Estimates for some square functions of littlewood-paley type

Pub . Mat . UAB Vol . 27 Ns 2 Juny 1983 ESTIMATES FOR SOME SQUAREFUNCTIONS 1 . Introduction One of the classicalresults of the LittlewoodPaley theorystates that the Lp-norm of a function is equivalent to the Lp-norm of the quadratic mean of its partial sums corresponding to the dyadic intervals, which form a decomposition of  1R n .  The precise statement can be seen in  101, Ch . IV . More recently,other types of partitions have been considered and, in particular, that obtainedfrom a fixed interval and its translates, i .e ., in theone-dimensional case : Af(x)  _  ~ ' 1 . 1  1  f(u)e 27r ixIdj 1 2  1/2 OF LITTLEWOOD-PALEY TYPE José L . Rubio de Francia For the operator A , one does not obtain equivalence of norms  (except for L 2 ), but only the inequality I~ A .1I p <_ Cp  11 f II p which, moreover, is only , valid in the range 2 <_ p < - . A sketch of proof for this result appears in [1], where it is shown to be a basic ingredient of A . Córdoba'sapproachto the estimates for spherical summation multipliers . A moredetailed proof is given in [2] . This paper grew out of conversations on this subject with A . Córdoba, to whom I am indebted for sharing his knowledge with me . The purpose is two-fold . First, we give in section 2 another proof of the result just mentioned and some slight generalisations . This new a simpler approach is based on the unexpected pointwiseinequality :  Gf(x) 2 1 Const . M(If1 2 )(x)  (where G is a smooth version of A), from which, weighted Lp inequalities for the operator A are also obtained almost immediately . Secondly, we explore some of the analoguesof A in order to get a deeper understanding of what is really going on withthesequadraticoperators . In particular, some continuous analogues of A considered in section 4 lead to a striking result on pointwise convergente of averages of ball multipliers . Further generalisation is gained in section 5, where we deal with Af and its smooth version Gf in locallycompact Abelian groups, a contextwhichrequires yet anotherdifferent proof of the Lp-inequalities . The notation used is fairlystandard . We denote by M = M 1 the Hardy-Littlewoodmaximal operator in and, moregenerally,we define M g f(x) = M(Ifl q )(x) 1/q = sup(IQI-1 J If(Y)Igdy)1/q XEQ Q IR n .  The classes of weights considered are :  A p (the usual Muckenhoupt's class) and Ap , whichconsists of all w(x) > 0 such that sup (III -1  w)(III-1 1 w l-p ) p-1 2 . Quadrati c Operators of Discr ete Type where 1 1 q 5 - and Q stands for an arbitrary cube in where I is an arbitrarybounded n-dimensional interval, and the usual modification is considered for p = 1 . Weights in A p correspond to products of operators which are bounded withrespect to wEA P , such as the strong maximalfunction or the double Hilberttransform (see [5j) . Given  mCL~(IR n ),  we denote by  Tm the multiplieroperator : (T m f)~ = f .m, which is well defined in L2 (IR n ) .  When  T m can be extended to a boundedoperator in Lp, we shall denoteagain this extension by T m . In particular, if  m  is a Schwartzfunction,  mG J(IR n ), we know that Tm can be defined in Lp for all 1 1 p The quadratic operator to be considered here is (1) Gf(x) = {y IT  f(x)I 2 } 1/ 2 ke2 n where m(k+ .) means the translation of our multiplier : m(k+ .)(E) = m(k+j) . Finally, if Q stands for the unit cube in  IR n ,  Q =  [-1,  1 ] n ,  we define the functions : q j (x)  =  I2)QI -1 X 2jQ  (x)  =  2 - i n q o (2 - )x) TheoremA : rp ecis ely : j-0 (2) Gf(x) 1  cj{qj*1f1 2 (x)} 112 < C M 2 f(x) ho1ds for every  fEL 1 +L . , where the constantsc j depend only on m and  c . - C < - . As a consequence , G is j-0 a bounded op erator in LP(IR (3)  f Gf(x)P  w (x)  dx  _<  CP (w)  f 1f(x)1P w (x)  dx or all  weA p/ 2 , 2 <__ p< then the pointwis e major isation 2< p an d more Proof :  Let  ge A IR n )  be the inverseFourier transform of  m .  For each finite sequence  a =  (a k )  of unit  R 2 - norm, we define G x f (x)  _  Z  ak  Tm(k+ .)  f (x) k k fe -21rik .y g (y)  f(x-y)  dy =  J  g (y)  h a (y),  f (x-y)  dy where  h ;k  (y)  is periodic :  h x (y+k)  = h x (Y)  (ks ZZ n ) and has unitnorm in L 2 (Q) . Now, we define : co = sup {Ig(x)I : xr .Q} c j =_  23n  sup  {Ig(x)I :  xs(2jQ)<2j -I Q)} k so that  2c j  < m  (because g GA, and IGa f(x)I < Z  cj 2-jn J . Iha(y) f(x-y)I dY j=0  2 ] Q 1 c j {2 - j n j=0 1/2 JJ If(x-y)I 2 dy} Q (j=1,2, . . .) Since G f(x) = sup IG a f(x)I, the firstinequality in a (2) is proved, and the second one follows because qJ . * f 1 Mf for every function f . Since  M 2 is a bounded operator in  L (IR n )  and in  LP(w)  when  p >  2  and  w s Ap /2,  only the case p = 2 of (3) remains to be proved . But this is a consequence of the firstestimate in (2) together with the observationthat J  ( 1 Q 1 -I  XQ )  *  f(x)  w (x)  dx  :5  C  f I f (x) 1  w (x)  dx for everycube  Q  and every weight  w s A I . The previous theorem is a - smoothversion of the Littlewood-Paley type inequalities we actuallywish to obtain, which are concerned with the partial sum operators SI, where I is an arbitrary n-dimensional interval and (S I f) , . = f XI . Now, there is a standard truncation argument to obtain the strong resultfrom its smoothversion, which is based on the inequality (4)  ¡SI .  fJ 12 )1~2  II  p  s  Cp ~fj ~2)1~2  ii p J  i which holds for arbitrary intervals I j and functions f s  Lp (IR n )  (see  Stein  [lo]) . The intervals {I j } are said to be almostcongru - ent (with constant C ? 1) if sjp  R i (I j)  <__  C  ijf  ti (I J )  (i  =  1,2,  . . .,  n) where t i ( " ) stands for the side length of an interval along the x i -direction . Theorem B :  If  2s p < w  and  m s Ap l2 ,  then, for every sequence {I j } of disjointalmost congruent inter - vals in  IR n,  the inequality (5)  II  (1  1S I .  f 1 2 ) 112 11  c  (w)  II  f II J  Lp (W)  p  Lp (w) ho1dsfo r all  feLP(w) . Proof : Supposefirst that all I j are bounded . Since everything is invariant under dilations in each coordinate, we can assume that Qi (I j)  :S  C  (1  :i  i  5  n) for every interval I j in our sequence . Then, each interval contains at least one lattice point : k j s Ii n zZ n . Take a Schwartzfunction m such that m(1) = 1 when s I = [-C, C]', so that SIJ  f  =  S IJ  (T m(-kJ+ .)  f)  (f  s  Lp) and an application of inequality (4) together with Theorem A yields : II (E ¡si . 82 ) 1/2 II p < Cp II G f11 p >> cp  II f 11 p This proves the case w(x) __ 1 . For a general w s AP /2 the argument is exactly the same, but we must usethe weightedversionof (4), namely, that suchinequality holds not only in  LP(IR n), but also in  LP(w)  if w s AP and 1 < p < °° . (This actually a rather straightforward consequence of a general theorem of Marcinkiewicz and Zygmund [6] together with the boundedness in LP(w) of the multipleHilbert transform ; see also Kurtz [5]) . Finally, it may be the case that, for some i  =  1,2,  . . .,  n,  we  have , Zi(I J)  _  for  all  Ij .  The necessary modifications to deal with this case are rather trivial, are are left to the reader . Given a sequence of disjoint consecutive intervals in IR, if they all have the lame length we have just proved that inequality (5) holds true, while in the case of lengths increasing at an exponential rate, the same inequality is obtained (and not only for pi2, but for all 1 < p < -) by classical Littlewood-Paleytheory . l t seems natural to .expect the same kind of result when the lengths increase arbitrarily . A partialpositive answer is contained in our next result which, for the sake of simplicity,will be stated in its one-dimensional version . TheoremC :  Let  {aej}j s ~Z  be an odd sequence of real num - bers (i .e . a-3 = -a j ), and assume that its positive part is co nvex - and slow1y increasing, i-e . j-1 j 2 ,~-1 ,~+1 a2 ~ j  :1 C a .  (j 1 1) Then, the quadraticexpression a . {  ~ 1 ?(C)  e 2uix~ d1 12  } 112 -~ a .,1-1 defines a bou nded ope rator in  LP(IR ),  2 1 p < ~ . GN f (x)  =  { f  1J  e -2Tril-y  g ( y )  f(x-y)  dy  12  dj}l/2 .  M<N  IR n When fsL 2 , the inner integral is absolutelyconvergent, and,  if we fix xs  IR n  and  N>  0,  there exists a function a(J) (depending on both x and N) of unit norm in L 2 (IR n )  such that GN f (x) 11 ¡<N J  a(I) e-2Tril-y g(y) I n = jh(y) g(y) f(x-y) dy where h s L 2 and jjh11 2 = 1 . Since K = (gl 2 s L 1 , we áppiy Scharz inequality to obtain G N f(x)  i  {1 K(y)  jf(x-Y)I 2 dy} 1/2 and part (i) is proved by letting N - - . Now, under the hypothesis of (ii), jK(x)j is dominated by a decreasing, radial, integrablefunction, and thus : (K * ¡f`2)1/2 < C M(Ifl 2 ) l/ 2 =C M 2 f In particular, the maximaloperator : f (x-y)  dy d1 8U0  {  J  I T m(u+d " )  f  -  m (u)  f12  du} 1/2 IR n is bounded in  LPOR n ),  2 < p <-_  and of weak type  (2,2) . By a standard technique, th epointwise convergence result will be proved for every f s Lp, 2 < p < -, if we establish this result for every Schwartzfunction . But, if f eJ(IR n ) : { JI T M(u+d .) f(x) - m(u) f)()¿)I 2 du}l/2 < <-_ {J{JIM(u+d~) - m(u)I  If(&)I  dj} 2 du} 1/2 < J  If(j)I{  JIM(U+61)  - m(u)I 2 du} 1/2  d1 Since m s L 2 , the inner integral in the last expression is bounded independently of d, 1, and it vanisheswhen d -> 0, so that an application of Lebergue's domináted . convergence theorem ends the proof . The followingparticularcase of TheoremD is worth mentioning : Let m = X B, where B is the unit 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 bothparts of the theorem can be applied . If we write S E for the partial sum operator corresponding to the multiplier  XE,  then the first part shows that it makes sense to define the operator : f --> {1 nISu+B f l2 du} 1/2 I for every  f s Lp(1R n ),  2 <__ p <__ -, even though, by Fefferman's theorem ([3]), each one of the operators Su+B can be defined only in L 2 . Moreover, the second part gives : Corollary 3 :  lf  f s Lp( .?n) ,  2 : p < -,  then S * f(x)  = sup  {J  ¡Sr(u+B)  f(x)12  du}1/2  C f9 2 f (x) 0<r<w  IR n and Zim  J  15 r(u+B) f (x) - f(x)I 2 du = 0  a .e . r->M (10) lim S rV f(x) = f(x)  a .e . Iu1<1 Given an open ball  V  in ]R n  containing the origin, it is not known whether is true for every fs L 2 . What Corollary 3 shows is that a certainaverage of the statements (10) for all balls containing the origin is true, but this is only a poor substitute which is far away from (10) itself . 5 :  The Case of Locallr Compact G roups An essential part of the results in sections 2 and 4 can be formulated in the context of locally compactAbelian (t .c .a .) groups, providing some sort of unified version of the discrete and continuous cases . Since there is no Hardy-Littlewood operator in context, we only obtain the Lp inequalities, pointwise estimates, and a different, slightly approach mustbe followed . Let X be a t.c .a . group with dual group X A generalelement of X (resp . X ) will be denoted by x, y, etc . (resp ., a, etc .), and ds, d1 will be the Haarmeasures on X and X , whichwe shallassume to be suitably normalised withrespect to each other . The letters H and P will be used to represent closed subgroups of X and  X , their Haar measures being m H and m r , respectively . The operator to be considered here is this general butnot the longer G  f (x)  I(ag)  -  f(x)I 2 dm,  (a)} 1/2 r When X =X= IR n , we can take r = 2z n , obtaining the operator defined in (1), or r= X = IR n, in which case we get the operator G of (9) (with g = m in both cases) . Theorem E :  Let  g s L 1 ( X)  be a positive function such that (11)  {  fIj(C,)12 dmr (a)}1/2 - M < r Then  the, operator  l :  defd red a bove  bound ea Lp ( X)  when  2  < p  < II Gfll p 5Mil  f11 p Proof : We define the closed subgroup of X 1 H  =  r  =  {x  s  X  :  < x,a>  =  1  for  all  as r} so that  r =  ( X/H) .  The Haarmeasure  m  in  X /H  is defined as the dual of  m r .  Then, a uniqueHaarmeasure mH can be fixed in H so that, defining a>g(y)  f(x-y)  dyl2 dm r  (a)} 1/2 and more precisely : (f s Lp ; 2 < p 5 -) (z)  _  1  f (x+y)  din H (y)  (f  s  L~ ( X) ) H (where z = (x +H) s X/H), the following identity always holds (12)  J  f(x)  dx  =  1  T(z)  dm(x) X  X/H A series of simplelemmas will be needed in the proof : Lemma 1 :  If g s L 1( X)  is positive, then, for every sX J 19-(CL-I)1Z dm,, (a) <_ 1 9(a)I Z  dm r  (a)  =MZ r  r Proof : Since (T) ^ (a) = f(a) for every a e r , Plancherel's theorem implies 1 9 12 din, (a) r J ji(a)j2 dm, (a) = 1  g(x) 2 dm(x) r  X/H But If(x)1 1 jfj ' ( x) for every f, and in our particular case, we have  1(1g) ' (x)j  :S g'(S¿),  whic hconclude sth e proof of the lemma . In our next result, we shall denote by B a Banach space (with dual B*) and by LB ( X)  the BochnerLebesgue space consisting of all (strongly) measurable B-valued functions F(x) defined on X and such that JI F(x) II B is integrable . Lemma 2 : Let K (x) be a measurable B*-valued function such that (13)  1  I K(x)-b1  dx  5  CII bII B  bEB Then, the formula (14) T F(x) = 1 K(x-y)- F(y) dy defines a bounded operatorfrom  L 1( X)  to  L 1 ( X)  mith norm 5 C . Proof :  It  suffices  to  verify  that  II TF II  1  1  C II F II L 1 for all simple integrable functions F, since these B are dense in LB . For each suchfunction F, (13) implies that TF(x) is well defined, 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 the next lemma, we take as our Banach space B =B * = L 2 (r 0 ) =L 2 (r 0 ; m r), where ro is a fixed compactsubset of r . Functions F s L 2 (X) are then isometrically identifiedwithfunctions f(x, a) = F(x) (a) in the product space  L 2 (X xr o ) . Lemma 3 : Let U : L 2 , LB be defined by Uf(x, a) (ag) * f(x) . Then U is a bounded operator withnorm 5 5 M, and its adjoint  T = U : LB , L 2 is given by (14) where the kernel is :  X(x) = k(x, a) = g(-x) <x, a> . Proof : The firstassertionfollowsfromPlancherel's theorem and Lemma 1 : ~I U f (x) 11 B dx x J  1 ^  ?(j)1 2  d1  dm r (a) rx 0 M 2 1 ^ I?(J)j 2 d1 = M2 J If(x)I2 dx The kernel of U * is obtained by a simple computation which is left to the reader (the fact that we take a compact subset r 0 c r makes all the integrals absolutely convergent) . We are now in a position to complete the pro .of of Theorem E . Let G of denote the operator defined as Gf after replacing r by ro . If it is proved that II Gof II p = MII f II p  (2 < p < -)  for an arbitrary compact subset r o of  r , then, letting ro increase to r , we get the desired result for Gf . But IGo f(X)1 = = IIU f(x)II B,  so that the inequalities to be obtained are (15) IIUfII LP ~MIlfll p (2<p5~) B For p = 2, this was proved in Lemma 3 . For p = -, (15) is equivalent, by duality, to 11T FII,1 5 MI¡ FIIT1 "  LB and this can be proved by verifying that the kernel K(x) defined in Lemma 3 satisfies (14) with constant C = M . In  fact,  given  b  =  b(n)  s  B  =  L 2 (r 0 ): IK(-x) .bl = I  g(x) <x, a>b(a) dm r (a)I r 0 =  Ih(x)I  g(x) with  h s  L 2 ( X /H)  and  II h II 2  =  II b II B, so that we can use the identity (12) to obtain 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, the case 2< p < - of (15) follows by interpolation, andthe proof is completed . When r = X , (11) simplymeans that g e L 2 (X) . Moreover, in this case, the assumption g i 0 is not necessary, and we have a statement completely analogous to Theorem D(i) . On the other hand, if r is discrete and g has compact support,then (11) is triviallyverified . The truncation argument used in TheoremB can also be applied to the smooth G-functionconsidered here : Given m s L w ( X),  we say that it is an Lp-multiplier if the operator  Tm defined (and bounded) in  L 2 ( X)  by (T m f) ^_ 1m admits a bounded extension to Lp( X) . Theorem F :  Let  me Lw ( X)  be - a compactly supported  Lpmultiplier, with 2<p < rg oup o,f  X ,  then,  the operato r ásrITM(a+ :)  f(x)I 2  } 1/2 is a discrete sub-