Pointwise smoothness, two-microlocalization and wavelet coefficients

Publicacions Ma emá iques, Vol 35 (1991), 155-168 . POINTWISE SMOOTHNESS, TWO-MICROLOCALIZATION AND WAVELET COEFFICIENTS S . JAFFARD In his pape , we shall compa e h ee no ions o poin wise smoo hness : he usual de ini ion, J .M . Bony's wo-mic olocal spaces Cxós,, and he co espond- ing de ini ion on he wa ele coe icien s . The pu pose is mainly o show ha hese wo-mic olocal spaces p o ide "good subs i u es" o he poin wise Hdlde egula i y condi ion ; hey can be e y p ecisely compa ed wi h his condi ion, hey ha e mo e unc ional p ope ies, and can be cha ac e ized by condi ions on he wa ele coe icien s . We also gi e applica ions o hese p ope ies . In Pa 2 some esul s on he mic olocal spaces con ained in [B2] will be ecalled . The- o ems 3 and 4 a e also essen ially con ained in [B2] . The s a ing poin o his pape was a no e ([J1]) he au ho had w i en on a compa ison be ween he Hdlde c i e ion o egula i y a a gi en poin xoand a co esponding p ope y de ined on he wa ele coe iicien s . Some easy p oc s a e omi ed o ab idged and can be ound in [J2] . 1 . Poin wise smoo hnessand wo-mic olocaliza ion Le s be a s ic ly posi i e eal numbe . Le us ecall he usual de ini ion o he Hdlde c i e ion a xo . A unc ion belongs o C" i h ee exis s a polynomial P(x) o deg ee equal o he in eg al pa o s such ha (x) = P(x) + O(Ix - xo js) . The ollowing p ope ies a e classical . I belongs o C2 o , no hing is implied on he de i a i es o . In dimension 1, he p imi i e o belongs o C'+' . In dimension la ge han 1, he ac ional in eg a ion o o de 1 maps C', in o C9+1 . x o The wo-mic olocaliza ion o J .M . Bony consis s in eplacing he p eceding no ion by ano he one which allows o de i a e and in eg a e . The classical pseudo-di e en ial ope a o s will ope a e on hese spaces, which will be spaces o empe a e dis ibu ions . These spaces a e de ined by condi ions on he Li lewood-Paley decomposi- ion . Le us ecall i s de ini ion . Le 0 be a unc ion in he Schwa z class such ha B(1) = 1 i 111 < 1/2 and B(1) = 0 i 111 > 1 . 156  S . JAFFARD Le hen Le Sj be he "low-pass il e ", which, a e a Fou ie ans o m, is a mul ipli- ca ion by 0(2 - j j) . De ine Aj = Sj+1 -S j . Thus The Fou ie ans o m o Oj(u) is suppo ed by he se The wo-mic olocal spaces can now be de ined . De ini ion 1 . Le s and s' be wo eal numbe s ; C"' is he Banach space o dis ibu ions such ha (2)  ISo(u)(x)I <_ C(1 + 1x)-9/ and 0(1) = 0(1/ 2 ) - B(1) . I=So+Do+Di + . . . 2'-1 <_ 111 < 2j+1 . láj(u)(x)j :5 C2 - j 9 (1 + j2jxj)-9' . The space Cxo 9 is hen ob ained h ough a simple ansla ion . I s' = 0, he space hus ob ained is he global HSlde space C9(Rn) . The e ec o s' is o accen ua e ei he he ole played by x o , when s' < 0, o he beha io a in ini y, when s' > 0 . A ew o he ema ks will gi e a be e unde s anding o hese condi ions . De ine uj = Oj(u) and Uj(x) = uj(2 - jx) . Then, he Fou ie ans o m o Uj is a dis ibu ion ca ied by he se 1/2 < 111 < 2, and (3) implies ha ( 4 )  l Uj(u)(x)I <C2 - j'(1 + Ix1)" . Such an es ima e is s able unde de i a ion and ac ional in eg a ion, mo e gene ally unde he ac ion o he ope a o s (I - 0) 9 / 2 o (-0) .9 / 2 , s E R, because hese ope a o s a e Fou ie mul iplie s which, once es ic ed o 1/2 <_ 111 <_ 2, coincide wi h a unc ion o he Schwa z class . Coming back o he "space a iable", we ge (-0)9~2Uj = K 9 * Uj whe e K s belongs o he Schwa z class . This con olu ion ope a o p ese es he polynomial inc ease o decay, as i appea s in (4) . So ha he ollowing equi alence holds (5)  u E C','  áu ~  E C' -1 " ' o 1 < j < n . zo  09x j  xo  -  - In he ollowing pa , we shall in es iga e he na u e o he elemen s o Cxó", whe he hey a e (e en ually smoo h) unc ions o dis ibu ions . We claim ha he elemen s o Ció ' a e (in gene al) dis ibu ions . To sho en he p oo , suppose n = 1, x o = 0 and 0 < s < 1 . De ine whe e he Fou ie ans o m o ~ belongs o he Schwa z class, anishes ou side [-1/2,1/2], and is equal o 1 on [-1/4,1/4] . So ha , o any N > 1, De ine hen Then Le j 0 be such ha hen I (x) - (0 )I < By de ini ion, Hence POINTWISE SMOOTHNESS AND WAVELET COEFFICIENTS  157 2 . The elemen so Ció" 8(x) = 0 * IxI', B(x) = IX],, + Q(IXI-N) . (x) = 1 : 2-i80(2ix)e'2' z = 1 : uj(x) . 0  0 uj(x) = 2 - J 9 u(2 1 x) and Iu(x)I < C(1 + IxI)', so ha belongs o Có' - ' . The es ic ion o o any in e al ]b/2, 8[, S > 0, is a dis ibu ion, because, i 5/2 < x < S, 00 (x) = IX I' 1 : e d21x + O(1) . 0 We claim ha he elemen s o Cs,~", when s' < -s, a e "hones unc ions" . In o de o p o e i , we shall suppose ha 0 < s < 1, x0 = 0, and ob ain ha I (x) - (0)I < Cixi' when 0 < IxI < 1 . 2 - (jo+1) < IxI < 2-jo I SOAX) - SO (0)I +  1 :  I u j( x ) - uj(0)I + E Iuj(x)I + E Iuj(o)I . 0<j<jo  j>ju j>jo Iuj(x)I <C2 - j"(1 +2jix1)" . 1 V uj(x)I < C2j(' - s)(1 + 2j1xI)-9, . So ha , i 0 < j < j0, Iuj(x)-uj(0)I :5 Cl 2 ' ( ' -J) IxI 158  S . JAFFARD and he o al con ibu ion o hese e ms is C2j(1 -9 ), which is equi alen o CIxl9 . The se ie E Iuj(x)I is bounded by C E 2-j9(2jjxj)-", which is equi alen 7>70  7>70 o C2 -1 and he same es ima e holds o E Iuj(0)I . Since, by Be ns ein's 7>7o inequali y, I S o (x) - S o (0) I < C o Ix1, he esul is p o ed . One can also easily check ha , i s > 0 and s' + s > 0, hen CxoC" ,- " isinclu de dinC x o . In he nex pa , we shall examine he egula i y o he elemen s o Cxó s a x o . and Chis esul is op imal . 3 . A compa ison be ween Ciá s and Cx o Le s > 0, we saw ha he elemen s o Cxó 9, e en es ic ed o R' - {xo} a e in gene al "wild dis ibu ions" o which (1) canno hold . Though, we shall p o e he ollowing esul . Theo em 1 . Le s and l be s ic ly posi i e numbe s, and u an elemen o Cid ' n CO(R n ) . The e exisis a polynomial P o deg ee less han s such ha , i IX - XOI < 1, (6)  Iu(x)-P(x)I :5 CIx-x0I 9 1og Ix 2 xol ' Rema k ha , in his heo em, we a e looking o egula poin s in an i egula backg ound, which is mo e sub le han he usual app oach ha consis s in inding i egula poin s in a C°° o analy ical backg ound (de e mina ion o he singula suppo s) . This heo em can be in e p e ed as a aube ian heo em . We ha e in o - ma ion on he beha io o a e ages o ( i s Li lewood-Paley decomposi ion) and a aube ian condi ion o minimal global egula i y, which allow o ob ain a poin wise esul . P oo o Theo em 1 : De ine j o and j l by 2-)0-1 < I x - xo I < 2 - io and Ji = s jo . Q Le us es ic o he case 0 < s < 1 and 0 < 0 < s . Then P(x) = u(0), and 7o u(x) - UMI 5 ISOU(x) -SOu(o)I + E Iu7(x) - Uj(0)I+ 0 00 E(IUj(x)I + IU ;(o)I) + 57(I Uj(x)I + IUj(0)I) = A+ B + C + D . 7o  h POINTWISE SMOOTHNESS AND WAVELET COEFFICIENTS  159 To es ima e A is a s aigh o wa d consequence o Be ns ein's inequali y . As conce ns B, we use u E C¿8'-"1 so ha ; whe e j j(x)j _< c(1 + jxj) - . The Fou ie ans o m o Nj anishes ou side he se 1/2 < 111 < 2, so ha hence ui(x) = 2-19 ij(21x) l pj(x) - j( 0)1 _< cixi i IxI < 1 ; l uj(x) - uj(0)j < c2jixi2-j9 . Adding up hese inequali ies, we ge ei he B < cixi 9 i s < 1, o , i s = 1, B < cixijo < c'ixllog I2I,  - x In o de o es ima e C, ema k ha luj(x)j < c( 2-j9 + jxj9), so ha C is a mos O(jxj 9 (jl - jo)) = O(Ixi' log 2 I ) . Because u is in CQ, 11 uj jj,,,,< c2 - jO, so ha D is a mos O(2 - j"Q) = 00x0, which ends he p oo . The case s > 1 is le o he eade . One easily checks ha , i (1) holds, hen u belongs o Czó s .  So ha , i s+s' > 0, Cio9 , C Ci o C Cx~ s . The necessi y o make he global CO asump ion and he op imali y o he loga i hmic e m in he esul ha e been p o ed by Y es Meye (pe sonal com- munica ion), using wa ele s and will be gi en in he nex sec ion . 4 . Wa ele coe icien s and Ció" spaces One o he in e es ing p ope ies o he space CxO' is ha i can be cha ac- e ized by condi ions on he wa ele coe icien s . The in ui i e eason o ha is because he wa ele coe icien s o a dis ibu ion a e gi en by a sampling ( ollowing Shannon's ule) on he il e ing gi en by he Li lewood-Paley de- composi ion . I is he e o e na u al ha spaces de ined by local condi ions on hei Li lewood-Paley decomposi ion can be hus cha ac e ized . We assume in he ollowing ha he o hono mal basis o wa ele s used has enough egula i y and decay . We use he usual no a ions 1 , j,k(x) = 2 nil2 0(2 j x - k), j E Z, kE Z n . Then, he ollowing heo em is e y easy o check . 16 0  S . JAFFARD Theo em 2 . A dis ibu ion u belongs o CxoC -" ,- " i ando nl yi 1 < u ,Oj,k > 1 < C2-(n/2+s)j(1 + Ik _ 2jxa1)-9' . The o he wo-mic olocal spaces can also be cha ac e ized by condi ions on he wa ele coe icien s . Recall ha wi h 1 :Icj 12 < oo . Then, u belongs o H9,9, i u EH9,9,  2 '9( 1 + 2j ix1) " uj JILI< cj L .2's(1 + 2i i~ -x01)29/¡Cj,k12 < oo . We now gi e he coun e -examples ha show he op imali y o Theo em 1 . Assume ha 0 is a compac ly suppo ed wa ele , as cons uc ed in [D] . One easily checks ha i is possible o suppose wl h Choose hen E ;  such ha 0(0) 7~ 0 . We i s p o e ha he global CQ asump ion is needed in Theo em 1 . Le m be a posi i e in ege and e,,, a eal numbe such ha 2'e  1 is an in ege , and E n -> 0 when m -+ oo . The p ecise alue o Em will be gi en la e . Le a be such ha 0 <a< 1 . The wa ele coe icien s o he coun e -example a e de ined by : i 2- < j < 2-+1 and k = 6,n27, Cj,k = 2-j/2Emi else, C j , k = 0 . Then de ine 00 x  = 1 : . (x) m=0 n(x) = Em  1 :  0( 2j ( x - cm.» . 2-<j<2-+ 1 The suppo s o he m a e disjoin , ¡ m(x)I < C2'em, and (0) = 0 . Then is con inuous (because 2 m e" 1 -+ 0) . Bu m(em) = C2'em, so ha Then POINTWISE SMOOTHNESS AND WAVELET COEFFICIENTS  161 limsup I (x)- (x°)I > limsupC2'E' - =+ooVy > 0 . xy - Hence is no in Có o any alue o y, al hough condi ion (3) holds a 0 . The ollowing coun e -example shows ha he loga i hmic e m is needed in (6) . Take he same cons uc ion as be o e, bu wi h E,,,, = 2 - Q 2m o a gi en /i > 0 . I (E-)- ( 0 )I >C2m > C'logjemi . m Hence he op imali y o he loga i hmic e m . I should be no iced ha o he condi ions simila o condi ion (7) can be in oduced in o de o be compa ed wi h o he ypes o poin wise egula i y condi ions . Fo example, a compa ison wi h poin wise di e en iabili y is gi en by he ollowing p oposi ion, he p oo o which is simila o he one o Theo em 1 . P oposi ion 1 . Leí be a unc ion di e en iable a xo wi h wa ele coe - cien s cj,k . Leí A be he poin (k2 - j, 2 - j) i a he uppe hal -plane . Then, he ollowing es ima e holds whe e 77(A) < 1 and l(A) = o(1) whenA ends o (xo, 0) . Con e sely, i is in C , (R") o a si icily posi i e ,l and i he e exis s a posi i e unc ion 0 de ined o posi i e alues o 1 such ha E B(j) < oo and hen is di e en iable a x o . Icj,kl :5 C~l(A)2-(2+')j(1 + Ik - 2'xol) Icj,kl :5 Cil(A)B(j)2 -cg + '> j(1 + Ik - 2'xo1), 5 . Pseudo-di e en ial ope a o s and wo-mic olocaliza ion We shall now s udy he ac ion o gene alized pseudo-di e en ial ope a o s on he spaces Ciós , . The ope a o s T ha we shall conside will belong o he algeb as Op(M7) (c . [DJ],[L] and [M2]) de ined by condi ions on hei dis ibu ion-ke nel K(x, y) as ollows . 16 2  S . JAFFARD De ine 0 7 o be he class o ope a o s such ha , o he main diagonal, hei dis ibu ion-ke nel K is a unc ion sa is ying he ollowing es ima es : o any in ege a such ha ce < y, I a is he in ege such ha y - 1 <a< y, C la°K(x,y)¡-< Ix-yln+« . 'Clx - x'1 7 -« l a«K(x, y) - a « K(x, y)l <  (x -yln+y i lx-x'I -< Ix 2 yI , lá'K(x,y)-a°K(x,Y,)¡<CI l x_yl+7a i Iy-y'I < Ix 2 y¡ , and he ope a o T is such ha T(X") =T * (X a) = 0 o a less han o equal o ^ y . The algeb a Op(M- 1 ) is hen he union o all he O^ o y' > y . The usual pseudo-di e en ial ope a o s o o de 0 a e he sum o such an ope a o and o a egula izing ope a o . Y es Meye p o ed ha he ollowing ca ac e iza ion holds (c . [M2]) . P oposi ion 2 . An ope a o T belongs o Op(M 7 ) i i s "wa ele coe - cien s" de ined by c(A, A') =< TOa10a , > saiis y ¡he ollowing condi ion : he e exisis y' > y, such ha Ie(A, A')I < w (A, A') wi h and w(A A') = C2-h-j'I(2+7)(  2-i + 2-j/  )n+-y' '  2-~ -l- 2 - j' + la - A'I A = (k2-i,2-) ) . We shall now p o e he ollowing esul . Theo em 3 . I belongs o Cio9, and T belongs o Op(M^~) wi h hen T( ) belongs o Cx Ó " . y> sup(IS + S' I, .s, IS'I, - n - S), I we keep Theo em 1 in mind, his heo em can be in e p e ed as ollows . The posi ion o he poin s o egula i y o a unc ion is essen ially p ese ed unde he ac ion o singula in eg al ope a o s such as he Hilbe ans o m . POINTWISE SMOOTHNESS AND WAVELET COEFFICIENTS  163 P oo o Theo em 3 : Le B(,1) = 2-(2'+9)j(1 + 2j IA - xoI)" . We mus p o e ha No ice ha , o any s' ,(A, A')B(, ) < CB(A') . a (8)  (1+2 j IA-xol) -9 ' <(1+2jIA'-xoI) - s , (1+2jIA- We spli he sum Ew(A,A')B(A) in o wo pa s . a) I j < j', hen and an w(A,A')9(A) < C  2-(j'-j)(Z+- ) 2-(2 +9)x( 1 +2jIA-xoI) -9a  a (1+2jIA- < C  2-(j'-j)(Z+- ) 2- (z+9)j( 1 + 2,IA' - x,1)-s' b y (8) -  (1+2jIa-~'I)n+7-~9'~ a < C2 - (z+9)j'  2-(j'-~)(7-9)  ( 1 + 2jIA' - xoI)-9, -  (1 + 2j IA a We in oduce now he wo ollowing subcases . i)I s'<0 hen (1+2 1. Ia'-xoI) -9 ' <-(1+2j'IA'-xoI)" ~w(a, a')B(a) < e(a') ~  2-cj'-jx7 -9) a  a  (1 + Ik - 2'a'Un+,-19'1 <C9(A')i -y>sand~y-Is'j >0 . ii) I s' > 0 hen (1 + 2j IA' - xoI) --" = 2(j'-j)9'(2(j'-j) + 2j' IAl - xoI)-9' < 2(j'-D,9'(1 + 2j' IA' - xo I) -9 d ~ w(a' a )B(A) < B(a) ~  2- (1 + 2j IA - a' I)-+7-191 <CB(,')i y>s+s'andy-Is'I>0 . b) I j > j', hen