Pointwise convergence of the fourier transform on locally compact abelian groups

Publicacions Ma emá iques, Vol 37 (1993), 45-55 . POINTWISE CONVERGENCE OF THE FOURIER TRANSFORM ON LOCALLY COMPACT ABELIAN GROUPS A bs ac MARIA L . TORRES DE SQUIRE * We ex end o locally compac abelian g oups, Feje 's heo em on poin wise con e gen e o he Fou ie ans o m . We p o e ha lim <pu * (y) = (y) almos e e ywhe e o any unc ion in he space (LP,l')(G) (hence in LP(G)), 2 <_ p< oo, whe e {cpp} is Simon's gene aliza ion o locally compac abelian g oups o he summabili y Feje Ke nel . Using his esul , we ex end o locally compac abelian g oups a heo em o F . Holland on he Fou ie ans o m o unbounded measu es o ype q . 1 . No a ion and P elimina y Resul s Th oughou , G is a locally compac abelian g oup, wi h dual g oup F, and Haa measu e m . By he s uc u e heo em, G is ep esen ed by Ra x GI, whe e a is a nonnega i e in ege and Gl is a g oup which con ains an open compac subg oup H . The se o basic neighbou - hoods o xeG is deno ed by JV,,(G) . We w i e C,(G),Co(G) o he spaces o unc ions on G ha a e con inuous, wi h compac suppo and anish a in ini y, espec i ely . We conside he amalgam spaces (Lp,lq)(G),(Co,l9)(G)(1 <_ p, q <_ oo) as de ined in [S] . The Fou ie ans o m (in e se Fou ie ans o m) o a measu e M, is deno ed by ( ,) . We le A,(G) be he se o all unc ions in C,(G) such ha e L' (F) . The cha ac e is ic unc ion o a subse E o G is deno ed by XE . The conjuga e p' o a numbe p is such ha 1/p + 1/p' = 1 . Fo each U e N o (G), A .B . Simon [Si] de ined a unc ion WU as he p oduc o wo unc ions aU and Q de ined on R' and on G1, espec i ely . *1980 Ma hema ics Subjec Classi ica ion (1985 Re ision) . P ima y 43A55, 43A25 . Resea ch suppo ed by NSERC g an 7914 . 46  M . L . TORRES DE SQUIRE The unc ion /3 U is con inuous, nonnega i e, wi h L l (G)-no m equal o 1, and (1)  sup ¡,Qu(x)j = BU <_ 2m(U)/1 - 2m(U)  ini e . G Hence BU --+ 0 as U -4 0 . The unc ion aU is de ined as ollows . Le (-b1, 51) x ... x (-b a , S Q ) x UH be a p oduc neighbou hood con- ained in U, whe e 6i > 0  (i = 1, . . . , a), and UH is an elemen o No (G) included in H . Fo i = 1, . . . , a  we se Ui = ( - bi, b i ), N i = 1/bi, and de ine he unc ion au n on R by 1 .1) Wu is con inuous, nonnega i e and bounded 1 .2) Wu is in eg able and 11 Wu 11 1 = 1 1 .3) cbu e C, (F) and 11O 11oo < 1 1 .4) Wu(x) = 0(y)y(x)dy by 1 .3) 1 .5) Fo e > 0 and U e No(G) gi en, we can ind a V such ha i V' <V and x ~  U, hen cp (x) < e and G_V cp ,(x)dx < e . 1 .6) 1 .7) aU¡ ( ) - 1 - cos(Ni ) 7 Ni Fo =  ( i , . . . , a )  in R a ,  he unc ion aU is gi en by au( )  = Há l au i ( i) . Clea ly aU is con inuous, nonnega i e, and i s L 1 (R a)-no m is equal o 1 . Each Wu has he ollowing p ope ies . Fo a p oo see [Si] . lime 0u (^Y) = 1 . he amily {WU1U e No(G)} is an app oxima e iden i y in L 1 (G) . We add o his lis he ac ha each W u belongs o he Wiene algeb a (C0, 11 0) P oposi ion 1 .1 . Fo each U in No(U), he unc ion aU belongs o (Co,11)(R") . P oo .. Since au n (i = 1, . . . , a) is an e en unc ion we ha e o n in Z- {0, -1} ha sup au n ( + n) = sup nu, ( - (1 + n)) <_  2  1 - . E[o,1]  E[o,1]  Ni7 n 2 I n e {0, -1}, hen he e exis s a cons an Ci such ha sup au n ( + n) < Ni - NiCi E [o,1l CONVERGENCE OF THE FOURIER TRANSFORM ON GROUPS  47 because he limi lim _ _n 1 - cos Ni(n + ) (N i (n + ))2 exis s . The e o e o all i = 1, . . . , a and all in ege n we ha e ha whe e sup  lceu a ( +n)( < can E [0,1] c = max (2/(Niz), NiCi/7 ) 1<i<a and an is equal o l/n 2 i n E Z- {0, -1}, and o 1 i nE {0, -1} . Finally, o i = 1, . . . , a we ha e ha Z F om he de ini ion o he no m 11111, i is easy o see ha IIaullool =Ilá 1liauilloo1 " Co olla y 1 .2 . Fo each U in N0(G), he unc ion WU belongs o (COI1 1 )(G) . P oo . By (1) we ha e o all ( , s) in G ha hence lau ; 11 .1 = 1 :  SUP  l aui ( ) l Z E[n,n+1] _ E sup lau ; ( + n) Z E[0,1] < C L : an < 00 . Wu( , s) = au( )Ou(s) < Buau( ), llWu1jool <_ Bu i  sup  Ia ( )1 = Bullaul i .1 . neZa +nE[0,1]a Fo he es o his pape Wu,'au, and Ru a e as indica ed in his sec ion . 48  M . L . TORRES DE SQUIRE 2 . Main Theo em In his second sec ion we wan o p o e ha (2)  limo Wu * (y) . = (y)  almos e e ywhe e o all in (LP, lw)(G)(2 < p < oo) . Fi s , we p o e wo lemmas . Lemma 2 .1 . Le V and K be wo elemen s o NO(G) o he o V = ( - s1, 61) x . . . x ( - sa Sa ) x VH and K = [--yl, `y1] x . . . x [ - ya, a] x KH, whe e Si > 0, -y2 > 0 (i = 1, . . . , a), VH and KH a e elemen s o No (G) con ained in H, and KH is compac . Fo 1<_ p < oo, we de ine li = min(S? P , ~ yi)  (i = 1, . . . , a), and we le WH be he in e io o KH . Then he se W = [ - 171, ll, ] x . . . x [-77a, l a ] x WH belongs o Vo(G) and o a ixed y = (yo, so) = (y,, ... , ya, so) in G, he elemen W y = y + W o N y (G) has he ollowing p ope ies : 2 .1) W y C y +KH 2 .2) I II . = [ - l1 + y1, 971 + y1] x . . . x [-?1 a + yaga + ya]i hen 1 1/P a [ Ia nu(yo - x)Pdx ] = Q(IIi-1bi) I 2 .3) Ra - II a C U I n , whe e {I ,} is a coun able amily o compac subse s o Ra, and 1/P I aa(yo - x)Pdx J  = O(1i i 1si)* In 2 .4) The e exis s a cons an C such ha SUPN C(I n ) <_ C, whe e C(In) is he ca dinali y o he se {j e Z a 1 (j + [0,1] a ) n in 7~ `N} . P oo . Se e al cons an s will appea du ing he p oo and since hei speci ic alue is i ele an o ou needs we jus w i e C1, C2 . . . . C Q . Pa 2 .1) is clea . Se Ji = [ - 7%i + yi, 77i +y¡](¡= 1, . . . , a) . Pa 2 .2) ollows om he con inui y o au, because (3) 11/P ~ i  11/P au n (yi - x)Pdx J =   aun (x)Pdx J  < C 1 N i 7l lp < C28i . espec i ely . Then and whe e Since  añ/p con e ges we conclude ha Simila ly Clea ly supN C (L(n, i)) and sup N C (R(n, i)) a e less han o equal o 2, hence o i = 1, . . . , a, he se R - Ji is equal o UIn , whe e each I n is compac , supC (I n ) < 2 and CONVERGENCE OF THE FOURIER TRANSFORM ON GROUPS  49 Now, o each i = 1, . . . , a, le L(n, i) and R(n, i) (nEN) be he in e als [ - n - 1 - ?7i+yi, -n- li+yi] and [n+i7i+yi,n+1+77i+yi] R - Ji = ( - oo, - 7i + yi) U (?7i + yi, w) C U L(i, n) U U R(i, n), N N cxu ; (yi - x)Pdx < C36pan L(n,i) 1  1 an _ (ni + n)2P -1  (ni + n + 1)2p-1 . 1/P aun (yi - x)Pdx J G C46i . N (ni) ,  l1/P //  au i (yi - x)Pdx J = Cssi . N  J ,(n,i) 1/P aun (yi - x)Pdx J = 0(6i) . N I n Since R = (R -J a ) UJ a , and J a is compac , by (3) and (4) we see ha R = UK n , wi h each K n compac , supC(K n ) < C 6 , and 1/P aua (ya - x)Pdx l = O(6a) . N K n 50  M . L . TORRES DESQUIRE We p o e p ope ies 2 .3) and 2 .4) by induc ion on a . The case a = 1 ollows om (3) . Suppose ha 2 .3) and 2 .4) hold o a - 1 . Tha is, Ra -1 -IIa-1 C_ UI n , each I n a compac subse o Ra-1, supC(In) < C7, and By (4) wi h i = a, we ha e ha R -J a <_ UIj, each Ij a compac subse o R, sup C (Ij) < 2 and Then 1/P Ha-1 a ~ (y¡ - xi)Pdx1  = O(II°=i Sz) N I 11/P a (yo - x)Pdx J  = I n xK  , 1/P [ ,, auo (ya - x)Pdx1 = O(6a)- R' - Ha = (R a-1 x R) - (II(a - 1) x Ja) = (R a-1 - II(a - 1)) x (R U II(a - 1)) x(R - Ja) < U (I n x K m ) U(II(a - 1) x I j) . n,m N The se s I n x K m , and II(a - 1) x I j a e compac subse s o R, o all n, m, j . Hence supC(I, x K  ,) <_ Cs and supC(II(a - 1) x Ij) < C9 . The e o e 2 .4) holds wi h C = max(C8, C9) . Finally, by (5) and (6) we ha e ha 11/P  11/P -  11~ II a_1 a ~ (y2 - x2)Pdx l [L  aUa (ya - x)Pdx ] = I  K  y = O(IIa 6i) We conclude om (3) and (7) ha 1/P a (yo - x)Pdx J = N [ a-IxIj 1/P  1/P = IIa=i  aun (y2 - x) P dx~  a d (ya - x) P dx  = h  N h =0 (IIQ 1S2) . CONVERGENCE OF THE FOURIER TRANSFORM ON GROUPS  51 Lemma 2 .2 . Fo each V y in N y (G)  (yEG) lim u_o o all in (LP, l') (G) (1 < p < oo) . G _ U cpu(y - x) (x)dx = 0 P oo . Le y = (y,, . . . , ya,, so) = (yo, so) be an elemen o Ra x Gl . We choose wo elemen s V and K o No (G) wi h he same o m as in Lemma 2 .2, such ha y + K C V y and V C U . Following he no a ion o Lemma 2 .2, we se 97z = min(S? P , -y2) (i = 1, . . . , a), and WH he in e io o KH . Then he se W = [-77 1 , 77 1 ] x . . . x [ - la, la] x WH sa is ies he p ope ies lis ed in Lemma 2 .2 . Hence by p ope y 2 .1) i is enough o p o e ha lim  Wu(y - x) (x)dx = 0 . U-0 c-w Since G - W y = (R a - Il a ) x G l UH a x (Gl - (so + WH)), we ha e by he de ini ion o he unc ion cpu, ha wu(y - x) = au(yo - ) PU (s0 - s) = 0 i so - s , and x = ( , s) in G . Hence Wu(y - x) (x)dx = J  Wu(y - x) (x)dx G-W y (Ra-IIa) x (so +H) + ax(so+(H-W,» WU(y - x) (x)dx . II Le {I,} be he coun able amily o se s gi en by p ope y 2 .3) . Fo each I n , we ha e by he Hólde inequali y and (1) I,x(so+H) 1 Wu (y - x) (x) ( dx <- Il . XI . x (so+x) I pBU <- <  au(yo - x)P dx 5 2  M . L . TORRES DE SQUIRE By p ope y 2 .4) SUP N ¡S(I" x (so + H))1 < C, whe e C is a cons an , and 1S(I, x (so + H))I is he numbe o K a 's (as de ined in [S]) such ha I, n x (so + H) 1 K a :,A 0 . This implies ha o all n s N II XInx(s,,+H)IIP <- IS(In x (so+H))I II lip . :5 CIi lipoo . Thus, we conclude om 2 .2) ha  ~ou (y - x) 1 (x) 1 dx < (g,a-II a )xGi 11/P , <_ CI i I ¡ .Bu  ~~ au(yo - x)P'dx J - I n Applying Hdlde 's inequali y we ge Lx(80+W-WH» Wu (y - x) I (x) I dx < 1P l < Bu 1l 1 ¡P .I S(Ha x (so +(H- WH)) I [ ila au(yo - x)P dx ] IIa No e ha lla x (s o +(H -WH)) is compac (H is compac and H-WH is closed), and because Bu -> 0  as U -> 0 Now, since lla -> y  as U - 0  and so + (H- WH) <_ so + H  is independen o U, we ha e ha 1 S(IIa x (so +(H- WH) -> 0 as U -~ 0 . The e o e by p ope y 2 .2) (10) J  cp u (y-x)I (x)Idx,0 as U,O . IIa x(so+(H-WH )) The esul ollows om (8), (9), and (10) . Theo em 2 .3 . Fo all in (LP, l')(G), 2 <p< oo, almos e e ywhe e . ló c Wu(y - x) (x)dx = (y) _ O(IIa 1biBU) " P oo .. Le V y be in N y compac . We ha e o show, by Lemma 2 .2, ha ló ~y ~ou (y - x) (x)dx Fu he CONVERGENCE OF THE FOURIER TRANSFORM ON GROUPS  53 con e ges o (x) almos e e ywhe e . Since he unc ion belongs o (Lp, l°°) C_ (L 2 , l °° ), he unc ion g = XV y belongs o L 2 (G), and by Co olla y 1 .2, 1 .1), and 1 .4) each WU also belongs o L 2 (G) . Hence by he Pa se al iden i y, we ha e ha Wu(y - x) (x)dx = Wu(y - x)g(x)dx IV"  G almos anywhe e . = ~bu(x)9(-x)[y,x]dx By he Lebesque Domina ed Con e gen e heo em (see p ope ies 1 .3 and 1 .6) we ha e ha (x)9( - x) [y, ¡]di = 9(-x)[y, x]dx = g(y) 3 . Fou ie Uans o m o Unbounded Measu es The space M 9 (G) (1 <_ p < oo) o unbounded measu es o ype q [S], consis s o Radon measu es M wi h ini e no m (lp , il q gi en by [j : j ¡p¡(Kc i) q ] 1lq . I G = R, hen he amily {K a } can be aken as {[n, n + 1] InEZ} . In his sec ion we gene alize o locally compac abelian g oups, he ollowing heo em due o F . Holland [H] . Theo em 3 .1 . Le 1 < q < 2 anda e M q (R) . Then as N - co 1 N e -¡x d[ ( ) 727 - N con e ges in he no m o (L q ~, l°°) o a unc ion i and h(x)i(x)dx =J h(x)dp(x)  (he(Lq, l l )(R)) . 27 (x) = (C .1) 1 e-ix dp( )