A SIMPLE NASH-MOSER IMPLICIT FUNCTION THEOREM IN WEIGHTED BANACH SPACES. Sanghyun Cho

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1 A SIMPLE NASH-MOSER IMPLICIT FUNCTION THEOREM IN WEIGHTED BANACH SPACES Sanghyun Cho Abtract. We prove a implified verion of the Nah-Moer implicit function theorem in weighted Banach pace. We relax the condition o that the linearized equation ha an approximate invere in different weighted Banach pace in each recurrence tep. During the lat everal decade Nah-Moer implicit function theorem helped to reolve everal difficult problem of olvability for nonlinear problem epecially nonlinear partial differential equation [611]. Uually nonlinear partial differential equation or nonlinear problem in general) can be tranformed into olving the problem : φu) = where φ involve the variable x the unnown function ux) and it derivative up to the order m. To prove implicit function theorem in infinite dimenional pace a pace of function uually are) we firt linearize the equation and then olve the linear equation o that we get the recurive olution with appropriate recurrence etimate. The implet one i nown a Picard iterative cheme. However when φ involve the derivative of u up to order m the Picard cheme can not be convergent unle the linearized equation get m derivative a doe elliptic equation). To overcome thi difficulty Nah[11] and Moer [1] propoed another cheme involving moothing operator o that the olution of the linearized equation could be etimated inductively in each Sobolev pace of order. Later Hörmander propoed improved cheme [678] to get optimal reult with repect to the regularity of the olution. However thee cheme are too complicated and rather frightening for the uninitiated reader. In [12] Saint Raymond etablihed a implified verion of C exitence theorem o that the number of derivative that are ued provided that it i finite) doe not matter and thi method i ueful when we are woring on C category [123]. Raymond ued the cheme propoed by Moer [1] which conit alternately in uing Newton cheme and Nah moothing operator cloer and cloer to the identity : v = ψu )φu ) u +1 = u + S v 2 Mathematic Subject Claification. 47J7. Key word and phrae. Nah-Moer Banach pace Sobolev pace. )Partially upported by R from KOSEF 1 Typeet by AMS-TEX

2 2 S. CHO and proved the convergence in C category. Here S i a moothing operator and ψu) i the right invere of φ/ u)u) that i 1) φ u)ψu) = I a an operator I=identity). Alo we need an etimate for v = ψu)φu) o called tame etimate ; 2) v C φu) +d + u +d φu) d ) in Hamilton [4]. In ome cae however we have to deal with the cae that a linearized equation ha right invere with error term of econd order c.f.[123] or in each recurrence tep we have to olve a linearized equation with different weight in each weighted Sobolev pace. For example when we try to prove an embedding problem of a Cauchy-Riemann tructure the approximate) linearized equation become an inhomogeneou Cauchy-Riemann equation on compact peudoconvex almot complex manifold cloe to being integrable). In thi cae we can not get an elliptic regularity for the olution up to the boundary. Therefore we have to ue weighted etimate for [59] with different weight in each Sobolev pace. We then ue thee weighted etimate for a in 5) below) in each recurrence tep in the proce of Nah-Moer iteration. In thi paper we prove the Nah-Moer implicit function theorem in weighted Banach pace. We relax the condition in 1) and 2) a mentioned above. That i in each -th recurrence tep φu) + φ u)v) = ha a olution with an error that depend on φu) ) )1+ε for ε > where d i a poitive integer and can be etimated a 3) v ) C t E) φu) ) +d + u +d φu) ) d ) where {t } i an increaing equence of poitive integer and the norm ) or ) ) defined in each recurrence pace are Sobolev norm of order with weight e t λ where λ i a C function and E) i an integer depending on. We tate the main theorem a follow. Theorem 5.1. Let {t } be a trictly increaing equence of poitive integer and uppoe that B ɛ and B ɛ Z < ɛ R are familie of Banach pace with the following propertie. i) For each fixed and ɛ B ɛ B ɛ t and B ɛ B ɛ t if > t. ii) For each fixed ɛ > and for each fixed B ɛ 1 B ɛ 2 and B ɛ 1 B ɛ 2 iii) If ) ɛ and ) ɛ denote the norm on B ɛ ) ɛ ) tɛ and B ɛ and ) ɛ ) tɛ if > t. if 2 1. repectively then iv) For each fixed ɛ > and et B ɛ exit an open et U ɛ B ɛ with = > Bɛ and B ɛ = > Bɛ. Then there U ɛ = {u B ɛ ; u u ɛ ) ɛ < δ}

3 NASH-MOSER IMPLICIT FUNCTION THEOREM 3 where d i a poitive integer u ɛ B ɛ and δ i a given poitive number. Furthermore there i a C 2 -map φ ɛ : U ɛ B ɛ uch that if u U ɛ and v w B ɛ then φ ɛ u) ) ɛ C ɛ Ed) 1 + u ) +dɛ ) d φ ɛu)v) ) ɛ C 1ɛ Ed) v ) ɛ φ ɛ u)v w) ) ɛ C 2ɛ Ed) v ) ɛ w ) ɛ Here E d) and Ed) are polynomial in and d. v) There i a poitive number a > with the following propertie : for each fixed ɛ > and for each and for all u U ɛ there exit a linear operator ψ ɛ u) : Bɛ uch that B ɛ +T ad)+ 1+a 5.1) φ ɛ u) φ ɛu)ψu)φ ɛ ɛ u) ) ɛ C 1 φ ɛ u) ɛ) ) and v ɛ := ψɛ u)φ ɛu) atifie for each d + T a d) + the following etimate o called tame etimate ) : 5.2) v ɛ ) ɛ C ɛ E1) t E 2) φ ɛ u) ) +dɛ + u ) +dɛ φ ɛu) ) dɛ ) where E 1 ) E 2 ) are polynomial in C i a contant and T a d) i an integer for example the mallet integer bigger than or equal to d+1) a + 3). vi) For each ɛ > and there are moothing operator S θ : = Bɛ B ɛ and S θ : = Bɛ B ɛ for all θ > 1 uch that for each real number t there are a contant C t and integer E 1 t) and E 2 t) uch that 5.3) S θ v ) ɛ C t ɛ E1t) t E 2t) θ t v ) tɛ t v S θ v ) ɛ C t ɛ E1t) t E 2t) θ t v ) ɛt t and the imilar etimate hold for S θ. Then there exit an integer D and a mall number b > uch that if φ ɛ u ) ) D < b for ome ɛ > and for ome u ɛ U ɛ then there exit an element u U ɛ uch that φ ɛ u) =. Remar 5.2. a). In many cae we approximate the non-linear problem up to econd order error term. Hence a = 1 in thee cae. b). If λ i a mooth bounded function we can modify λ o that 1 λ 1 + a 4. Let B B be the weighted Sobolev pace of order on a bounded domain Ω C n with weighted norm 5.4) f ) ) 2 = α then 5.3) hold with θ = e. Ω D α f 2 e t λ dv f B

4 4 S. CHO Theorem 1. Let {t } be a trictly increaing equence of poitive integer and uppoe that B and B are familie of Banach pace with the following propertie. i) For each fixed B Bt and B Bt if > t. ii) If ) and ) denote the norm on B and B repectively then ) ) t and ) ) t if > t. iii) For each fixed if B = > B et U B with and B = > B then there exit an open U = {u B ; u u ) < δ} where d i a poitive integer u B and δ i a given poitive number. Furthermore there i a C 2 -map φ : U B uch that if u U and v w B then φu) ) φ u)v) ) φ u)v w) ) C 1 + u ) +d ) C 1 v ) C 2 v ) w ) d when one deal with nonlinear) partial differential equation of order m thee etimate claically hold for d > m + n/2). iv) There i a poitive number ε > with the following propertie : for each and for all u U there exit a linear operator ψ u) : B B+T ɛd)+ uch that 4) φu) φ u)ψ u)φu) ) C 1 φu) ) )1+ε and v := ψ u)φu) atifie for each d + T ɛ d) + the following etimate o called tame etimate ) : 5) v ) C t E) φu) ) +d + u ) +d φu) ) d ) where E) i a polynomial in C i a contant and T ɛ d) i an integer for example the mallet integer bigger than or equal to d+1) ɛ + 3). v) There i θ > 1 uch that 6) θ 1+ ɛ 4 )t t +1 ) ) +1) θ t t +1 ) θ 1+ ɛ 4 )t t +1 ) ) +1) θ t t +1 ) where ɛ > i the number in iv) atifying 4) and hence B B +1 B B +1 for each.

5 NASH-MOSER IMPLICIT FUNCTION THEOREM 5 vi) For each there are moothing operator S θ : = B B and S θ : = B B for all θ > 1 uch that for each real number t there are a contant C t and an integer E t) uch that S θ v ) v S θ v ) C t t Et) θ t v ) t t C t t Et) θ t v ) t t and the imilar etimate hold for S θ. Then there exit an integer B and a mall number b > uch that if φu ) ) B < b for ome u U then there exit an element u U uch that φu) =. Remar 2. a). Since φu) ) 1 we may aume that < ɛ 1. b). In many cae we approximate the non-linear problem up to econd order error term and hence ɛ = 1 in thee cae. c). If λ i a mooth bounded function we can modify λ o that 1 λ 1 + ɛ 4. Let B be the weighted Sobolev pace of order on a bounded domain Ω C n with weighted norm : f ) ) 2 = D α f 2 e tλ dv f B. α Ω Then 6) hold with θ = e. By chooing a ubequence if neceary we may aume that the equence {t } atifie 7) t t and t i ufficiently large. Alo we will ue a equence of real number {θ } defined inductively a follow : 8) θ = θ t 1 and will ue the correponding moothing operator S θ. In the equel we et τ = 1 + 2ɛ/3) > 1 and τ = ) ɛ = and hence 1 < τ 5 < τ 4 < τ 3 < τ 2 < τ 1 < τ. For a convenience we et 9) T := T ɛ d) := [[ d + 1) + 3)]] ɛ where [[Γ] denote the mallet integer bigger than or equal to Γ. We firt prove the following Lemma which i a crucial tep in the proof of Theorem 1.

6 6 S. CHO Lemma 3. With the ame aumption a in the theorem and with the moothing operator S θ of the remar the equence v = ψ u )φu ) u +1 = u + S θ+1 v are well defined if φu ) ) θ 2t for ufficiently large t ; more preciely there exit contant U t ) t d and V independent of ) uch that for i) ii) iii) u u ) v ) +3 V θ τ 3 < δ and φu ) ) θ τ 1 + u +1 +1) + ) U θ +1/ u ) + ) d + T +. Proof. Since the property i) implie that the equence u and v are well defined it i ufficient to prove i) ii) and iii) inductively. The property i) i true by aumption. Proof of ii) +1. The tame etimate 5) give for every d + T + that 1) v ) C t E) For = d and uing i) and 1) we have 11) v ) d C d t Ed) ) φu ) ) +d + u ) +d φu ) ). ) 1 + u u ) + u ) φu ) ) V θ τ 1 becaue t Ed) θ τ 1 τ i bounded. Let T be the number defined in 9) and et N = 4+1). From the tame etimate 5) and the propertie of φu ) ) tated in iii) of Theorem 1 it follow for d + T + that ) v ) C t E) φu ) ) +d + u ) +d φu ) ) d ) 12) C t E) C +d 1 + u ) + ) + C d1 + u ) ) u ) +d ) C t E) C +d + C 1 + δ + u ) ) 1 + u ) + ). The etimate 13) 1 + u j j) T + ) 1 + u ) 1 T + )θn 4 j hold obviouly for j =. Moreover if it hold for ome j < we obtain from 7) 13) and iii) j that 1 + u j+1 j+1) T + ) U T θ +1/2 j u j j) T + ) U T θ 1/4 j u ) T + )θn j θ j+1 U T θ 1/4 j u ) T + )θn 1 4 j+1

7 NASH-MOSER IMPLICIT FUNCTION THEOREM 7 becaue θj N θ 2N/3 j+1 and + 1) + 2N/3 < N. Therefore by induction for j that 13) hold provided t and hence t 1 ) i ufficiently large o that θ t 1/4 U T 1. Than to 13) we may write 12) a : 14) v ) T C T t E) C T +d + C 1 + δ + u ) ) ) 1 + u ) T + )θn 1 4 V 1 θ N becaue t E) θ 1/4 i bounded for d + T. Combining 9) 11) 14) and the propertie vi) in Theorem 1 of the moothing operator the interpolation formula with θ = θ τ 1 τ 2 +3) can be written a 15) v ) +3 S θ v ) +3 + v S θ v ) +3 C +3d t E+3d) θ +3 v ) C +3d V t E+3d)+Ed) θ τ 2 V θ τ 3 d + C +3T t E+3T ) + C +3T V 1 t E+3T ) θ +3 T v ) T θ 4+2)+N becaue t E+3T ) θ 4+2)+N+τ 3 ii) +1. Proof of iii) Now we want to etimate u +1 +1) + u + S θ+1 v it follow for d + T that u +1 +1) + u +1) + Thu one obtain from 13) that 1 + u +1 +1) + and t E+3d)+Ed) θ τ 2+τ 3 are bounded. Thi prove + S θ +1 v +1) + in term of v ). Since u +1 = u ) + + C +t E+) W t E+)+E) θ+11 + u ) + ) U θ +1/ u ) + ) θ +1 v ). becaue t E+)+E) θ d + T + i bounded. Thi prove iii) with contant U doe not depend on. Proof of i) +1. Since u u = j< S θ j+1 v j t [ 1] one can write from 6) and ii) that u + ts θ+1 v u +1) j S θj+1 v j +1) C θ t t +1 t E) j C V θ t +1/3 t E) C V Sθ t +1/4 j v j j) θ τ 3 j

8 8 S. CHO where S = j θ τ 3 j < i a contant. By chooing t and hence t 1 ) ufficiently large o that C V Sθ t 1/4 < δ we have u +1 u +1) = j<+1 and thi i the firt part of i) +1. By virtue of Taylor formula we can write ; where φu +1 ) = φu ) + φ u )S θ+1 v + 1 S θj+1 v j +1) < δ 1 t)φ u + ts θ+1 v )S θ+1 v S θ+1 v )dt = φ 1 + φ 2 + φ 3 φ 1 = φu ) + φ u )v φ 2 = φ u )S θ+1 v v ) φ 3 = 1 1 t)φ u + ts θ+1 v )S θ+1 v S θ+1 v )dt. Firt we etimate φ 1. For thi we ue the following two etimate ; 16) φu ) ) θ 1+2ε/3) = θ τ and φu ) ) T +d AθN. Note that the econd inequality come from the propertie iii) in Theorem 1 and 14) with = T +d. From 4) we have φ 1 +1) = φu )+φ u )v ) +1) C 1 φu ) +1) ) 1+ε. Setting θ = θ 2ε it follow from 16) that φu ) ) = S θ φu ) ) + φu ) S θ φu ) ) C t E) θ φu d ) ) + C T +dt = C t E) θ 1 = C t E) Hence it follow from 6) that φ 1 +1) C 1 φu ) +1) ) 1+ε + C T +d t + C T +d t ET +d) ET +d) Aθ 1 ET +d) = C 1 θ 1+ε)t t +1 ) φu ) ) )1+ε C 1 C t E) C 1 C t E) 1 3 θ 1+2ε/3) +1 + C T +d t + C T +d t ET +d) ET +d) A)θ 1. θ T d φu ) ) T +d ) 1+ε 1+ε)t A θ t +1 ) θ 1+ε) ) 1+ε t θ +1 )1+ε) A

9 NASH-MOSER IMPLICIT FUNCTION THEOREM 9 by chooing t ufficiently large. Next let u etimate φ 2. From 6) 15) and the property iii) of Theorem 1 we obtain that φ 2 +1) θ t t +1 C 1 S θ+1 v ) C 1 C +3 t E+3) θ 3 +1 v ) +3 θt t +1 C 1 C +3 V t E+3) θ 3 +1 θ τ 3 θ t t θ 1+2ε/3) +1 provided t i ufficiently large. Finally we etimate φ 3. By chooing t ufficiently large we have from 15) that φ 3 +1) S θ+1 v +1) ) 2 θ 2t t +1 ) S θ+1 v ) ) 2 C t E) v ) 2t θ t +1 ) ) 2 C V t E) 2t θ t +1 ) θ 2τ θ 1+2ε/3) +1. )2 If we combine the etimate of φ 1 φ 2 and φ 3 the econd part of i) +1 follow. We define a contant M = τ 4+N+1 τ 3 τ 4 = 15 ɛ 6 15ɛ ) + 1) and for each integer et 17) Λ) = + T M)d 1. M + 1 The proof of the property ii) of Lemma 3 can be modified to prove an etimate for v ) for every d. Lemma 4. There exit contant V ) d uch that the equence {v } of Lemma 3 atifie for every and d Λ) that v ) where D = [[ d) τ 3 τ 4 ) τ 4 + N + 1) + ]]. V t maxed)ed)) θ τ 4 Proof. Keeping the value N = 4 + 1) we obtain from 7) and iii) of Lemma 3 that 1 + u +1 +1) + )θ N +1 U θ +1/ u ) + )θ N +1 U θ 1/ u ) + )θ N U θ 1/ u ) + )θ N

10 1 S. CHO for d + T +. Therefore for each fixed the equence 1 + u ) bounded and hence there exit a contant K uch that 1 + u ) + )θ N K. Subtituting thi into 13) we obtain for d + T + that + )θ N 18) v ) C C +d + C 1 + δ + u ) ))K t E) θ N = W t E) θ N W θ N+1 where N = 4 + 1) doe not depend on. Now if d Λ) the definition of Λ) in 17) how that D + T +. Therefore for d Λ) we rewrite our interpolation formula with θ = θ τ 3 τ 4 ) d) ; v ) S θ v ) C d t Ed) + v S θ v ) θ d v ) d + C D t ED) C d t Ed) V θ τ 4 + C D t ED) W D θ τ 4 V t maxed)ed)) θ τ 4 where we have ued the etimate 18). θ D v ) D Proof of the Theorem 1 Let u and v be a in Lemma 3. From lemma 4 we have for any j and d Λj) that S θj+1 v j j) C t E) j v j j) C V t E)+maxEd)ED)) j θ τ 4 j A θ τ 5 j. By 6) one thu obtain for d Λj) that 19) S θj+1 v j ) θ 1+ɛ/4)t j t ) S θj+1 v j j) C A θ 1+ɛ/4)t j θ τ 5 j = C A θ ɛ/12 j becaue 1 + ɛ/4) τ 5 = ɛ/12. By virtue of 17) we alo have 2) Λj) < if and only if j < Λ 1 ) = M + 1) 2 + M)d T + 1. Now for each fixed the equence u = u + j< S θ j+1 v j i convergent in B becaue S θj+1 v j ) S θj+1 v j ) + S θj+1 v j ) < j<λ 1 ) j Λ 1 ) where the firt um in the right i finite um by 2) and the econd um in the right i finite by 19). Moreover the limit u B of the equence u atifie φu) ) φu ) ) + 1 φu ) ) + C 1 u u ) and by 6) one obtain that φ u + tu u ))u u )dt ) φu ) ) θ1+ɛ/4)t t ) φu ) ) θ 5ɛ/12 for all. Therefore by taing limit for = it follow that φu) = and thi prove Theorem 1 with B = and b = θ 2t. i

11 NASH-MOSER IMPLICIT FUNCTION THEOREM 11 Reference 1. Catlin D. Sufficient condition for the extenion of CR tructure J. of Geom. Anal ) Catlin D. and Cho S. Extenion of CR tructure on three dimenional compact peudoconvex CR manifold preprint). 3. Cho S. Embedding of peudoconvex CR manifold with Levi-form one degenerate eigenvalue Pacific J. of Math. to appear). 4. Hamilton R. The invere function theorem of Nah-Moer Bull. of Amer. Math. Soc ) Hörmander L. L 2 etimate and exitence theorem for the operator Acta Math ) The boundary problem of phyical geodey Arch. Rat. Mech. Anal ) Implicit function theorem Lecture at Standford Univerity Summer Quarter 1977). 8. On the Nah-Moer implicit function theorem Annale Acad. Sci. Fenniae Serie A.I. Math ) J. J. Kohn Global regularity for on wealy peudo-convex manifold Tran. Amer. Math. Soc ) Moer J. A new technique for the contruction of olution of nonlinear differential equation Proc. Nat. Acad. Sci ) Nah J. The imbedding problem for Riemannian manifold Ann. of Math ) Raymond S. X. A imple Nah-Moer implicit function theorem L Eneignement Mathematique t ) Department of Mathematic Sogang Univerity Seoul Korea addre: ):hcho@ogang.ac.r