Estimates for some square functions of littlewood-paley type

Pub . Ma . UAB Vol . 27 Ns 2 Juny 1983 ESTIMATES FOR SOME SQUAREFUNCTIONS 1 . In oduc ion One o he classical esul s o he Li lewood- Paley heo ys a es ha he Lp-no m o a unc ion is equi alen o he Lp-no m o he quad a ic mean o i s pa ial sums co esponding o he dyadic in e als, which o m a decomposi ion o  1R n .  The p ecise s a e- men can be seen in  101, Ch . IV . Mo e ecen ly,o he ypes o pa i ions ha e been conside ed and, in pa icu- la , ha ob ained om a ixed in e al and i s ans- la es, i .e ., in heone-dimensional case : A (x)  _  ~ ' 1 . 1  1  (u)e 27 ixIdj 1 2  1/2 OF LITTLEWOOD-PALEY TYPE José L . Rubio de F ancia Fo he ope a o A , one does no ob ain equi- alence o no ms  (excep o L 2 ), bu only he inequali y I~ A .1I p <_ Cp  11 II p which, mo eo e , is only , alid in he ange 2 <_ p < - . A ske ch o p oo o his esul appea s in [1], whe e i is shown o be a basic ing e- dien o A . Có doba'sapp oach o he es ima es o sphe ical summa ion mul iplie s . A mo ede ailed p oo is gi en in [2] . This pape g ew ou o con e sa ions on his sub- jec wi h A . Có doba, o whom I am indeb ed o sha ing his knowledge wi h me . The pu pose is wo- old . Fi s , we gi e in sec ion 2 ano he p oo o he esul jus men ioned and some sligh gene alisa ions . This new a simple app oach is based on he unexpec ed poin wiseinequali y :  G (x) 2 1 Cons . M(I 1 2 )(x)  (whe e G is a smoo h e sion o A), om which, weigh ed Lp inequali ies o he ope a o A a e also ob ained almos immedia ely . Secondly, we explo e some o he analogueso A in o de o ge a deepe unde s anding o wha is eally going on wi h hesequad a icope a o s . In pa icula , some con inuous analogues o A conside ed in sec ion 4 lead o a s iking esul on poin wise con e gen e o a e ages o ball mul iplie s . Fu he gene alisa ion is gained in sec ion 5, whe e we deal wi h A and i s smoo h e sion G in locallycompac Abelian g oups, a con ex which equi es ye ano he di e en p oo o he Lp-inequali ies . The no a ion used is ai lys anda d . We deno e by M = M 1 he Ha dy-Li lewoodmaximal ope a o in and, mo egene ally,we de ine M g (x) = M(I l q )(x) 1/q = sup(IQI-1 J I (Y)Igdy)1/q XEQ Q IR n .  The classes o weigh s conside ed a e :  A p ( he usual Muckenhoup 's class) and Ap , whichconsis s o all w(x) > 0 such ha sup (III -1  w)(III-1 1 w l-p ) p-1 2 . Quad a i c Ope a o s o Disc e e Type whe e 1 1 q 5 - and Q s ands o an a bi a y cube in whe e I is an a bi a ybounded n-dimensional in e al, and he usual modi ica ion is conside ed o p = 1 . Weigh s in A p co espond o p oduc s o ope a o s which a e bounded wi h espec o wEA P , such as he s ong maximal unc ion o he double Hilbe ans o m (see [5j) . Gi en  mCL~(IR n ),  we deno e by  Tm  he mul i- plie ope a o : (T m )~ = .m, which is well de ined in L2 (IR n ) .  When  T m can be ex ended o a boundedope a o in Lp, we shall deno eagain his ex ension by T m . In pa icula , i  m  is a Schwa z unc ion,  mG J(IR n ), we know ha Tm can be de ined in Lp o all 1 1 p The quad a ic ope a o o be conside ed he e is (1) G (x) = {y IT  (x)I 2 } 1/ 2 ke2 n whe e m(k+ .) means he ansla ion o ou mul iplie : m(k+ .)(E) = m(k+j) . Finally, i Q s ands o he uni cube in  IR n ,  Q =  [-1,  1 ] n ,  we de ine he unc ions : q j (x)  =  I2)QI -1 X 2jQ  (x)  =  2 - i n q o (2 - )x) Theo emA : p ecis ely : j-0 (2) G (x) 1  cj{qj*1 1 2 (x)} 112 < C M 2 (x) ho1ds o e e y  EL 1 +L . , whe e he cons an sc j depend only on m and  c . - C < - . As a consequence , G is j-0 a bounded op e a o in LP(IR (3)  G (x)P  w (x)  dx  _<  CP (w)  1 (x)1P w (x)  dx o all  weA p/ 2 , 2 <__ p< hen he poin wis e majo isa ion 2< p an d mo e P oo :  Le  ge A IR n )  be he in e seFou ie ans o m o  m .  Fo each ini e sequence  a =  (a k )  o uni  R 2 - no m, we de ine G x  (x)  _  Z  ak  Tm(k+ .)  (x) k k e -21 ik .y g (y)  (x-y)  dy =  J  g (y)  h a (y),  (x-y)  dy whe e  h ;k  (y)  is pe iodic :  h x (y+k)  = h x (Y)  (ks ZZ n ) and has uni no m in L 2 (Q) . Now, we de ine : co = sup {Ig(x)I : x .Q} c j =_  23n  sup  {Ig(x)I :  xs(2jQ)<2j -I Q)} k so ha  2c j  < m  (because g GA, and IGa (x)I < Z  cj 2-jn J . Iha(y) (x-y)I dY j=0  2 ] Q 1 c j {2 - j n j=0 1/2 JJ I (x-y)I 2 dy} Q (j=1,2, . . .) Since G (x) = sup IG a (x)I, he i s inequali y in a (2) is p o ed, and he second one ollows because qJ . * 1 M o e e y unc ion . Since  M 2 is a bounded ope a o in  L (IR n )  and in  LP(w)  when  p >  2  and  w s Ap /2,  only he case p = 2 o (3) emains o be p o ed . Bu his is a conse- quence o he i s es ima e in (2) oge he wi h he obse a ion ha J  ( 1 Q 1 -I  XQ )  *  (x)  w (x)  dx  :5  C  I (x) 1  w (x)  dx o e e ycube  Q  and e e y weigh  w s A I . The p e ious heo em is a - smoo h e sion o he Li lewood-Paley ype inequali ies we ac uallywish o ob ain, which a e conce ned wi h he pa ial sum ope a o s SI, whe e I is an a bi a y n-dimensional in e al and (S I ) , . = XI . Now, he e is a s anda d unca ion a gu- men o ob ain he s ong esul om i s smoo h e sion, which is based on he inequali y (4)  ¡SI .  J 12 )1~2  II  p  s  Cp ~ j ~2)1~2  ii p J  i which holds o a bi a y in e als I j and unc ions s  Lp (IR n )  (see  S ein  [lo]) . The in e als {I j } a e said o be almos cong u - en (wi h cons an C ? 1) i sjp  R i (I j)  <__  C  ij  i (I J )  (i  =  1,2,  . . .,  n) whe e i ( " ) s ands o he side leng h o an in e al along he x i -di ec ion . Theo em B :  I  2s p < w  and  m s Ap l2 ,  hen, o e e y sequence {I j } o disjoin almos cong uen in e - als in  IR n,  he inequali y (5)  II  (1  1S I .  1 2 ) 112 11  c  (w)  II  II J  Lp (W)  p  Lp (w) ho1ds o all  eLP(w) . P oo : Suppose i s ha all I j a e bounded . Since e e y hing is in a ian unde dila ions in each coo dina e, we can assume ha Qi (I j)  :S  C  (1  :i  i  5  n) o e e y in e al I j in ou sequence . Then, each in e al con ains a leas one la ice poin : k j s Ii n zZ n . Take a Schwa z unc ion m such ha m(1) = 1 when s I = [-C, C]', so ha SIJ   =  S IJ  (T m(-kJ+ .)  )  (  s  Lp) and an applica ion o inequali y (4) oge he wi h Theo em A yields : II (E ¡si . 82 ) 1/2 II p < Cp II G 11 p >> cp  II 11 p This p o es he case w(x) __ 1 . Fo a gene al w s AP /2 he a gumen is exac ly he same, bu we mus use he weigh ed e siono (4), namely, ha suchinequali y holds no only in  LP(IR n), bu also in  LP(w)  i w s AP and 1 < p < °° . (This ac ually a a he s aigh o wa d consequence o a gene al heo em o Ma cinkiewicz and Zygmund [6] oge he wi h he boundedness in LP(w) o he mul ipleHilbe ans o m ; see also Ku z [5]) . Finally, i may be he case ha , o some i  =  1,2,  . . .,  n,  we  ha e , Zi(I J)  _  o  all  Ij .  The necessa y modi ica ions o deal wi h his case a e a he i ial, a e a e le o he eade . Gi en a sequence o disjoin consecu i e in e als in IR, i hey all ha e he lame leng h we ha e jus p o ed ha inequali y (5) holds ue, while in he case o leng hs inc easing a an exponen ial a e, he same inequali y is ob ained (and no only o pi2, bu o all 1 < p < -) by classical Li lewood-Paley heo y . l seems na u al o .expec he same kind o esul when he leng hs inc ease a bi a ily . A pa ialposi i e answe is con ained in ou nex esul which, o he sake o simplici y,will be s a ed in i s one-dimensional e - sion . Theo emC :  Le  {aej}j s ~Z  be an odd sequence o eal num - be s (i .e . a-3 = -a j ), and assume ha i s posi i e pa is co n ex - and slow1y inc easing, i-e . j-1 j 2 ,~-1 ,~+1 a2 ~ j  :1 C a .  (j 1 1) Then, he quad a icexp ession a . {  ~ 1 ?(C)  e 2uix~ d1 12  } 112 -~ a .,1-1 de ines a bou nded ope a o in  LP(IR ),  2 1 p < ~ . GN  (x)  =  {  1J  e -2T il-y  g ( y )  (x-y)  dy  12  dj}l/2 .  M<N  IR n When sL 2 , he inne in eg al is absolu elycon e gen , and,  i we ix xs  IR n  and  N>  0,  he e exis s a unc- ion a(J) (depending on bo h x and N) o uni no m in L 2 (IR n )  such ha GN  (x) 11 ¡<N J  a(I) e-2T il-y g(y) I n = jh(y) g(y) (x-y) dy whe e h s L 2 and jjh11 2 = 1 . Since K = (gl 2 s L 1 , we áppiy Scha z inequali y o ob ain G N (x)  i  {1 K(y)  j (x-Y)I 2 dy} 1/2 and pa (i) is p o ed by le ing N - - . Now, unde he hypo hesis o (ii), jK(x)j is domina ed by a dec easing, adial, in eg able unc ion, and hus : (K * ¡ `2)1/2 < C M(I l 2 ) l/ 2 =C M 2 In pa icula , he maximalope a o : (x-y)  dy d1 8- U0  {  J  I T m(u+d " )   -  m (u)  12  du} 1/2 IR n is bounded in  LPOR n ),  2 < p <-_  and o weak ype  (2,2) . By a s anda d echnique, h epoin wise con e gence esul will be p o ed o e e y s Lp, 2 < p < -, i we es a- blish his esul o e e y Schwa z unc ion . Bu , i eJ(IR n ) : { JI T M(u+d .) (x) - m(u) )()¿)I 2 du}l/2 < <-_ {J{JIM(u+d~) - m(u)I  I (&)I  dj} 2 du} 1/2 < J  I (j)I{  JIM(U+61)  - m(u)I 2 du} 1/2  d1 Since m s L 2 , he inne in eg al in he las exp ession is bounded independen ly o d, 1, and i anisheswhen d -> 0, so ha an applica ion o Lebe gue's dominá ed . con e gence heo em ends he p oo . The ollowingpa icula case o Theo emD is wo h men ioning : Le m = X B, whe e B is he uni ball in IR n . Then II(x)1 = Ixi-n/2 IJn/2 (2-nIx1)I 5 C(1+Ix1)-n/2-1/2 and bo hpa s o he heo em can be applied . I we w i e S E o he pa ial sum ope a o co esponding o he mul iplie  XE,  hen he i s pa shows ha i makes sense o de ine he ope a o : --> {1 nISu+B l2 du} 1/2 I o e e y  s Lp(1R n ),  2 <__ p <__ -, e en hough, by Fe e - man's heo em ([3]), each one o he ope a o s Su+B can be de ined only in L 2 . Mo eo e , he second pa gi es : Co olla y 3 :  l  s Lp( .?n) ,  2 : p < -,  hen S * (x)  = sup  {J  ¡S (u+B)  (x)12  du}1/2  C 9 2 (x) 0< <w  IR n and Zim  J  15 (u+B) (x) - (x)I 2 du = 0  a .e . ->M (10) lim S V (x) = (x)  a .e . Iu1<1 Gi en an open ball  V  in ]R n  con aining he o igin, i is no known whe he is ue o e e y s L 2 . Wha Co olla y 3 shows is ha a ce aina e age o he s a emen s (10) o all balls con- aining he o igin is ue, bu his is only a poo subs i u e which is a away om (10) i sel . 5 :  The Case o Locall Compac G oups An essen ial pa o he esul s in sec ions 2 and 4 can be o mula ed in he con ex o locally compac Abelian ( .c .a .) g oups, p o iding some so o uni ied e sion o he disc e e and con inuous cases . Since he e is no Ha dy-Li lewood ope a o in con ex , we only ob ain he Lp inequali ies, poin wise es ima es, and a di e en , sligh ly app oach mus be ollowed . Le X be a .c .a . g oup wi h dual g oup X A gene alelemen o X ( esp . X ) will be deno ed by x, y, e c . ( esp ., a, e c .), and ds, d1 will be he Haa measu es on X and X , whichwe shallassume o be sui ably no malised wi h espec o each o he . The le e s H and P will be used o ep esen closed sub- g oups o X and  X , hei Haa measu es being m H and m , espec i ely . The ope a o o be conside ed he e is his gene al bu no he longe G  (x)  I(ag)  -  (x)I 2 dm,  (a)} 1/2 When X =X= IR n , we can ake = 2z n , ob aining he ope a o de ined in (1), o = X = IR n, in which case we ge he ope a o G o (9) (wi h g = m in bo h cases) . Theo em E :  Le  g s L 1 ( X)  be a posi i e unc ion such ha (11)  {  Ij(C,)12 dm (a)}1/2 - M < Then  he, ope a o  l :  de d ed a bo e  bound ea Lp ( X)  when  2  < p  < II G ll p 5Mil  11 p P oo : We de ine he closed subg oup o X 1 H  =   =  {x  s  X  :  < x,a>  =  1  o  all  as } so ha  =  ( X/H) .  The Haa measu e  m  in  X /H  is de ined as he dual o  m .  Then, a uniqueHaa measu e mH can be ixed in H so ha , de ining a>g(y)  (x-y)  dyl2 dm  (a)} 1/2 and mo e p ecisely : ( s Lp ; 2 < p 5 -) (z)  _  1  (x+y)  din H (y)  (  s  L~ ( X) ) H (whe e z = (x +H) s X/H), he ollowing iden i y always holds (12)  J  (x)  dx  =  1  T(z)  dm(x) X  X/H A se ies o simplelemmas will be needed in he p oo : Lemma 1 :  I g s L 1( X)  is posi i e, hen, o e e y sX J 19-(CL-I)1Z dm,, (a) <_ 1 9(a)I Z  dm  (a)  =MZ  P oo : Since (T) ^ (a) = (a) o e e y a e , Plan- che el's heo em implies 1 9 12 din, (a) J ji(a)j2 dm, (a) = 1  g(x) 2 dm(x)  X/H Bu I (x)1 1 j j ' ( x) o e e y , and in ou pa icu- la case, we ha e  1(1g) ' (x)j  :S g'(S¿),  whic hconclude s h e p oo o he lemma . In ou nex esul , we shall deno e by B a Banach space (wi h dual B*) and by LB ( X)  he Bochne - Lebesgue space consis ing o all (s ongly) measu able B- alued unc ions F(x) de ined on X and such ha JI F(x) II B is in eg able . Lemma 2 : Le K (x) be a measu able B*- alued unc ion such ha (13)  1  I K(x)-b1  dx  5  CII bII B  bEB Then, he o mula (14) T F(x) = 1 K(x-y)- F(y) dy de ines a bounded ope a o om  L 1( X)  o  L 1 ( X)  mi h no m 5 C . P oo :  I  su ices  o  e i y  ha  II TF II  1  1  C II F II L 1 o all simple in eg able unc ions F, since hese B a e dense in LB . Fo each such unc ion F, (13) implies ha TF(x) is well de ined, and JITF(X)j dx 5 1 1 IK(x-y) " F(y)I dy dx = = J {J  ¡x (x) -F(y) j dx} dy <_ c 111F(y) 11 Bdy . In he nex lemma, we ake as ou Banach space B =B * = L 2 ( 0 ) =L 2 ( 0 ; m ), whe e o is a ixed compac subse o . Func ions F s L 2 (X) a e hen iso- me ically iden i iedwi h unc ions (x, a) = F(x) (a) in he p oduc space  L 2 (X x o ) . Lemma 3 : Le U : L 2 , LB be de ined by U (x, a) (ag) * (x) . Then U is a bounded ope a o wi hno m 5 5 M, and i s adjoin  T = U : LB , L 2 is gi en by (14) whe e he ke nel is :  X(x) = k(x, a) = g(-x) <x, a> . P oo : The i s asse ion ollows omPlanche el's heo em and Lemma 1 : ~I U (x) 11 B dx x J  1 ^  ?(j)1 2  d1  dm (a) x 0 M 2 1 ^ I?(J)j 2 d1 = M2 J I (x)I2 dx The ke nel o U * is ob ained by a simple compu a ion which is le o he eade ( he ac ha we ake a com- pac subse 0 c makes all he in eg als absolu ely con e gen ) . We a e now in a posi ion o comple e he p o .o o Theo em E . Le G o deno e he ope a o de ined as G a e eplacing by o . I i is p o ed ha II Go II p = MII II p  (2 < p < -)  o an a bi a y compac subse o o  , hen, le ing o inc ease o , we ge he desi ed esul o G . Bu IGo (X)1 = = IIU (x)II B,  so ha he inequali ies o be ob ained a e (15) IIU II LP ~MIl ll p (2<p5~) B Fo p = 2, his was p o ed in Lemma 3 . Fo p = -, (15) is equi alen , by duali y, o 11T FII,1 5 MI¡ FIIT1 "  LB and his can be p o ed by e i ying ha he ke nel K(x) de ined in Lemma 3 sa is ies (14) wi h cons an C = M . In  ac ,  gi en  b  =  b(n)  s  B  =  L 2 ( 0 ): IK(-x) .bl = I  g(x) <x, a>b(a) dm (a)I 0 =  Ih(x)I  g(x) wi h  h s  L 2 ( X /H)  and  II h II 2  =  II b II B, so ha we can use he iden i y (12) o ob ain IK(x)-bl dx =  Ih(x)I g(x) dm(z) < X  X /H ~~ h 11 2  11 8 11 2  b 11 BM Finally, he case 2< p < - o (15) ollows by in e pola- ion, and he p oo is comple ed . When = X , (11) simplymeans ha g e L 2 (X) . Mo eo e , in his case, he assump ion g i 0 is no neces- sa y, and we ha e a s a emen comple ely analogous o Theo em D(i) . On he o he hand, i is disc e e and g has compac suppo , hen (11) is i ially e i ied . The unca ion a gumen used in Theo emB can also be applied o he smoo h G- unc ionconside ed he e : Gi en m s L w ( X),  we say ha i is an Lp-mul iplie i he ope a o  Tm de ined (and bounded) in  L 2 ( X)  by (T m ) ^_ 1m admi s a bounded ex ension o Lp( X) . Theo em F :  Le  me Lw ( X)  be - a compac ly suppo ed  Lp- mul iplie , wi h 2<p < g oup o,  X ,  hen,  he ope a o ás ITM(a+ :)  (x)I 2  } 1/2 is a disc e e sub-