Generalized Lipschitz Spaces on Vilenkin Groups

zyxwvutsr zyxw Math. Nachr. 132 (1987) 67-80 zyxw zyxwv Generalized LIPSCHITZSpaces on VILENKINGroups Dedicated to Professor Hans Triebel on his fiftieth birthday By WALTERR. BLOOM of Perth and JOHN J. F. FOIJRXIER*) of Vancouver (Received April 23, 1986) .ibstmet. It is shown t h a t , for certain choices of the defining indices, the generalized LIPSCHITZ spaces on VILEWKIJgroups are incIudec1 in certain FIGA-TALAXAYCA spaces A,, and t h a t the FOURIER series of functions in the letter spaces converge uniformly. This result includes a n extension of the classical DIN test, and endpoint versions of BERNSTEIN'S theorem on absolute convergence of FOURIER series. zyxwv zyxwvu zyxwv zyxwvuts zyxwv zyxwv VILEYKIN System 1. The groiips G and I' 1.1. The analysis will be carried out on an infinite zero-dimensional compact abelian group G satisfying the second axiom of countsbility. It is well known that the dual group I' of G is a discrete countable abelian torsion group admitting a sequence {T,L};=o of finite subgroups with the following properties : (i) Po= {xo},where xo is the identity for I'; (ii) T,,cI',,,; (iii) 1'= - U T,,; n=O (iv) Pn+JF,, is ofprime orderpn+l; (v) there exists ~ J ~ € F ~ +with ~ : I 'y?+' , , Ern. Such a pair (G, r)is called a VILENKIN system. Furthermore if sup pn -=- then n theVILENKIN system is termed bounded. Write G, for the annihilator of I', in G, so that each G, is a, compact open subgroup of G satisfying Go=G, GnxGn+l and - n Gn={O}. For each n there m .. =n- exists x,€G,,\G,+~ such that Q),(x,,)=exp ( 2 7 ~ i p ; ; ~and ) , each xEG has a unique representation b,x, where 0 sb, <pn + l . n =O *) Research partially supported by NSERC Grant # 4822. 5' zyx zy ti8 Jlnth. Xachr. 131 (198’7) zyxwvuts zyxwvutsrqp zy zyxwvuts zyxwvutsrqponmlkjihgfedcba zyxwvu zyxwvutsrq zyxwvu for C , -1) T zt and observe t h a t each nntural number can be uni(1uely eq)ressed as wit,h 0 S X , ~ p , (the + ~ slim is always finite). Now defitie CC,~,, m . The following facts are evident: (a) /’,={x,,:osn-=m,J; (1)) ~ , = ~ ~ ~ , ~ ~= w ~,m h ,e- t r- kewith n O~k-=n,,. I n l)articular,xnl, = y, . All of the above notions were introduced by \ ‘rr.mnx [13]. It is well known that every compnct group satisl‘ying the secon(l m i o m of countability can be mapped onto the interval [0,1] with the preservation of measure. In the case of a VILENKINsystem such a mapping can he given quite explicitly. Defjnee: G-[O,ll by - -I d.2)= n2 h,#%‘t I 7 - =(I where X E Gis represented as in Section 1.I above as bllz,,.The uniqueness ofthe 14 representation entails that ~(c,,) = [ O , m,‘] for each 17. Q is well defined. I t is clear t h a t We note that e sa (pn + n=k I =n 1) x,,and zk- I are mapped t o m,’ Q is onto a i d that ( is not one-to-one for example, both 1 , b u t the set of these “anornolous” points is countable, hence of measure zero. The map Q is also measure preserving because of the fact that ;(GI,)=m,’ =A(e(Gll))for each I I . 1.3. DIRICHLET kernel on G The DIRICHLET kernel of order n is defined by zyxw zyx zyxwvut zyxwvu zyxw zy zyxwvut zyxwv zyxwvutsr zyxwvut zyxwvu BIoo~ii/Fournier,Generalized Tipschitz Spaces Writing 71. = - (J.3.1) we have ( C ~ S S E L I Y CL,.~~. r=U - ur ti9 [el, p. 345 : see also VILENKIN [ 131, p. 5) -I x %,GrUr(cA). D,,=%r,r=O J =I) It should be obberved that for each T , D,,,,. = m , t,;,. VILENKIX[13], 3 2.6 showed that there exists a positive cwiistant B such that liD,illi~Blog )L for infinitely many 1 1 . \Ye need the following strengthening of that conch sion. 1.3.1 Lemma. There i d U T L nhsolirte positive comtaiit B ,YO thrrt / o r P c w h pobititv: k t e g e r 3 there i.3 o n iiiteger 16 5 A T f o r ?rihiefzl/Blt/iiZ B logs. P r o o f . The corresponding conclusioti for uniformly-bounded o r t h o n o r i d systems on the interval [0,1] is a special case of known results [3]. To prove the lemma, use the measure-preserving mapping 0 of Section 1.2 to transfer the results on [0,1] to VILENKINgroups. // 3. HEsov Spaces 2.1. Moduliis of coiitirtuitj The motlulus of continuity off E L’’(G) is tlefiiiod by w1,(J;t ) =SUP {litJ-J l l p : rZ(r/. O) ~ t , ] where t z 0 , t,f:z-f(.e-a), and the metric d is given by d(.r. y)=m,;;, for .c-y CC21h\G,c+ [. Alternatively one can make use of the measure-preserving mapping 0 introduced in Section 1.d above and write oJ;(f; t ) =sup {lIt&-f[lp: p(al st; fortE[O,J]. Tt is easily verified that furtE[)ih;,!l, m,’], c g k ( f ; t ) s t u j ( f ; 111, I ) =w,(f; m,;+!J z c 9 , ( f ; t ) , and that the t-ivo moduli are equivalent if and only if G is bounded. LVe will follow convention and use cop. 70 zy zyxwvutsrq zyxwv zyxw zyx zyxwvuts Math. Nmhr. 139 (1987) zyxw zyxw and define the generalized LPSCHITZ space A ( a ,p , q ) by 4 a 7 P>4 ) = { f € - W G ) :llJlln;p,p~m) where, in the case p =a,it is further assumed that the members are continuous. Given an integrable function f and a non-negative integer n, let f , be the differspace B ( a , p , q ) is defined by ence Dmll+,*f - Dmn*f.The BESOV r - B ( a , p >-) = { f € L P ( G ) :sup ~ ~ + 1 l l f f l I l p - = 4. 1) Both of these spaces were studied by QUEK and YAP [lo] and OMBE [8]. The BESOVspaces, for the case when the underlying group is EucLIDean space, are discussed by TRIEBEL[ 11J for the extended range ( - -,m) of the index a. Quxxi and YAP [lo] gave the following characterisation of the generalized LIPSCHITZ spaces. 8.2.1 Theorem. F o r a ~ ( O , - ) ,qE[l,-),p€[l,=], zyxw zyxwvut I f€L”(G): - 2[r4+PJp(f; n 731,51)19 <-}. =U We note that the result can be extended to the case CI. = 0 or q = 00. 2.8.2 Theorem (1) For qE[l,-), p€[l,m] , 4 0 , P ?q ) c i f EWG): - cb J f ; 17L;+9Iq . ?k=o In the cme that G is bounded, t8e incEasio7L cun be replaced by equality. (ii) ForaE[O,m),p € [ l , = ] , A ( a ,p,-) = {fcL”(G): sup nzi+lw,(f; 7n;;l) ,-} . n P r o o f . The equality in (ii) is easy to show. The inclusion (i) follows from I in,; +I For bounded G, log m,,log p , + IS C , giving the reverse inclusion. // mn The above results lead to the equivalence of the generalised LIFSCHITZ spaces and BESOV spaces; this was done by OMBE [ 8 ] , Theorem 1.2.5 for bounded VILENKINgroups. zyxwvuts zyxwvu zyxw zyx zyxwv zyxwvu Bloom/Fournier, Generalized Lipschitz Spaces 71 2.2.3Tlieorern(i) F o r c r ~ ( O , I ) , q ~ [ l , ~ ] , lp lE([~l,,pq~,)]=~ B ( ~ , pq ), . 4 0 , p , q) c B ( 0 ,I), (1) , (ii) P r o o f . Similar argunieiits work when q=m and q-=.g. We present the details for the latter case. Let fc A('A,p , q ) , 'A z 0 . Then No \v - zyxwvu zyxwv zyx (since t,,Dm,l - I), = 0 for all .n S T ) where Or,,(y) =sup { Iz(y) - 1I : x < T,*),and we have used BERNSTEIN'S inequality. El],Theorem 1 (proof) we obtain Referring to BLOOM since y E G,, and thus 72 zyx zy zyxwvutsrqp 31at.h. Nnrhr. 139 (1!1X7) zyxwvut zyxwvuts zyxwvu zyxw zyxwvzy zyxwvut a s ~ n i . , , + , I m , , ~ ? f o r e a1 1c. h Now define sequences ( n k ) .( b k )by Then ( ~ ~ ) c (h,<)EE', l", an(l hence (b,) * ((L~)EZ'~. Thus ( 2 . 2 . 0 gives thiLt, jE-'l(~.p,fl), using Theorem 3.2.1 // For luonndecl VILEXKIX g r o u p the ahove result,s were nicistly given by 0 3 1 1 3 ~ [Y]. The first p art of the proof also works where a s 1, so that. L~l(.;., p , q ) c B ( z ,p . q ) for such values of X. OMBE[ 8 j , Theoreni 3.1.3 also proved the following inclusic!n.. between certain EESOV slmces which, in view of Theorem 2.2.3 ( i ) , also holds for generalized h W X I T 2 spaces. 2.2.4 Theorerri. L P C~ 6c n Thrn VILENKIX group. Gizwi iiir1,icc.y p I rrrcil p 2 tcith. thc i n t e n d ( 1 / p l - 1 / p 2 ,I ) . let /Y=z-(I/p,- l i p ) . Doiitirietl 1 =plcp2-=wtr i n c l m irides x ill * I ( @ , p i , y ) c ; l ( $ , p., q ) for nll i,)tclices q in thc iutfwctrl [ 1 , -1. In prticular, .f(1/p,,p1,l)c:I(t/p:. f1.2. I) if C: is hounded a n d 1 4p,--rp2-==,.Moreover, as noted above . I ( l , t , t ) c B ( l . l , t ) , which is included in all the spaces .I( 1/p,p , 1) with p c ( l , - ) . There is n o such inclusion, however, hetmeen t.he spaces .!(l / p , p , 1) and A(B,.o, 1) for any p z1. To see this i t is sufficient, to show t h a t t h e smallest of these generalized LII~SCHITZ sliaces, .1( 1, 1, I ) , is not c o n t ~ ~ i i i eind il(o,m, 1 ) . Indeed, writing k,,=nlJ,'L),,,, it is not difficult t o check t h a t from which it follow-st h a t I, and we have -I zyxwvutsr zyxwv zyxwv zyxwv zyxwvut zy zyxwvut zyxwvu zyxwvutsrqpon B loom/Fournier, Generalized Lipscliitz Spaces 73 Thus the sequence ( k n ) is hounded in -1( I , I , I ) but not in .I(O,-, 1 ) , and this takes care of the assertion. It is also the case t h a t Ll(O,=, 1 ) is not included in any of the spaces -,1(1/p,p , l ) , p z 1 when G is bounded; this will follow from Example 2.3.5 below. ’3.3. Puiictioiis with iinifortrily coiivergent Fourier series The FOURIER series of f € L1(G) is said to converge uniformly if U,, * f converges to f uniformly. The collectioii of d l such (continuous) functions is written U C . with norin given by !if/iUc 2: su1) * . 71 zyx There are various conditions known for the circle group that ciL11 be plticed oti it function f to guarantee uniform convergence of its FOC’R~F:R series. Such conditions are usually given i n terms of tho modulus of continuity. For the L”-norm HOHSON’S uniform vekion of the 1 3 1 ~ 1test (see % T G ~ I T N . D[ 141, Theorem 0.8) is the central result. mid interesting versious of this for the L” spaces have been given by GARSLA \ 51. 3otli zLppronches can be viewed as titi inc~lusionbetwoen the iLppropriate generalizecl L W S ULTZ ~ sl)ilcc iintl L’C. We exillore this idea in tho setting of VILXWKLN groups. Let p’ be the i n c h conjugate t o p . Consider the FIG~-TALASISNCA space i C yk * hk with gkcL?’(G‘), I it,((;)= /€L”(C:):/= I.=l h , c L ” ’ ( ~ ) and C 1/yk&, [+k/’l,, k=l <-l . A suitable norm is obtained by setting [[f[iLip equal to the infiiiiutn of the 11011negative sums appearing in the above definition, taken over all possible representations off as an infinite sum of convolutime ofg,: arc1 ht. It is easily checked t h a t A t(G)= C(G\ (see HEWITTand Ross [7], Theorem 32.31) and A,(G)=A(G) (see HEWITT and ROSS[7], Corollary 34.16); here A ( G ) denotes series. In fact- one the set of functions on G- with absolutely convergent FOURIER has the inclusion A,,(G)cA,,(G) whenever l/plis closer to 1/2 than 1 / p ; in particular A,(G) = A , , , ( G ) . The spaces A,(G), p E (I,-) provide a, rich source of functions with uniformly convergent FOURIER series, as is shown in the next result for bounded VILENKIN groups. zyxw 2.3.1 Theorem. Let G Be a bozincled VILENKIN group. Then A,(G)c UC for pE(l,-). P r o o f . Wefirst show t h a t Lp * L p ’ c U C .L e t f = g * A , whereg€LP(G),hELp‘(G). Now the M. Rmsz theorem (see GOSSELIN[B], p. 352) shows that IP, *gll, ~~plli711, I P N *8-9/lp’O zy zy zy 74 Natli. Nachr. 131 (1987) and zyxwvut zyxwvut zyxwvutsrqp zyxw zyxw zyxwvut 7 I k>K c I!Ql zyx * (a& *A,/&.)- u k y k * hJ- z k >x jakl I gp-gkll,, lIhkj!p’2 4 2 . Also there exists 7)” such t.hat \IDlt* ( a k g , * h,) - CTtgk * /?,kI!- %E/2K forallnzw,,, ksK.ThuulID,,,*f--fll,~&ff0ritll71.~nl,, asrequired. // We now show that the LIPSCEITZ space A( 1/p,p , 1) is contained in A,(Q) for each p z 1. zyx 2.3.2. Theoreni. Let G be (I VILENKINgroup. Th.e?i,f o r p E [I,-) 4 U p , p, l ) c A , ( G ) . P r o o f . First note that f o r f c d ( l / p ,p , l ) , llfnllAp and since 4lf7tllp llRl,t+ L - ~ “ , c ! l pr ’ 11 D,n,t+,- Dmn1$ =m:’ (mi’- m,;: for some constant, C‘, we have - ,) + (m, + - 7 ~ 4 ~ ) ” ’na;+,I s C ‘ 7 r ~ ~ ~ L ’ - - 2 I I ~ ~ I sI ~c, 2 ~ L P ; T ’ ) ’ ~ ’ I II/~ =c C d’?,i,iifnilP-==o 78 n rr=o p n=ll using Theorem 2.2.3(i). In particular, f € A , ( G ) , as required. // When 1 s p -=2, and G is bounded, the conclusion of Theorem 2.3.2 can be Strengthened by first using Theorem 2.2.4. Indeed, for such indicesp, -4(l/p,p, 1 ) ~ A ( l / 2 2, , l ) c A ? ( G ) = A ( G.) This improves on Theorem 2.3.2 for these indices because A2(G)SA,(G)when p + 2 . On the other hand,’ as p increases through the interval (2,-), the spaces Ap(G)become larger, and the conclusion of Theorem 2.3.2 cannot be strengthened in this way. At the end of this section we will comment further on the connection between our results and BERNSTEIN’S theorem. Next we prevent an analogue for VILENKINgroups of HOBSOX’S uniform version of DINI’Stest. zyxwvu zyxwvutsr zyxwvu zyxwvut zyxwv zyxwvu zyxw zyxw Bloom/Fournier, Generalized Lipschitz Spaces 75 2.3.3. Theorem. Let G be a bounded VILENKINgroup. Theti il(O,-, 1)c UC . P r o o f . LetfEA(O,-, l ) ,ZCQ.Then, for eachinteger Now, writingn= 11, 2 ' zrmr w h e r e ~ s u , . ~ p ,wehave. +, foryEG,\G,,+, , r=ll zyxwvutsrqpo using (1.3.1.) and the property that D m r ( y = ) 0 for r =-s. Thus S Y ),I - rfl m-1 r+1 = Qllfll",.=,I * It follows that Ilflloc =SUP llQa n * fll..~~llfllo,-,l zyxw * It is easy to show t h a t the subspace of A(O,-, 1) consisting of all trigonometric polynomials is dense with respect to the norm ~~*~~,,,,,l. Moreover, every trigonometric polynomial belongs to UC. Given a function f in L I ( O , m , l),express it as a limit in A ( o , - , 1) of a CAUCHY sequence (Pn);=l of trigonometric polynomials. The inequality above guarantees the polynomials PI,also form a CAUCHYsequence in the space L'C. Therefore f C UC. I/ 76 zyxwvutsrq zyx zyxw zyxw zyxwvuts zyxwvu zyxwvutsr JtRtll. Saol1r. 132 (1987) Theorem 2.3.3 should be compared with the D I S I - ~ ~ Y S C ~test H I Tgiven Z for g r o u p by ONNG\VEI"X a n d 1 Y A r m m u N In], Corolliiry 2. Taking bounded VILENX~S Theorems 2.2.4 and 2.3.1-2.3.3 together we obtain IX l'hei~ 2.3.4 Theorem. Let C be boiiiirlcrl Y L L ~ X I ~qroup. zyxwv zyxwvut zy -I(l/p,p,1 ) c c I C forallp€[l.-l. It follows from our [iroof of Theorein 2 , 3 . 2 thilt if fe&J(l/p,p, 1) fol.so11iepE[ 1 . a ) the11 (; is hour (led u n c l if 111 view of Theorem 2 . 3 . 3 , it miglit be Iiopecl that the samc conclusioii lioltls for all f in A ( 0 , - , 1).We shorn that this is not possible. 2.3.5 Exartiple. Let is There esists g in - I(U,-, be n, hounded V r r , ~ c , u ~group. 1) 2 jjgll//UC =-, In 1)artiCuliLr the functioii 9 beloiigs to none of' the sl)aces =u A( l/p, p , I ) witlipE[l,-). with IL P r o o f . For each positive integer LV, proceed as follows. First notc that for each integer n between 1 anclX, the tlual group ofGI, '3s" l'/i',,.J3y Leinrria 1.3.1 ;Lpl)lied to the group G,,, some DIRICHLET kernel of degree a t inost 2"' hits LI-norm tLt least B N . The t.r,znsform of this lreniel vanishes outside the subgroup f,, .kLv/T',l.It, follows that there is 5 trigonometric polynomial, P,,say, on a,, such that fj, vanishes outside l',,+s/I',kand such that Ill',,[[..= 1 but ~ / P , JZ~B r cX . Extend P,,to all of G by making it 0 off GTk,and den0t.e this esteiision by Flk. Then the spectrum of F,,is included in I:,+, because P,,(y)= PTL( y + f 'lt)/mrh for all y in I'. Noreover llF,ll-= I and ~ ~ l f ' , , , ~ ~ c T These c~BX properties . of FILare not affectecl if it is translated by an element of G. For each positive integer n s N carry out such a translation of El, by z I 1 - , , multiply the result by rpll+s, and denote the function thus obtained by Q,,. Then &,, vanishes outside r,l+-v+,\T,t+S and l/Qn/lm= 1, but ~ ~ Q ~ L ~ ~Let uczBil~. and, as before, let f,,=D,,z 20=0 +l * f - D,,,, * f. Then / 7 E + x = Qand ,k 7Z-l On the other hand, the effect of the translation by in forming F,, is that Q, vanishes outside the set Gl,-l\Gn. I n particular, the functions &, have disjoint supports, and IlfIl. 5 1. 60 co,(f, t ) s2 for all t . But also w,(f, t ) = 0 whenever d ( t , 0) c l/mLJ,+lbecause f vanishes outside rLAV+,. It follows that llfll",ca,l SCIV . zyxwvu zy zy zy zyxwvut zyxwvu BIoomiFoiirnier, Genernlizecl Lipwhitz Spaces If in fact every function 9 in A ( O , - , 1 ) had tlic property that then the rlosed-graph theorem would yield a constant A so thRt Ti7 - 3 ~ ~ c J ~ ~ ~ zyxwvutsrqp c ”=” Q~ll&-,! ll~,~lIcrC n=I) zyxw zyxwvuts zyxwvuts for all functions g in A ( O , - , 1 ) . The analysis in the previous paragraph shows thnt I ) with there is 110 such constant K . Hence there must be tz function q in ;I(O.-, JVe now conunent on the relufioiiship betweeii Theorem 9.3.2 nncl ext.enaior~s of :KERNSTEIX’S theorem for LIIWHITZsppaces. Firstly observe that ;l(a, p,-)={fELY(O): q , ( f ; t ) =O(t“)) is just the usual LWXHITZspace with euponciit p . It wa.s s h o w n by t h o [ I ? ] f o r h u n t l e d VILESKIS groups, nntl ext,entlecl to the general case by J?i.uoht [2], that, for p t 2 an cl u =- 1 / p , 1l(a. &I, c-L” ) * L”‘(G). In p t r t icu lxr it. follows that A(.*,p , 1) c A,(@ for a > lip. In t,liis form it is clear that ‘I.’heorem 2.3.2 provklex the n,pprop;j:tte endpoint,.result. An anslogy can itlso be drawn with the clnssical EERYSTE~X theorem. This was given for ‘ILEYKIN groups by BLOOM [21, who showetl t h t for a >1/2 (2.3.1) il(x, ~,-)cA(G). zyxwvu By Theorein 2.2.4, the same inclusion holds for ~ l ( u , p , - ) when pc[1,2) anti a =- l/p. As indicated in the discussion after Theorem 2.3.2, the endpoint result whenpc [1,2) is that -4(I./p,p, I ) c A ( G ) . Finally, we remark t h a t our conditions on the indices are best-possible. As in BLOOM [ 2 ] , Theorem 3, there is ZL function that belongs to all the spaces A ( a , p, p) with u -= l / p or a = 1/p and q > 1, but which does not belong to L‘(G). Also, when p 2 2 , the smallest of the spaces A, that include A ( l / p ,p, 1 ) is A,. 5.4. htegrahility theorems In [4] results similar t o those in the previous section are proved for the circle group. It is also shown that, in that context, there is a connection with certain. conditions on the coefficients of trigonomet,ric series that guarantee the series represents an integrable function. The results in this section will serve a5 a basis for a similar study of such conditions on the coefficients ofseries on VILENKIN groups. 78 zyxwvutsrq zyx zy zyxwvuts zyx zyxwvut zy Xiath. Xachr. la? (1YS7) 2.4.1 Theoreni. L e t G' be a bounded VILENKINgroup: pE[1, -) nxd f E A ( l / p ,p , I ) (zuith f ( 0 )=O). Then the fwzction z-f(x)/d(z,0) beEongs to Ll(G) if and onZg if zyxwvuts P r o o f . I n view of Theorem 2.2.4, we can assume from the outset that 23-1. First consider Now dz d(z. 0 ) 91 = I J and lip S Q IVnIlp m n + l ; hence, as f E A ( l / p ,p , 1), the sum (2.4.1) is finite. Thus . - n Bloom/Foornier, Generalized Lipschitz Spaces zyxwvu zy zyxwv zyxw zyxw zyx zyxwvutsrqpo zyx zyxwvut zyxw iff 2 ID,,,*f(o)i . i== ,l =O It remains to establish the equivalence between the latter sum in (2.4.3) and th;Lt appearing in the statement of the theorem. For this it suffices to show that 2. max n i , s - - kc r n , t or equivnlen tly . (2.4.4) 2 * f ( o )- D,,~ * {(o)1 c= +L max *f,,(O)-n,,,, *f,,(O)i IDk n=u " , ~ % k < m , + + i m -=- . Now f n =/,I :l2 (n, - Dm,,) and, using the Jf. RIESZtheorem (GOSSET~IN [GI, p. 321). + Il(%-~,nJ .b AaIL 41(Q-D,nn) ~Qd':*llfnllp *fA, Il&, +, -DmJIp, 7 so the inequality (2.4.4) a n d the theorem are established. // I n the proposed applications of these idem w e will need the fact that the first two steps in our proof of Theorem 2.4.1 are valid under weaker hypotheses. Observe first that the equivalence (2.4.2) holds provided that Also, the equivalences in (2.4.3) hold for all functions f in the space B(0, a,1). We summarize these observations. 2.4.2 Lemma. Let f be a fiinctio.n in B(0, a,1) satisfying condition (2.4.5). Then the fundion x - f ( x ) / d ( x , 0 ) belongs to Ll(G) if and only if - C IDmn* f(o)l <a. n=O 3. References [i] WALTERR. BLOOM,A choracterisation of Lipschitz classes on 0-dimensional groups, Proc. Amer. Math. SOC.53 (1975) 149-164 [2] WALTERR. BLOOM, Factorisation of Lipschitz functions on zero dimensiohal groups, Bull. Austral. Nath. SOC.23 (1981) 215-226 so zyx zyxwvutsrqp zyxwvut zyxwv zyxwvu zy zy zyxwvutsrqp zyxwvutsr Xath. Xachr. 132 (1987) [3] S. V. BOCK-~IZIEV, -1method of averaging in the theory.of ort,hogonal series slid soine problams in the theory of bases, Trudy. Mat. Inst. Steklov 146 (1978) 1-92; trnnsleted 8s Proceedings of the Steklov Institute of Mathematics 1980, #3 [4]JOHN J. F. FOURXIER and WILLIAJI81. SELB,Some sufficient conditions for uniform convergence of Fourier series, J. Moth. Anal. Appl. (to appear) [ 8 ] ADRIANO31.OARSIA, X remarkable inequality and the uniform ronvergence of Fourier series, Indiana Univ. Math. J . 25 (1976) 85-102 [ri] JOHN GOSSELIX,Almost everywhere convergence of Vilenkin-Foirrier series, Trans. Amer. Xath. SOC.1% (1973) 345-370 [7] EDWINHEWITT,and KEWNETIIX. Boss, Abstract hnrinonic analysis, volu. I. 11. SpringerVerlng 1963,1970 [Y] H. OXBE, Besov-type spaces on certain groups, P h . D. thesis, UXTM (hlbtiqtierque) 1984 [3] C. W. OXXEWEER, and D. WATERAXAN, Uniforin convergence of Fourier series on groiips Tt Xi&. Math. J. 1s (1971) 265-273 [ 101 TONG-SEXG QUEHand LEONARD T.H. YAY, Foarier transforms of Lipschitz fiinctioiis and Fourier multipliers on compact groups, Nath. Z. lS? (1983) 537-548 [ 111 HAKS TRIEBEL,Theory of function spaces, Monograph in Math. is, Uirl<hiiiiser 1983 [ 121 I'osrrrrr.~zu Uxo, Lipschitz fnnctions nncl convolution on 1)oundcd Vilenkiii gronps, Sci. R,ep. I C a n s z u w Univ. "8 (1978) 1-6 [In] N. Ja. VILPNKIN,On a class of complete orthonoritial systems. .liner. AIatli. Sor. Trnnsl. 2 s (1963) 1-35 [ 141 .L ZYGw.lXD, 'I'rigononietric series. vols, I, I1 2nd ccl.. ('ainliri(lge Univ. Prws 19ti8 School of Yothe mtcticol (&nil Physical Sciences ilIurdoch Uniuersity Perth, WA 6150 Australia Department of Yathemutics University of British Columhici V(tncower, B. C . Cmadu V 6 T 1 YB