a-priori-estimates for the derivatives of generalized q-holomorphic vectors (crodel)

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A-priori-estimates for the

derivatives of generalized q-holomorphic vectorsAndreas Crodel

a

aMartin-Luther-Universitt Halle, Sektion Mathematik,

Universittsplatz 6, Halle/S., DDR, 4020

Published online: 29 May 2007.

To cite this article:Andreas Crodel (1994): A-priori-estimates for the derivatives ofgeneralized q-holomorphic vectors, Complex Variables, Theory and Application: AnInternational Journal: An International Journal, 25:1, 1-10

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Complex Variables, 1994, Vol. 25, pp. 1-10Reprin ts available directly from the pub lisherPhotocopying permitted by license only1994 Gordon and Breach Science Publishers S.A.Printed in the Unlted Stares of America

A-Priori-Estimates for the Derivatives ofGeneralized q-Holomorphic VectorsANDREAS CRODELMartin-Luther-Universitat Halle, Sektion M athema tik, Universitatsplatz 6 HallelS.,DDR-4020

It is well known, that the derivative of a holomorphic function can be estimates by the sup-norm of thefunction itself:

r. the preser.t paper we investigate the va idi y nf ~i rn ila r stimates for generalized analytic vectors (lnthe sense of B. W Bojarski [I]), i.e. solutions of the elliptical systemw ~q - - e w - fw* Oaz z

For q-holomorphic vectors (i.e. in the case = f ) we will prove estimates in the C-, C a and L p -norms by means of a Cauchy-integral representation. The general case can be reduced to the considera-tion of q-holom orphlc vectors by applying a modified T -opera tor.M No. 30C 35Communicated: W. TutschkeReceived December 10 1987

1. TH INHOMOG N OUS BELTRAMI EQUATIONAt first it is useful to study the inhomogeneous Beltrami-equation

dw wq =dz dzin a bounded domain D e assume q to be a function with compact support insatisfying / q / qo < 1 and belonging both to Wj C) or some p > 2 and to Ca(C)for some E (0 , l ) . From the theory of the (homogeneous) Beltrami-equation it iswell known (cf. [8] that there exists a "generating" homeomorphic solution p tothe (homogeneous) Beltrami-equation with p(z) z E C1@ C)n W;(C). A changeof the independent var iab ~ r , ( I .?) t~ t = p z) yie ds 2 2 i n h ~ ~ o g e n e o u sauchy-Riemann-equation

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2 A CRODELwhich can be solved by applying the T-Operator

where @ is holomorphic function (cf. [8], too). From this we obtain the generalsolution to the original problem as

with

and a function satisfying the homogeneous Beitrami-equation. From this it foliows.that the operator ?;, defined by

Gf [zl = ~,,,,f[io(z)lmaps each function f on a particular solution of (1.1). Moreover, we define

*Df = a,(D)J'[~(z)~(where a denotes the classical T-operator). Using the relation (d/dt)Tf = ~f thederivatives of TDf may be represented as

d - dV - * anddFDf[Z] = -TD f [z] = -TDf [z]dz dz 191

From this, as a consequence of the well known properties both of the classical T-and w-operators and of the generating solution we have that the following mappingsare bounded:

FD LLp(D) Lp(D) for p _ 1 even

(and hence FD Cff@)- CO(D))), (l.2)

* df , d ' ? ~ L,(D) - L, (D) for p > 1, andD, F ~ , d ' f ~Ca(D) Ca(D), if D belongs to the class C2

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q-HOLOMORPHIC VE TORS 3Moreover, if f belongs to w j S o c ~ )L p D ) or some p > 2, then it follows froma similar property of the T-operator that f f possesses second order Sobolev-derivatives: T f E w~,~,(D).2. q-HOLOMORPHIC VECTORSFd owing t h e notation of R W. Boiarski a vector-valued function w(z) (w l,.w , ) ~ s said to b e q-holomorphic, ;f it fulfils the system

where q is a matrix of the quasidiagonal type

satisfying an ellipticity condition (somewhat weaker than in [I]) /qf)j go < I V jWe should remark, that the quasidiagonal matrices of a given structure form acommutative subalgebra of the algebra of n x n-matrices.For considering q-holomorphic vectors it is always useful and (without loss ofgenerality) possible to assume that q consists only of one single diagonal block q(j),and henceforth we will do so.Corresponding to the theory of the Beltrami-equation we can construct a "gener-ating solution" p to the system (2.1), if q has a compact support in 43 and belongsboth to W,'(C) for som e p > 2 and to Ca(C) for some cr E (0, I See also [4]This generating solution can be constructed as follows: Assuming to be a matrixof the sa me structure as q we can rew rite (2.1) as

which are inhomogeneous Beltrami-equations for the cpk (k ) and a homoge-neous one for 91 We choose the generating solution of this homogeneous equationfor cpl and define recursively (cf. [3 41

where D is an arbitrary, bounded C2-domain with suppq D. The following con-sideration yields the correctness of this definition: Suppose that pj i s in c~,"(D)w&~(D) (which is true for cpl) for 1 j < k. Then both the q j and the dp j dz (and




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4 A CRODEL

hence their products) are in ca (D ) f l w ~(D ). T he propert ies of the operators TDa;jb and a* guarantee that yk belongs to C 1sU(D)flw&(D), too. By inductionit is clear, that every v k possesses this property. Since q has compact support themark fEnc?iGn is independent =f D and s it is coEecr y defined. ? {Greover - 7Ysom e considerations at infinity it can be shown that v(z ) z E C1sO(C)n W ~ ( C ) .Du e t o B W Bojarski a Cauchy integral representation can be proved for con-tinuous q-holomorphic vectors (see (2.2) with k 0 .Now we define a differentiai operator D y the relatlon

In a p aper by B. Goldschmidt [5] it is shown that this opera tor maps the space of q-holomorphic functions into itself and may be interpreted there as a n (in som e sense)ordinary derivative. Moreover, for continuous q-holomorphic vectors a Cauchy in-tegral representation of the derivatives V k is valid:

with dqC I dC q d * (cf. [5] oo).Using the estimates

we can prove the following theorems:THEOREM.1 Let D be n bounded domain and w E Lp Z?)-holomorphic. I f K isa compact subset of D , then

with C independent of K , D and wProof Since is locally continu ous in D (cf. [5 ] , we have from the Cauchyintegral representation for z E K and z 6ei with some d < d ( K , D )




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q- H O LO M O RPH I C V ECTO RS

and from the Holder inequality

Integrating this over K and interchanging the order of integrations we have

This inequality is valid for all d K,BD), and, by considering the limit 6K, 3D ), it holds for 6 d K,a D), too. rRemark In the holomorphic case n 1 q 0 D djdz) the constant re-duces tn 1 but C is not sharp in this case).

THEOREM.2 Let K be a compact subset ~ the hounded domain D, and let w bea q-holomorphic vector in D. f w is contimorrr Holder-continuous) in D then

where C does not depend on K , D and wProof Est im~ting oth the Cauchy integrals

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6 A CRODELfor z K and d K , B L ) ) we obtain the inequalit~es

(cf. [2], and for the holomorphic case [ 7 ] ) ,with C1depending only on and y. Forthe estimation of the Hoider constant iet l and 2 2 bc two points u l zl zzjwith lzi 2l < d / 2 at first. From the Cauchy integral representation it follows that

From /Dw(zl)-Dw(zz)l IDw(zl)l + lDw(zz)l and (2.4) an appropriate estimatefollows for lzl 2( 012. Thus

and with the first inequality o f 2.4) the proof is complete. r

3. GENERALIZED q HOLOMORPHIC VECTORSFor our purposes it is useful to modify the T-operator constructed by B. W Bojarski[ I ] so that the derivatives of Tf can be estimated in a relatively simple way. For thisreason we construct a particular solution to the equation

in a bounded domain D by applying the TI-operator recursively to the components


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q-HOLOM ORPHIC VECTORS

and we define 7TD j= w = ~ i... w ~aw awfnf = and ~ ' f p f ,f.d z

Since

the correctness of this definition for f L, D) ( p > 1 ) follows directly from theproperties of the operators FD 8% and d"TD.Moreover, we can estimate the com-ponent functions wk, d w k / a z and a w k / d z * recursively by (1.2). This yields the fol-lowingTHEOREM.1 The operators TD and a h deBned by (3 .2) are iinear andhounded mappings of L, D) into itself. he .same property holds in Ca(D) , f Dbeiongs LO the c i ~ ~ s2 'r~' tichmplies hn is riii~plv m nected). He m e f D mapsL , ( n ) into wif.9 jand nto ~ ''jii;br p i , d = mir? p ) , / p , ~ ) )es,~ectivelvC (E)n f o c ~ , ~ ~ ( D ) .oreover, the junction w = fof is a s o f ~ ~ r i o no 3.1). At thlspoint we ~ h o d dUMLITL ~ o m eifferences between the op e ru m and B. W.Bojurski ;ipD-operator defined by

i) The difSerence PD f ~ ~ ) fs q-lzolomorphic i n D a d ts first component van-ishes i r l~nt ical&.

ii) In general the difference n T D f is not zero in D : Even Jor simpie exampleslike f = f ~ ,2)' = ( 1 ,O),

and hence p(z) =the second com ponent

does not vanish, if p is suitably chosen outside D.iii) In opposite to iT , the furtction f D f is not q-holomo rphic outside D

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8 A CRODELFor the following considerations q may have the full quasidiagonal form again. It

is clear, that in this case operators TD, 8 and d * f D or the whole system can becompounded of the appropriate operators for the diagonal blocks q(j ) .. .Tcinn tho - ~ ~ P V O ~ T O nrrr. - ha l l . . : n .....-.-. -,.&:--& ,... C ..V U L 6 LIIC U ~ L U L V v v b UU ~ J L W Y L I I L L I I I I J W I I I ~ ~ -pl I U I I - L ~ L I I I I ~ L L >U ~ L I I L I ~ I -ized analytic vectors:THEOREM.2 et be a b ~ u d e d omain, D D und w E L, ( D ) ( p > I j geiier-u l i x ~ -hn n .mnyh ic , i.e. a sn l ?ic?.l n ?FR ~ ? h ~ ~ ~ ? G i - ? p ~ t i o . n

dw dwe w = - - q -- e w - f W *d z * 8.2in D. If K is a compact subset of D , then

in D. Since w is a solution to (3.3), = w W is q-holomo rphic in D. Moreover,w e can estimateI I $ I I L , ( D ) I I w I L , ( D ) + IIWIIL,(D)

< [ I + l f ~ 1 l L ~ ~ 6 ~ ( l l e l l i ~ ~ f i )llf l l ~ ~ ~ ~ ~ ~ l l l ~ l lFrom w = W J we obtain

which yields with 8$/dz = pz D $ and Theorem 2.1

and, since d( K , d D ) d i a m ( ~/2 ,a wI I

An appropriate theorem is valid for Co-solutions of (3.3) under some restrictionsto D and the coefficients, and it can be proved in the same way as Theorem 3.2.In order to obtain bounds for W = pi,(ew f w * ) and for d W /d z we have to c la im

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q HOLOMORPHIC VECTORS 9

e f E P ( D ) and D EC2. Furthermore, we have to continue w outside D in the firststep of the proof. This can be done by applying the construction described in [6] toeach component of the real and imaginary parts of w:

and for the norm of the continuation we obtain //wl a 5 f i l \ w l ~ ~ , ( ~ ~indepen-5dent of D). Then we haveTHEOREM.3 Let D be a bounded domain belonging to C2,D nd w E c"(D)solution to (3.3). If K is a compact subset oj D and e f t ~ " ( b, ihen

with C and C independent of w , K and D.In particular cases we can prove even estimates in the C-norm by factorisation.

More exactly, we formulateTHEOREM.4 Let D be a bounded c2-domain D c D and w E C(D) a solution to

where s a quasidiagc~ulmatrix of he same structure like q) belonging to C P I ) )for some j3 E (0,l). If K is a compact subset of P then

with C and C independent of K, D and w.Proof At first we set H exp(fhe), where the exp-function is defined as usualby its power series (and hence it possesses all the well-known properties if its argu-ment is quasidiagonal). Now we set H - l w exp(-f e)w, and from (3.5) and

eH it fol ows th2t is q-holomorphic with

For the derivative dw / /d7we have

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10 A CRODEL

where r ? f b j e E c ~ 5 ) .ence: by Theorem 2.2.

Thc estimate for dw/dz can be shown in a quite similar way.References[ I ] R. W Rojarski: Theory of generalized analytic vectors, Ann. Polon. Math. 17 (1966). 28-3 20 fin

Russian).[2] A . C rodel, Nichtlineare Evolutionsgleichungen fur q-ho iomorp he Ve'ktoren, Z. Anai. Anwendungen(iYu?j, 49-59.131 R. P. Gilbert, Constructive metho ds for elliptic equations, Lecture notes in mathematics 365 Springer-Vcrlag, Bcrlin-Heidelberg-PI~GY ~ r k :'???, ch. .

R F, Cilbc;t 2nd 3 L. B-ch-nen, ".is: c;;ce e ii7::c sj?;;rm; cz-cc t h e ~ r p t i r pprfiar'., .A.rl-demic Press, NCWYore, 1Y t t j .

[5j F.. Ga dschrr?idt. T;'.nkt:~nenthr,or~tischcEige nschafren vera ocmeinc?rter ana v iqcher Vek nrcn,Annth .,--l n rh r.,. Q (19791, 57-90,

[ h ] E. J McShane, Exten sion of range of functions, Bulletin AMS 40 (1934). 8374342.[ 7 ] W Tutke W. Tutschke), A problem with initial values for generalized analytic functions depending

on tim e (generalizations of the Cauchy-Kovalevski and Holmgren th eorem s), Soviet Math. Dokl. 251 (198 2), 201-205.81 I. N. Vekua, Verallgemeinerte analytische Funktio nen, Akademie-Verlag, Berlin, 1963.

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a-priori-estimates for the derivatives of generalized q-holomorphic vectors (crodel)

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