Dilations associated to flat curves

Publicacions Ma emá iques, Vol 35 (1991), 251-257 . and DILATIONS ASSOCIATED TO FLAT CURVES STEPIIEN WAINGER I would like o gi e an exposi ion o ecen wo k o Tony Ca be y, Mike Ch is , Jim Vance, Da id Wa son, and mysel conce ning Hilbe ans o ms and Maximal unc ions along cu es in R 2 [CCVWW] . Thus we le ( ) _ ( , y( )) be a cu e in R 2 wi h y(0) = y'(0) = 0 . Fo a unc ion in Có (R 2 ), we se H (x) = J 1 (x - ( ))~ , i h M (x) =  s upi 1  I (x - ( ))I d . We a e in e es ed in he p oblem o ob aining es ima es o he o m 1)  IIH 1IL . < A(p, )II JIL, and 2)  IIM lIL , < A(p, )jI JIL, Posi i e esul s ha e been known o a long ime o 1) and 2) unde an app op ia e cu a u e hypo hesis . The cu a u e hypo hesis is ha y(k)(0) 7~ 0 o some k > 2 . Thus i 3)  ( ) = ( , k)  k > 2, k a posi i e in ege , o 4)  ( ) = ( , k - k+ 1 )  k > 2, ka posi i e in ege , he cu a u e condi ion is sa is ied . I 5)  ( ) he cu a u e condi ion is no sa is ied . 25 2  s . WAINGER An impo an p ope y o cu es o which he cu a u e hypo hesis holds is ha i is almos homogeneous . We say F( ) is homogeneous i he e exis s a g oup o linea ans o ma ions A(A), de ined o A > 0, such ha , 6)  (A ) = A(A) ( ), A(Al ) = A(A)A(p), and 8)  A(A)x --~ 0 as A -+ 0 o e e y x Thus F( ) = ( , k ) is homogeneous wi h A(A) _  A  0l - (0 ~k /' and P( ) = ( , k - k + 1 ) is almos homogeneous in ha In 3) and 4) abo e he dila ions A(A) a e s a ing one in he ace . The ques ion we a e in e es ed in is whe he o no he e could be a use ul amily o dila ions o cu es in which he cu a u e condi ion ails, a cu e like In discussing his ques ion, i is well o keep in mind ano he example o a homogeneous cu e, namely 9)  F( ) = ( , log 11D . He e I'(A ) = A(A)F( ) + small e o . F( ) = ( , e _l / " ) . A(a) a o = (AlogA A) . Thus i we wish o ha e a heo y ha includes he example 9), we would no wan o ake _ 1 0 A(a) -  0  Y(A) in gene al . Be o e desc ibing he amily o dila ions ha we ound, le us y o decide wha we migh hope o p o e using hem . Le us assume ha y( ) is odd and con ex o > 0 . Then mos known esul s a e exp essible in e ms o a unc ional h( ), h( ) = y'( ) - y( ) . Geome ically h( ) ep esen s he dis ance om he o igin o he y-in e cep o he line angen o I' a I'( ) . We hen ha e he ollowing Theo em : DILATIONS ASSOCIATED TO FLAT CURVES  253 Theo em A . Assume y( ) is odd and con ex o > 0, hen IIH JIL2 < Ali JILZ i and only i 10)  h(c ) > 2h( ) o all > 0 o some c > 0 . Also i o some c > 0 and all > 0, See [NVWW1] and [NVWW2] . h(c ) > 2h( ) IIM ilLI < CII lIL2 . Fu he i is known ha 10) does no su ice o 1) o 2) o all p, 1 < p < oo . See [CCNWW] and [C1] . Using he amily o dila ions we cons uc we a e able o p o e he ollowing : Theo em B . Assume y( ) is odd and con ex o > 0 .  Then i o some e>0, 11)  h( ) > e h ) , IIH JIL , <_ A(p, )II JILI, 1 <p < 00, and ~IM JIL, :5 A(p, )II JILp,1 <p < oo, (Oihe su cien condi ions a e gi en in [CCCDRVWW] and [CW] .) We may iew 11) as an in ini esinal e sion o 10) . In pa icula 11) implies 10) . I ums ou ha he amily o dila ions we use a e B(A) - (^/(A)  h(A)) No e ha in he case o he cu e F( ) = ( , log I l) B(A) and A(A) a e he same . In he case ha F( ) = ( , k ) 254  S . WAINGER A(A) and B(A) a e no he same, bu hey a e equi alen in he sense ha i K is an open con ex se con aining he o igin in i s in e io , he e is ano he such se K 1 such ha A(A)K 1 C B(A)K and isa e sa . The dila ions B(A) do no o m a g oup, ha is condi ion 7) is iola ed . Howe e , one can dispense wi h 7) i one p o es 12)  JIB(s) -1 B( )jj < C(s)E o some posi i e e . I u ns ou ha 10) implies 12) . The dila ions a e used in he p oo o Theo em B in 2 ways . The i s applica ion is o ob ain uni o my decay es ima es o measu es suppo ed on I'( ), and he second is o be able o de elop a Calde on-Zygmund heo y . The Calde on-Zygmund heo y gi es ise o a Li lewood Paley Theo y om which Theo em B can be de i ed using ideas de eloped in [DR] and [NSW] . The Calde on Zygmund heo y is de eloped by using a a ian o he a gumen o Coi man and Weiss conce ning spaces o homogeneous ype [CW] . He e we shall conce a e on he decay o he Fou ie asn o m o measu es suppo ed on I' . We e e he eade o [CCVWW] o he Calde on Zygmund heo y, Li lewood Paley a gumen s and he es o he p oo o Theo em B . We de ine measu es dp  suppo ed on I' as ollows : Fo a es unc ion So 2 2 - ^ dl 2 n(O) = 2n  '~  0( ( )) d . - _  2 2 - ^ dU"(~) = 2'  d . 2-^ We wish o say in a p ecise way ha dp,,(~) decays as 1 ends o oo "uni o mly" in n . (No e ha he condi ion 10) does no imply ha -> 0as n --> oo because F( ) can con ain s aigh line segmen s while he condi ion 11) a leas insu es ha F( ) is ei he pa o he x-axis o has no s aigh line segmen s . The uni o m es ima e we ob ain is (i 11) holds and ]P( ) is no a segmen o he x-axis 13)  Idl2"(« < clIB*(2`)J11` o some e > 0 . The use o he dila ion is ha we don' ha e o make a close examina ion o he cu a u e o I' in he in e al (2-n,2-2-"), bu ins ead we can "no malize" o he in e al (1, 2) . Le us be mo e p ecise . Thus o p o e 13) i su ices o show 14)  11 2e 4n' n( ) d 1 < Cli,711-1~2 whe e 11 is a ec o in R 2 and n( ) = B-1(2-n)1'(2-n ) . A calcula ion shows ha I . ( ) = ( , Y . ( » whe e y n( ) has he ollowing p ope ies : 15)  yn( ) is con ex So DILATIONS ASSOCIATED TO FLAT CURVES  255 dun(1) = 2n 2 .2-n es . ( ) d - J 2 e'£' (2-n ) d 2-n  1 2 _  e¡£ .(B(2-n) .B_1(2-n) (2-n )) d 1 2 e¡B'(2-n)1 .B-1(2-n) (2-" ) d . 1 ?'n( 1 ) = 1  yn( 1 ) = 0 17)  h' (s) > e hn(s) s (He e hn (s) = s-/n(s) - -Yn(s)), and - Yn(S) >  -Y , (S) 17) ollows immedia ely om 11) while 16), 17), and he con exi y o - ,, imply 18) . One can hen p o e 14) by adap ing he ideas o Van De Co pu , see [SW] . To see 18) no e ha yn( ) = ?'n(s)d s < ( - 1)y ;,( )_ 1 2 yn( ) C hn( ) C hn( ) -  yn( ) 25 6  S . WAINGER Thus o 1 < < 2 7n ( )  47n( )' Finally 14) is ob ained om Van De Co pu 's Lemmas, Lemma 4 .3 . o [CCVWW] . I 17721 >_ 111,1, we use he "second de i a i e es ima e" oge he wi h 18) and 16) . I 17711 > 17721 we use he " i s de i a i e es ima e" o 's such ha 171217n( ) < 2` and he "second de i a i e es ima e" oge he wi h 18) o 's such ha 1 7 12 17n( ) >  17711 . We ema k ha he ope a o s conside ed abo e a e special cases o mo e gen- e al ope a o s whe e x is in R n, x - 1'( ) is eplaced by a k-pa ame e su ace S(x, ), ( ER') wi h S(x, 0) = x, and 1/ is eplaced by a Calde on Zygmund ke nel on R' . In his mo e gene al se ing posi i e esul s ha e been ob ained in [C2] and [CNSW] p o ided S(x, ) sa is ies an app op ia e cu a u e con- di ion . We would hope ha a be e unde s anding o he ope a o s H and M abo e wi hou he assump ion o he cu a u e condi ion would e en ually ca y o e o he mo e gene al ype o ope a o wi hou assuming he cu a u e condi ion . Le us ema k ha he a gumen s in [CNSW] s ongly depend on amilies o dila ions . Re e ences [CCVWW] A . CARBERY, M . CHRIST, J . VANCE, S . WAINGER AND D . WATSON, Ope a o s associa ed o ia plane cu es, Duke Ma h . J . ( o appea ) . [CCCDRVWW] H . CARLSSON, M . CHRIST, A . CORDOBA, J . DUOANDI- KOETXEA, J .L . RUBIO DE FRANCIA, J . VANCE, S . WAINGER AND D . WEINBERG, Lp es ima es o maximal unc ions and Hilbe ans o ms along la con ex cu es in R 2 , Bull . Ame . Ma h . Soe . 1 4 (1986), 263-267 . [CW] H . CARLSSON AND S . WAINGER, Maximal unc ions ela ed o con ex polygonal lines, Indiana Uni . Ma h . J . 34 (1985), 815-823 . [C1] M . CHRIST ( o appea ) . [C2] M . CHRIST, p ep in 1985 . [CNSW] M . CHRIST, A . NAGEL . E .M . STEI N AND S . WAINGER ( o appea ) . [CW] R . COIFMAN AND G . WEISS, "Analyse ha monique non-commu a i e su ce ains espaces homogenes," Lec u e No es in Ma hema ics 242, Sp inge -Ve lag, New Yo k, 1971 . [DR] J . DUOANDIKOETXEA AND J.L . RUBIO DE FRANCIA, Maximal and singula in eg al ope a o s ia Fou ie ans o m es ima es, In en . Ma h . 84 (1986), 541-561 . DILATIONS ASSOCIATED TO FLAT CURVES  257 [NSW] A . NAGEL . E .M . STEI N AND S . WAINGER, Di e en ia ion in lacu- na y di ec ions, P oc . Na l . Acad . Se¡ . 75, USA (1978), 1060-1062 . [NVWW1] A . NAGEL, J . VANCE . S . WAINGER AND D . WEINBERG, Hilbe ans o ms o con ex cu es, DukeMa h . J . 50 (1983), 735-744 . [NVWW2] A . NAGEL, J . VANCE . S . WAINGER AND D . WEINBERG, Maxi- mal unc ions o con ex cu es, Duke Ma h . J . 52 (1985), 715-722 . [SW] E .M . STEIN AND S . WAINGER, P oblems in ha monic analysis ela ed o cu a u e, Bull . Ame . Ma h . Soc . 84 (1978), 1239-1295 . Depa men o Ma hema ics Uni e si y o Wisconsin-Madison Madison, WI 53706 U .S .A .