THE WEIGHTED WEAK TYPE INEQUALITY FO THE STONG MAXIMAL FUNCTION THEMIS MITSIS Abstract. We prove the natural Fefferman-Sten weak type nequalty for the strong maxmal functon n the plane, under the assumpton that the weght satsfes a strong Muckenhoupt condton. Ths complements the correspondng strong type result due to Jawerth. It also extends the weghted weak type nequalty for strong A weghts due to Bagby and Kurtz. Let f be a locally ntegrable functon n 2. The strong maxmal functon M s defned by Mf(x) = sup f, where E denotes the two-dmensonal Lebesgue measure of a set E 2, and the supremum s taken over all rectangles 2 wth sdes parallel to the coordnate axes, such that x (from now on by the term rectangle we wll always mean a rectangle wth sdes parallel to the coordnate axes). By a classcal result of Jessen, Marcnkewcz and Zygmund [7], M s bounded from L( + log + L) to weak L, that s ( f () {Mf > } C + log + f ), > 0, whch mples that M s bounded on every L p, p >. The dea of ther proof was to domnate M by terates of the usual one-dmensonal Hardy- Lttlewood maxmal functon actng n dfferent drectons. A drect geometrc proof was gven much later by Córdoba and. Fefferman [2]. The dffculty n a drect approach s that the Bescovtch coverng lemma fals 2000 Mathematcs Subject Classfcaton. 42B25. Key words and phrases. Strong maxmal functon, weghted nequalty. Ths research has been supported by EPEAEK program Pythagoras.
when appled to a famly of rectangles havng arbtrary eccentrctes. The man contrbuton of [2] was exactly the dscovery of a sutable substtute for the Bescovtch coverng lemma. As far as weghted nequaltes are concerned, t s known that f w s a strong A p weght, p >, that s, f there exsts a constant C > 0 such that for all rectangles we have ( ) ( w then M s bounded on L p (w), namely (Mf) p w C p w /(p ) ) p C, f p w. Ths, agan, follows by an appeal to the one-dmensonal theory. A dfferent proof of a more general result may be found n Jawerth [5]. The endpont case (p = ) has been treated by Bagby and Kurtz []. They proved that f w s a strong A weght, namely, f there exsts a constant C > 0 such that for all rectangles we have then (2) {Mf>} w C f w C essnf w(x), x ( + log + f ) w, > 0. The results above suggest an analogy between weghted nequaltes for the strong maxmal functon and weghted nequaltes for the usual Hardy- Lttlewood maxmal functon. However, ths analogy cannot be pushed too far unless we put some restrctons on the weght. For example, f we consder the weghted verson of M,.e. M w f(x) = sup rectangle x w() f w, where w() = then. Fefferman [4], usng the dea of [2], has shown that f w belongs to a fxed strong A r class, r >, then M w s bounded on L p (w) for all p > (see Jawerth and Torchnsky [6] for the endpont). Note that f M s the 2 w,
Hardy-Lttlewood maxmal functon then M w s bounded on every L p (w), p >, wthout any restrcton on w. So, n ths case, the analogy breaks down. Under the same assumpton on the weght as n [4], Jawerth [5] proved, by dfferent methods, that M s bounded from L p (Mw) to L p (w), for all p >,.e. (3) (Mf) p w C p f p Mw. As before, f M s the usual Hardy-Lttlewood maxmal functon then (3) holds true for arbtrary w. Ths s due to C. Fefferman and Sten [3], and actually, (3) may be thought of as the prototype weghted maxmal nequalty. The purpose of ths paper s to prove the endpont case (p = ) of (3) whch, as expected, turns out to be the weghted verson of (). Namely, we shall show the followng. Theorem. Let w be a strong A r weght for some fxed r >. Then ( f (4) w({mf > }) C + log + f ) Mw, > 0. Proof. As usual a b means a Cb for some constant C > 0 not necessarly the same each tme t occurs. Let M d be the dyadc strong maxmal functon M d f(x) = sup f, where the supremum s taken over all dyadc rectangles (cartesan products of dyadc ntervals) wth x. Frst, we shall prove the correspondng weak type estmate for M d : (5) w({m d f > }) f ( + log + f ) M d w, > 0. So, pck a pont x n {M d f > }. Then there exsts a dyadc rectangle x contanng x such that x f > x. 3
Wthout loss of generalty we may assume that { x } x s a fnte famly { } L =. Now, fx a number 0 < ε 0 < to be determned later. By the Córdoba -. Fefferman coverng lemma [2], there exsts a subfamly { }M = { } L = such that (6) j ε 0, =,..., M, j< and (7) L = {Mχ S M = ε 0 }. Snce w s a strong A r weght, M s bounded from L r (w) to L r (w). So, (7) mples that L w( = M ) w( = ), where the mplct constant depends on ε 0, r and the A r -constant of w. Now, wrtng = M, 2 = M,..., M = and applyng the Córdoba -. Fefferman coverng lemma to { } M = we get a subfamly { } N = { }M = such that (8) j j ε 0, =,..., N, and (9) M = As before, (9) mples that Therefore {Mχ S N ε 0 }. = M w( = ) w( = ). L (0) w({m d f > }) w( ) w( = 4 = ).
Now, let µ and µ w be the multplcty and the weghted multplcty functons, respectvely, assocated to the famly { } N =,.e. µ(x) = = χ (x), µ w (x) = w( = ) and fx a number 0 < δ 0 < to be chosen after ε 0. Then w( = ) = = δ 0 Usng the elementary nequalty w( ) δ 0 w( = ) f µ w (M d w) δ 0 M dw. st e s + t( + log + t), s, t 0 χ (x), f δ 0 we get () Now, let w( = ) δ 0 Q = S N = f + δ 0 exp(µ w (M d w) )M d w ( + log + f δ 0 S N = + ( log δ 0 ) ) M d w exp(µ w (M d w) )M d w f ( + log + f ) M d w. exp(µ w (M d w) )M d w. S N = We clam that f we choose ε 0 small enough then (2) Q w( = ). To see ths, we expand the exponental n a Taylor seres. Then Q = µ k w(m d w) k = µ w µ k w (M d w) k. 5
Snce w( ) χ M d w, we have Q = µ w ( N = µ w µ k = χ M d w Q k. ) k (M d w) k To estmate Q k we ntroduce the followng notaton: For I {,..., N} we put A I = I \ / I. Then the famly {A I : I {,..., N}} s dsjont and moreover, for all, n wth, n N we have (3) {µ = n} = I {,...,N} I =n / I A {} I. So Q k = n= I {,...,N} I =n A I µ w µ k. Note that f I = n, then on A I we have µ = n and µ w = I w( ). Therefore Q k = n k n= I {,...,N} I I =n 6 w( A I. )
earrangng the terms and then usng (3) we get Q k = = n= n= Snce the rectangles n k n k N = N = w( ) w( ) I {,...,N} I =n / I A {} I {µ = n}. satsfy (8), the argument n [2, p. 00] (ths s the only pont where we use the fact that the rectangles are two-dmensonal and dyadc) shows that {µ = n} ε n 0, where the mplct constant depends on ε 0 (t s, actually, equal to (ε 0 ( ε 0 )) ). Consequently Now = w( ) = = Q k = = n k ε n 0 n= w( w( = w( ). j ) + j< = j ) + w( j< = w( ). \ j< j ) Snce w s a strong A r weght, there exst constants c 0 > 0, η 0 > 0 such that for every rectangle and every E we have ( ) w(e) E η0 w() c 0. In partcular (6) mples Therefore w( j< j ) w( ) = w( ) c 0 ε η 0 0 ( c 0 = 7 j< j w( ) + w( = ) η0 c 0 ε η 0 0. ).
So, f ε 0 has been chosen small enough we have Ths mples that = w( ) w( = ). Q w( = ) n, ε n 0 nk w( = ), whch proves the clam, for approprately small ε 0. Combnng () and (2) we obtan ( δ 0 C ε0 )w( = ( f ) ( log δ 0 ) + log + f ) M d w. Choosng δ 0 small enough and then usng (0) we get (5). We now show that (5) holds wth M d replaced wth M. Indeed f x {Mf > } then there s a rectangle contanng x such that < f. Notce that there exst four dyadc rectangles, 2, 3, 4 wth measure comparable to the measure of so that s contaned n ther unon. Then < 4 k= k whch mples that for some k we have k Therefore Hence Consequently k k k f. k {M df }. f, {M d f }. Mχ {Md f }(x). 8
We conclude that {Mf > } {Mχ {Md f }(x) }. Snce M s bounded on L r (w) we get that w({mf > }) w({m d f }), whch completes the proof. Note that, by nterpolaton, (4) mples (3). Moreover, t mples (2) snce a strong A weght s a strong A r weght, for every r >, and also satsfes M w w almost everywhere. So, our result extends the correspondng results n [] and [5]. eferences []. J. Bagby and D. Kurtz. L(log L) spaces and weghts for the strong maxmal functon, J. Analyse Math. 44 (984/85), 2-3. [2] A. Córdoba and. Fefferman. A geometrc proof of the strong maxmal theorem, Ann. of Math. 02 (975), 95-00. [3] C. Fefferman and E. M. Sten. Some maxmal nequaltes, Amer. J. Math. 93 (97), 07-5. [4]. Fefferman. Strong dfferentaton wth respect to measures, Amer. J. Math. 03 (98), 33-40. [5] B. Jawerth. Weghted nequaltes for maxmal operators: lnearzaton, localzaton and factorzaton, Amer. J. Math. 08 (986), 36-44. [6] B. Jawerth and A. Torchnsky. The strong maxmal functon wth respect to measures, Studa Math. 80 (984), 26-285. [7] B. Jessen, J. Marcnkewcz and A. Zygmund. Note on the dfferentablty of multple ntegrals, Fund. Math. 25 (935), 27-234. Department of Mathematcs, Unversty of Crete, Knossos Ave., 7409 Iraklo, Greece E-mal address: mtss@fourer.math.uoc.gr 9