Weighted norm inequalities for maximal functions from the muckenhoupt conditions

Publicacions Matemátiques, Vol 38 (1994), 25-36 . WEIGHTED NORM INEQUALITIE S FOR MAXIMAL FUNCTION S FROM THE MUCKENHOUPT CONDITION S Y . RAKOTONDRATSIMB A Abstract For some pairs of weight functions u, v, which satisfy the wellknown Muckenhoupt conditions, we derive the boundedness of th e maximal fractional operator M S (O < s C n) from L~ to Lú wit h q C p • O . Introductio n Let u, v weight functions on R n , n ~ 1 (i .e . nonnegative locally integrable functions) . The fractional maximal operator M S (O Ç s < n) i s given b y (M S f)(x) = sup {  I Q ~ f 1 ; Q a cube with Q x } Throughout this paper Q will denote a cube with sides parallel to th e co-ordinate planes . Let 1 C p, q < oa, with l p — 4 Ç ñ . It is fundamental in analysis t o give a characterization of the pairs of weights (u, v) which satisf y (o) II M sfIILú ç C I I f I I Ly for all functions f, C = C( s, n, p, q, u, v) ~ O . Here II g II L1 ., denotes (f lan 11Tw dx) , with dx the Lebesgue measure o n R n . In the case of 1 C p Ç q < oo Sawyer[Sa2] showed that the inequalit y (0) holds if and only if (u, v) E S(s, n, p, q)i . e II~ M ~ v ~11~~~~~IIL~ ç C II v p1 1~QIILv = II I [QIILP 1 Coo 26  Y . RAKOTaNDRATSIMB A for all cubes Q, here S = S ( s, n, p, q, u, v) > O . A known necessary bu t nat sufficient condition for (0) [Mu] is (u, v) E A(s, n, p, q) i . e - 1 s '  1  1  1 Q + -t,  (J v ) P - ~ - ~ 1  ç A for all cubes Q , IQI Q  IQI Q with A = A(s, n, p, q, u, v) > O . As we will recali in Section 2, this condition is verified more easily tha n the first one . Pérez [Pe] (see also [Sal]) proved that (u, v) E A(s, n, p, q ) implies the inequality (0) whenever d a = v - 7 ± 1 dx E A, i .e . far som e 6 > 0 : El, < I - E - J - for all cubes Q and for all measurable sets E C Q (21,  1Q I here I denotes f Q u . In fact the equivalence between (0) and (u, v) E A(s, n, p, q) is also valid with a weaker condition on da, far instance i n [Ra3] it was proved that it is sufficient da E Bs i .e . b ~< (lQ'l1 for all cubes Q, Q' with Q ' C Q with [1 - - Ç S . As we will see in Section 2, measures dp, can be foun d such as dp, E B 6 but dp « A 00 . The condition p - < 1 can be derive d from the inequality (0) by the Lebesgue differentiation theorem . Henc e for s = 0 (Mo is the Hardy-Littlewood maximal operator), the inequalit y (0) must only considered for q Ç p . The case p -- q was studied by Muckenhoupt [Mu] for u = v and by Sawyer [Sa2] for general weight s u, v . Far q < p, a characterization of the pairwise of weights (u, v ) satisfying the inequality (0) was given by the author [Rai] ; but th e condition used is difficult to check . Therefore 1 < q < p < oo a natural question is : " does (u, v) E A( s , n, p, q) imply (0) whenever do- E A,, . In this paper we give a positiv e answer with the additional assumptions u dx E B v , v - P 1 1 dx E B P wit h 0C v,p and [1---  Cp 1-p +v- . We state our main result inSection 1 . In Section 2 we give some usefu l remarks and observations about the weight condition B . The proof o f our main result is inSection 3 . A paper of Verbitsky [Ve] concerning th e characterization of the problem (0) with q C p and for general weights u, v appeared when this manuscript was written . Acknowledgement . The author wauld like to thank the referee fo r his helpful comments and suggestions . WEIGHTED INEQUALITIES FOR MAxIMAL FUNCTI4NS  2 7 1 . The main resul t To include many classical maximal functions, we deal with the operato r (Mf)(x) = sup {QQ1 —1  If 1 ; Q a cube with Q ~ x where ~ is a map defined on the set of cubes, taking its value in JO, oo [ and satisfying the following growth conditions : 7-t1 there is C ~ o such a s . P( Q 1 ) C ■ D( Q 2 ) for all cubes Q 1 , Q 2 with Q 1 C Q 2 ; ?-t 2 there are C 1 , C 2 > 0, a, 77 ~ 0 such a s C 1 t"1 . (Q) Ç . 1)(tQ) Ç C2t n% D( Q) for all cubes Q and all t > 1 . When (19(Q) = 1 the Hardy-Littlewood maximal operator is obtained . The fractional maximal operator M s (O < s < n) is given by . ¿(Q) = IQ 1n s . Maximal operators connected to the Bessel potential operato r [Ke -Sal are defined by (Q) =  ço(s) ds ; and generally M~ arise s in studies of other potential operators [Ch—St—Wh] . Let 1 C p, q < oo . We say that the inequality B(M, D , p, q, u, v) hold s for a constant C > o whe n f H L?, ç C 11 f 11 141 for all functions f and we write (u, v) E A((I), p, q) if for some constant A > o x4  ) 1— * (Q)lQI -1— 11 -,  1 Ju) (fv ) T r ,= 1  <A for all cubes Q . IQ1 Q  1~1 Q In this paper we always adopt the convention 0 .00 = O .  B y B(M I ., p, q, u, v) and the Lebesgue theorem, we see that if u  o it i s necessary to suppos e ('H 3 )  lim (ii*) C oo . IQH q For instance x3 is satisfied if p — Ç A . For ~ (Q) = 1, the hypothesi s ?-( 3 implies q Ç p, and for •EQ) = lQi ñ it means - -- Ç p 4  ñ . Let > o and w be a weight function . As in Section o, we writ e w dx E B e if there is C~ osucha s 1Q'Iw < fQ'I ~ for all cubes Q, Q ' with Q' C Q . lQl w  IQ I Now our main result can be stated : 28  Y . RAKOTONDRATSIMB A Theorem . Let 1 C p, q < oc and (1) be a function which satis fies  7-t 2 , 7-~C 3 . A) If the inequality P(Mcp ,p, q, u, v) holds for a constant C > o the n (u, v) E A( . P, p, q) with the constant A = C . ~ B) Let 1 C q C p < oc and da = v-P—1 dx E A . Moreover assum e u dx E B u , vp x 1 dx E B e with D < v, o and (1-a) <p( 1 - + v ( 1 ) . If (u,v) E p, q} then the inequality P(M~, p, q, u, v ) holds for a constant Ac, c = c((T . , n, p, q, u, v) > O . The part B) i s also valid when do- E B o with 1-A < o . Actually the constant c depends on the fact that u dx E B u , v T P-1 dx E B e but not directly on u and v . The result stated in the introduction is now easily derived from the theorem by taking 1 . (Q ) = IQ* . Let o < s < n and Is the fractional integral operator defined b y I s =  1x - yI s-n f ( y ) dy . f n In the case of 1 < p  q < oo it is known [Pe] that the inequalit y P(I s , p, q, u, v), i .e . 11 Is f II Lz ç C~ f 11 1,1 for all nonnegative functions f holds if and only if (u, v) E A (s, n, p, q ) whenever u dx, v - p 12x d x E A . By the results in [Ra2] and [Ra3] this equivalence also holds if u dx E B, n dx E B e with 1 - C v and 1- < (see also [Pe ] far such a result) . The condition w dx E D o() means : I2QIw Ç C IQ iw for all cubes Q . 2Q is the cube with the same center as Q but the edge lenght expande d twice . As a consequence of our theorem, for 1 C g C p we hav e Corollary . Let 1 < q < p < oc, o < s < n and udx, v - P 1 1 dx E A~ . Moreover assume u dx E B u , v - P~ 1 dx E B e with o < v, o an d (i-) < a 1 - p + v ( 1 ) . Th en th e inequality P ( I s , n, p, q, u, v ) holds if and only if (u,v) E A(s, n, p, q) . This equivalence also hold s 1 when u dx E B, n D~, v - P-1 E B e with 1 -- < v and 1- Ç O . For seeing this, it is sufficient to remind that the Muckenhoupt - Wheeden inequality [Mu-Wh ] 11 1 5f ~~L~ Ç cilmsf IILI WEIGHTED INEQUALITIES FOR MAXIMAL FUNCTIONS  29 holds whenever u dx E A~ . This is also the case when u dx E B u n D ~ with 1 — ñ C v (see [Pe] or [Ra2J) . 2 . On A((I), p, q) and B e condition s Now we also assume the functions (I) defined on the set of balls b y (B) — (Q) whenever Q is the smallest cube which contains the bal l B . A weight function w satisfies the condition C when there are constant s c, C > osothat C  sup w(x) Ç n  w(y) dy . IRC IxIÇ4R  R  ~y~ Çc R Many of usual weight functions w satisfy this growth condition, sinc e nonincreasing and nondecreasing radial functions are included . Condi - tion (u, v) E A((P, p, q) for uand v satisfying C can be easily realized , mainly for radial weights . Indeed we hav e Proposition 2 .1 . Let 1 C p, q C oo and — Ç A . Assume u, v satisfying the growt h condition C . Then (u, v) E A(1>, p, q) for a constant A > o if and only i f (u, v) E Ao ( . P , p, g) ; i . e c 1)(B(O,R))R nP j - 1 ) R n 1_ 1 1  p ~ v - ~ - 1  A o Rn f i y iC R u ) for ah R> o, whereAo--A xn, p, q, u,v} . As an example for o Ç s C n, p Ç —n C a < n(p - - 1) , ps — n < a, [3 = 1(n + a) — qs — n, u(x) = Ixi 0 , v(x) = lxr the n (u, v) E A(s, n,p, g) . Now let us discuss how we can verify in practise, for usual weights the condition w dx E B e , g > O . To do this, we first recall sorne know n classes of weights . The Muckenhoup class A p . Let us recall that w dx E A p (1 < p < oo) if and only if (w, w) E A ( o, n, p, p) . It is known [Ga-Rb] that A cc , = U r > 1 A r . 30 Y . RAKQToNDRATSIMB A The reverse Holder class RH r . We write w dx E RH r (1 < r < oo) if and only i f (I ` ~~ fQw r ) R (- 1  w) for all cubes Q C = C(w) > O . IQI f The classes Rll r and A p are related ; for instance w dx E Rllr if an d only if w -1 dx E A r r A . If w dx E A p then it is known [Ga-Rb] tha t w dx E RH I + p for some p ~ 0 ; the converse is also true . The reverse doubling class RD e . We write w dx E D e (0> 0) if and only i f Ct "1 Q l ,,, < ItQi w for all cubes Q and all t > 1, C = C(w) > 0 . If w dx E RH - r then, by the Holder inequality, w dx E RD I . Suppos e r-1  r w dx E D~ with the doubling constant D, i . e 1 2 Q L ~ D I n w w for all cubes Q D = D(w) > 1 , then [St- To] w dx E RD p for some > 0 . Precisely [Ra3] we can tak e = ln 2 n, In D D1 1 where c = 4 + i r-A- . But the reverse doubling conditio n RD,, is weaker than the doubling condition D oo (take for instance w(x) - = e !xI ) Thus it is clear that w dx E A~ implies w dx E B e for some O . On th e otherhand we can stat e Proposition 2 .2 . lf w dx E B e far some o > 0 then w dx E RD e . Conversely if w dx E RD, n D~ then w dx E B e . So in practice to obtain w dx E B e it is sufficient to get w dx E RD e n D o . . By the above condition w dx E RD e (0 < ç 1, with th e precise value of o) can be realized from w dx E RH 1 1 or w dx E D o c . e Consequently, it is interesting to know when we have w dx E D,, . It i s well known [Ga-Rb] that w dx E D~ when w dx E A oo . But we can fin d w dx E D oo with w dx [Wi] . As a tool for w dx E D,,,,, Stromber g and Wh eeden [St-Wh] proved that lxl ` ~u(x) , (4) c' u(x) E D~ whe n u dx E RD e n D,,, and a > -np . By adapting an argument in [St - Wh], this result can be extended for weights w(x) = = O( I x Du (x) wher e u dx E RDe n D o0 ~ o > 0), and B essentially constant on annuli an d satisfying a condition like : E k > 02 -1n Q0(2 -k L) Ç O(L) far all L ~ 0 . WEIGHTED INEQUALITIES FOR MAXIMAL FUNCTIONS  3 1 3 . Proofs of result s Our main theorem is a direct consequence of the inequalities (3 .1) , (3 .2), (3 .3) in the following propositions . Proposition 3 .1 . Let 1 C q C p < flo . Assuume 1 . be a function which satis fies hypotheses H 1 , n 2 , n 3 . Let defin e O(x) = sup{1 . (Q)IQ1 -1 IQ a cube with Q ~ x} , da » = v — P 1 1 dx and ú(x) = O—P(x)u(x) . Then ú E LL(l[8 n , dx) an d (3 .1)  II M fIILú  11MfIILIeIIL ú for all functions f , where r =p P Q . Proposition 3 .2 . Let 1 C p < oa and ú defined as above . Assume da- E A~ or do- E B e with 1 — a Ç o (a is the exponent in the hypothesis n2) . Then there i s c = c(T , , n, p, q, u, v) > 0 such tha t (3 .2)  IIMfI1Lú < c~~ f II L v for all functions f . Proposition 3 .3 . Let 1 C q C p C oo . Assum e i) n : I) be a function which satisfies 7-t 1 , 7-í 2 , 7--L 3 ; ii} Z .LdxEBy,dxEB 2 with 0 C v, oand ( 1—a ) C o 1—~ + v ( l ) ; P iii) (u, v) E A ( ■ : D , p, q) for a constant A ~ 0 .  T hen there i s C( .: I :■ ,nap, q, u, v) > Q so tha t 1 O IILú < CA r = Proof of Prop os1 tion 3 .1 : Let us first observe the locally integrabilityof the function ú . Indee d for each cube with i 1 — 1 1 ~~~  > 0 and for each x E Q : o—1 ( x ) 5 (Q)IQ1Q i Ia PIQIú)  > O q p p—q 32  Y . RAKOTONDRATSIMB A and s o (3 .4)  IQk = fo x)1d[lx < (oQ)Q1-'IQI~w)< 00 . Note that for I Q I ~ = O, by the convention O .00 = 0, we immediatly hav e IQIÜ =D . Inequality (3 .1) comes from the Holder inequality, indeed for 1 C q < p < oa and r= P-q we ge t IIM~hfllj  fR n -  { (Mf)eu]d x  ~  I  9 PIILT = II M ~fIILúII o IILú . ■ Proof of Proposition 3 .2 : First let us note that by (3 .4), (ii,v) E A( <D, p, p) i .e . (Q)IQI1IQ 1 QI) > 0 for all tubes Q . For da- E  an easy modification of the proof in [Pe] yields to th e conclusion (3 . 2) . For do- E B e with 1 - - a Ç o, we get  v} E S (1 . , p, p ) [Ra3] and then by a similar argument as in [Sa2] the inequality (3 .2 ) holds for a constant c =  n, p, ic} > Q . n Proof of Proposition 3 .3 : For each R ~ O, let us defin e O R (x) = sup{1 . (Q)IQ1 -1P IQIú ; Q a cube with Q ~ x, IQI ñ R} . The conclusion appears once we obtai n (3 .3')  ~~ ~ R II ~ú Ç cA c -= c(1 . , n, p, q, u, a) > O, r _ .  q p p - Then in order to prove (3 .3' ), we take a cube Qo with 1 Q 0 1* = R . The n Il~ll~ t =8 1,R + e 2, R where el , R = f Qo %U dx, e2,R = fan\Q . T R u dx . n WEIGHTED INE(aUALITIES FOR MAXIMAL FUNCTIONS  3 3 Estimate of el , R . Let x E Q0, Q a cube with Q ~ x and IQI* Ç R . Note that Q C (3Qo). Now using i), ii}, iii) we ge t A (Q) =  P IQI Z < c(~n, cr, u) \ IQ() I ~ /  n ( 3 Qo) ~ < c( 4), n, ~ u ) n ( 3 Qo) • Thus O R (x) c(~, n, Q, u)A(3Q 0 ), and consequentl y e l,R ç c ' (1 . , n, u, u) (A(3Qo)I3QoI) T ú = (3 .5)  1_?  1 r =  a, u} ( 3 QOI 3 Qo '3 QOII 3 Qo1, 1 Ç c' ez1) ,n, o- ,u}A r . Estimate of e 2,R . First we can writ e e2 , R = E  eRUdx . k >o L 1 Q O )\(2 k Q O ) Let k E N, x E (2 k+1 Qo)\(2 k Qo) and Q ~ x with lQI-1 < R . The n Q C (32k+1Qo) _ (6Qo) . As the aboye computation we hav e Ç c ' N> ® A (Q)  c' el>, n, u, u)2- kn E A +e( 1 -1)+ v I i,lA(62 k Q 0 ) . Next, since 1 - a C  1-  + v , the n (3 .6 ) e2,R c /(,p, n, cr, u ) E 2-  ~  ~ kn[a +e ( 1-- 1 }+U1] (A(62kQo) (6kQ0)  r ç  lu  ~ k ~ o na d, u}A T E 2—n[a+e(1-- p )+v p l < k ~ o Ç  n, o- ,u}A T . Inequalities (3 .5) and (3 .6) yield (3 .3'), and consequently by a limitin g argument we get (3 .3) . <