THE WEIGHTED WEAK TYPE INEQUALITY FOR THE STRONG MAXIMAL FUNCTION

Transcription

1 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.

2 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,

3 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

4 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 = ).

5 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

6 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. )

7 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. ).

8 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

9 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), [3] C. Fefferman and E. M. Sten. Some maxmal nequaltes, Amer. J. Math. 93 (97), [4]. Fefferman. Strong dfferentaton wth respect to measures, Amer. J. Math. 03 (98), [5] B. Jawerth. Weghted nequaltes for maxmal operators: lnearzaton, localzaton and factorzaton, Amer. J. Math. 08 (986), [6] B. Jawerth and A. Torchnsky. The strong maxmal functon wth respect to measures, Studa Math. 80 (984), [7] B. Jessen, J. Marcnkewcz and A. Zygmund. Note on the dfferentablty of multple ntegrals, Fund. Math. 25 (935), Department of Mathematcs, Unversty of Crete, Knossos Ave., 7409 Iraklo, Greece E-mal address: mtss@fourer.math.uoc.gr 9