (ak)C_ k . k=-oo  C ak +l  k = —o o Now we can use Lemma 4 with ak = f (ak)C —k and obtain b k ~ a k wit h KC 2' + 1 E b k <  —  E —~ C—12 - 1 We can define a new nonincreasing function g with g(x) ~ f(x) and g(c) = C kbk. Obviously Mog ~ M o f . Jensen ' s inequality give s ak  a k M o9( a k) =exp f ing(x)dx c ( 1 gP(x)dx) p < 0  0 1 oa i  1 O°  ~  ¡¡  ¡  ¡  (  m+ 1  C l  E g P ( a m +1 )l a m — Cxm+1 ))P Ç L  P bm +I am )P • ak m =k  ~~ m =1 c By (3 .8), the terms in the last series of this estimate decrease geometrically with a quotient that is at most C 1 T ; 212 —1 . Thu s ¡¡  1 M®f(ak) Ç Ck+1bk+1(- 1 C~ i 2 E 2 —I )P y — ~ if p is large enough . We are still free to choose E and p . We can fo r example choose E = and p = = 3 in C if C ~ e 8 . If C < e 8 we Cak e p = 10 . Some elementary calculations then show tha t MO f (ak) < DC k+1 bk+1C 3 . 7 K~ 150  C . SBORDONE, I . WI K where D is an absolute constant . Therefor e f Mo f(x)w(x)dx ~ DC C , 1 ~ Mo .Í(ak+i)C–kcic+lbk-I-1C3 .7 = o  k=–o o = DC 3 ' 7 (C — 1) E b k EC 4 .7 K , -0 0 where E is an absolute constant . We deduc e m ' ,,(w ) ç BC 4 . 7 and the theorem is proved . ■ Now that we have the tools, it is tempting to prove theorem (1 .7) i n [11, for 0 C p < oo . We will use Theorem 4 and the technique of Theore m 6 . Theorem 7 . For 0 C p Ç oo,  is a bounded operator on L d (w) i f p arad only ifw E B p . Proof : p = oo is already treated in Theorem G . In the easy necessity part, we have nothing new to offer . It follow s directly by chosing f = x ( o ,$) in (3 .1) . For the sufficiency part we suppose that w E B p with B p (w) = B . I n Theorem 4 we take E = 4( $ +1) , and put p 1 = p - 2E and p 2 = p - E . Th e conclusion is that BP1 (w) Ç 2B and J  1 w(u)du > 2B + 1 fX  = r2 2 B  J w(u)du for r 1 . 0 We now choose ro < 1 so small that 71 2 ' (2B + 1) = 1 . This give s r Q x f J w (u)du > rw(u)du, b'x > 0 . o a k Put r~ 2 = C 1 and choose {ak}oc so that f w(u)du = C - k . Then w e o have «k + 1 a,  aic +i  r a fw(x)dx=co  w(x)dx ~  w(x)d x q  o  o r X P~  rP 1 - rp1  ~ p2  ' 2B - ~ -~- ~ - 1 0 MAXIMAL FUNCTIONS AND RELATED WEIGHT CLASSES  15 1 and therefor e (3 .9) a k + 1 Ç ro ak . Now we can proceed as in the proof of Theorem 6 (with C replaced b y Co ) to find 1 °O  m+1 1 Mif( a k) ç Mlg(ak) < (- E C o P bm+ 1 G1Gm) P . P  p  a k m = k By the definition of Co and (3 .9) we deduce that the terms of this serie s 1— u decrease geometrically with a quotient that is at most ro ~ 2 - 7 : . . W e have not yet decided what E > 0 (in Lemma 4) should be . We just hav e to take E < (p2 — p) Il to be sure of obtaining geometrical decreasing . Take for instance E equals half that quantity . Then we hav e M 1 f (ak) C ( B , p ) C ó +1b k+1 , p where C(B, p) is a constant, depending only on the indicated quantities . This give s 0o  o o f M P .Í ( x ) W ( x ) dx C(B,p) E Có +2 b,+zco ; k < C i( B , p ) K= o  -00 o o = C 1 (B, p) f (x)w (x)dx , 0 by which we have proved the sufficiency part of the theorem . ■ 3 .4 . A~ and nondecreasing weights . We end this paper by proving two theorems, the first of which is a n extension to g = oo of Theorem (1 .10) in [11 . The second is an analog y with Theorem 5 far nondecreasing weights w . Theorem S . If w E A' o‹) , then m p ' (w) C oc for p large enough . A nonwdecreasing w lies in A~ if and only if m' .(w) < oo and the n m' co (w) ~ . A' oo (w) . Theorem 9 . For w non -decreasing we hav e wEA ' . <=> w E U Ap . The proofs of these two theorems are based on the following lemma :