The article focuses on the hardy space of solutions of the generalized Beltrami system. The Beltrami equation is considered the first-order system on the plane wherein its eigenvalues lie in the upper half-plane with a continuous coefficient. Moreover, the matrix function is a solution of the generalized Beltrami system wherein any differentiable supported function specifies the solution of Beltrami equationyesBelgorod State Universit

Soldatov, A. P.

Vashchenko, O. V.

DSpace at Belgorod State University

T he H ardy Space o f Solutions  o f th e  G eneralized B eltram i SystemO. V . V ashchenko and A . P. SoldatovBelgorod State University, Belgorod, RussiaWe consider the first-order systemon the plane, where J  £ <Clxl is a constant m atrix  whose eigenvalues lie in the upper half-plane Imz/ >  0. In the scalar case I = 1, Eq. (1) w ith an (in general, continuous) coefficient J(z) ,  Im  J  > 0 ,  is referred to  as the Beltram i equation [1, p. 72].The m atrix  functionE(z)  = — z 7 1 (2)V 7 2vr i j  V 7[here and throughout the  following, we use the m atrix  notation Zj = x x l  + y x J  for z  =(x + yi) G C] is a fundam ental solution of the generalized Beltram i system (1). In other words, forany continuously differentiable compactly supported function F(z) ,  the  integral(TF) (z )  = ^ ~  j \ t -  z ) j 1F ( t ) d t1dt2 (3)cspecifies a classical solution of Eq. (1).Indeed, T F  is a continuously differentiable function, and its derivatives are given by the formulasd (T F ) 1 i  -1 d F , , , , d(TF) 1 f _ < d F ,  , , ,=  J  t:’ t e  ' +  t )d t ' dU-  —  = 2 7 i J t ^ i!: + t ) d t l d h ■ (4)dxConsider the two-dimensional singular integral(SF)( z )  = — [  t j 2F ( z  +  t )d t1dt2 (5)2 m  Jctreated  as the limit of integrals over {|i| >  e} as e —>■ 0. Since[  t j 2dst = 0, (6)1*1 = 1it follows th a t the necessary condition for the existence of such integrals is satisfied. To verify relation (6), it is most convenient to  use the functionVASHCHENKO, SOLDATOVwhich is analytic in the upper half-plane Imz/ >  0. On the one hand, the integral on the left-hand side in relation (6) coincides w ith the value x ( J )  of this function on the m atrix  J . On the other hand, the function % of the m atrix  and all of its derivatives vanish at the point v  =  i, whence we have %(J) =  0. Let us rewrite the integral (3) as the limit of integrals over {|i| >  e} as e —> 0. Then, in the usual way, we obtain the relations= (S F ) ( z ) +  a .F i z ) ,  ^ p  =  J ( S F ) ( z )  +  a2F{z) ,  (7)where the coefficients a k e C lxl are given by the formulasGk = 2 \ i  /  t J lrikdSt’ k = 1>2>\t\ = lhere n  =  (n1, n 2) stands for the unit inward normal of the cycle |i| =  1, so th a t27T 27r<Ti = --------- / (cos 9 +  J  sin 9)~ l cos 9d9,  a 2 = --------- /  (cos 9 +  J s in # ) -1 s in9 d9.2 tti J  2 tti J0 0One can readily see th a t27T<t2 — J o \  =  — [ ( — sin 9 +  J  cos 6)(cos 9 +  J  sin 0)~1d0 = 1. (8)2 m  JoIndeed, the left-hand side is the value on J  of the function27T, , 1 f — sin 9 +  v  cos 9 inXo{v) = 7) 7 /  ------ 7T,---------- ~d9,27xi J  cos 9 + vs \n  90which is analytic in the half-plane I m u  > 0. Simple com putations show th a t Xo(*) =  1, k = 1 , 2 , ,  and hence Xo(^) =  1-By combining (7) w ith (8), we find th a t the function T F  indeed satisfies Eq. (1).The singular operator (5) belongs to  the Calderon-Zygm und type. By [2, p. 52 of the Russian translation], it is bounded in the space L p{C), p > 1; moreover, if F  e  L p, then the integral exists for almost all z. This, together w ith (7) and (8), implies the following result.Theorem 1. Let a domain D  C C lie in a finite part of  the plane, and let F  e  L P(D), p > 1. Then the integral (3) defines a funct ion T F  that lies in the Sobolev space W 1'P(D) and whose derivatives are given by (7). Moreover,|T F |tvi.p(D) <  C \F\LP{yD'), (9)where C  > 0 is a constant depending only on p and D.Proof. The estim ates\ {TF)x \Lp +  \ {TF)y\LP <  C \F\lpform the contents of the Calderon-Zygm und theorem  for the singular operator S.  The estim ate \TF\Lp(D) < C \F \Lp(D) for the integral (3) can readily be derived from the Holder inequality. These estim ates imply (9).DIFFERENTIAL EQUATIONS Vol. 43 No. 4 2007THE HARDY SPACE OF SOLUTIONS OF THE GENERALIZED BELTRAMI SYSTEMLet D  be the  dom ain bounded by a piecewise sm ooth contour T. Consider a sequence of contours r „  C D, n  = 1 , 2 , . . . ,  approxim ating T in the following sense: for each n, there exists a piecewise continuous differentiable homeomorphic m apping a n : T —> T„ such th a tIa n(t) -  t |c(r) +  IOLn -  l | c(r) —► 0 as n  —► oo. (10)In particular, if the contour T„ bounds the dom ain D n, then  every compact set K  C D  lies in all D n for sufficiently large n.Let 0  G W ^ ( D ) - ,  i.e., th is function belongs to  the class W 1,p in each dom ain D n, n  = 1 , 2 , . . .  Then, by the embedding theorem  [3], the estim ate|0Up(r„) <  Cn\(f)\w1’P(Dn) (11)is valid for each n.We introduce the following notion. A function (f> e  W ^ ( D )  belongs to  the H ardy class H P(D)if(12)sup|(/>|LP(rri) <  oo. (13)nNote th a tW hp(D)  C H P(D).  (14)Indeed, in this case, the norm  on the right-hand side in the estim ate (11) can be com puted in W 1,P( D ); therefore, it is only necessary to  show th a t the constants Cn occurring in this estim ate are uniformly bounded. By using condition (10), one can readily obtain this fact from the proof of the  embedding W 1,p C L P(T) (e.g., see [3, p. 420]).It follows from (14) and Theorem  1 th a t T  is a bounded operator in the spaces L P( D ) —> H j ( D ) and each function <fi e  H p can be represented in the form<f> =  T F  +  0o, (15)where F  e  L P(D ) and 0 O satisfies the homogeneous equation (1). Solutions (f) of this homogeneous equation (1) were dubbed Douglis analytic functions or, briefly, J-analy tic  functions in [4].An example of such functions is given by the Cauchy type integral(Itp)(z) = —  f ( t -  z ) j 1dtJtp(t), z e D ,  (16)iri Jrwhere d t j  stands for the m atrix  differential dt± +  J  dt2, t  = ti  +  i t2 £ T, and, to  be definite, we assume th a t the contour T has the positive sense w ith respect to  D  (i.e., the dom ain D  lies to the  left when moving along it).By [5], the integral (16) w ith a vector function ip e  L P(T) defines a function lip e  H P(D).  More precisely, the  following assertion is valid.T h e o re m  2. (a) The operator I  is a bounded operator in the spaces L P( D ) —> H P(D).(b) Let <p G L P(T), p > 1. Then for  the integral, there exist angular limit values (f)+ (to) for  almost all t 0 £  T, and the Sokhotskii-Plemelj formula4>+ (to) =  p> (to)  +  —  [  ( t  -  t o ) ^ 1 d t j ( p ( t )  iri Jris valid, where the integral on the right-hand side is a singular integral treated in the sense of the Cauchy principal value. In  addition, \4>+\Lp <  C\p \LP.In particular, it follows from Theorem  2 th a t H P(D ) equipped with the norm101 =  I0v -  +  suPl0Unr„)  (17)nis a Banach space. By using Theorem  2, one can readily give the following equivalent description of the space H p.T h eorem  3. In the class of  smooth functions in D, the norm (17) is equivalent to the norm101 =  \4>V ~  +  \4>\lp{T)- (18)P roof. By virtue of the expansion (15), it suffices to  verify the equivalence of the norms (17) and (18) for J-analy tic  functions.By C +0 (D)  we denote the class of Holder continuous functions. If (f) is a J-analy tic  function in C +0 ( D ) , then, by applying Theorem  2 to  the Cauchy formula2<f>(z) = ~  f ( t ~  z ) j 1dtJ(f)(t), iri J  rwe obtain the estim ate \<P\h p ( d ) <  C\<fi\LP(r)- Conversely, let (f) e  H P(D)  n C +0 (D) .  Then, by (10), the  sequence a n uniformly converges to  (f) on T, whence we obtain \4>\lp{t) <  Ci\<P\hp(d)- This implies the desired equivalence of the norms.Theorem  3, together with its proof, implies th a t the space H P(D)  can be obtained as the closure of the class C 1^ 0 (D) in the norm (18). In particular, to  each function <fi e  H p, one can assign its limit value (f)+ e  L P(T).The expansion (16) can be complemented as follows.T h eorem  4. Let D  be the domain bounded by a simple piecewise Lyapunov contour without return points, and let J  be a triangular matrix. Then each funct ion (f) e  H P(D) can be uniquely represented in the form<p = TF  +  lip +  *£, (19)where £ e K 1, F = <py — J<px , and (p e  L P(T) is a vector funct ion taking real values.P roof. By replacing 0 by <f> — T<f>, w ithout loss of generality, one can assume th a t F  = 0. In this case, it suffices to  use the  results in [4, 6].ACKNOW LEDGM ENTSThe work was financially supported by the Program  “Universities of Russia” (project no. UR04.01.486).REFEREN CES1. Vekua, I.N., Obobshchennye analiticheskie funktsm (Generalized Analytic Functions), Moscow: Nauka, 1988.2. Stein, I., Singular Integrals and Differentiability Properties of Functions, Princeton: Princeton Univ. Press, 1970. Translated under the title Singulyarnye integraly i differentsial’nye svoistva funktsii, Moscow: Mir, 1973.3. Nikol’skii, S.M., Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya (Approximation of Functions of Several Variables and Embedding Theorems), Moscow: Nauka, 1969.4. Soldatov, A.P., Izv. Akad. Nauk Ser. Mat., 1991, vol. 55, no. 5, pp. 1070-1100.5. Soldatov, A.P. and Aleksandrov, A.V., Differ. Uravn., 1991, vol. 27, no. 1, pp. 3-8.6. Soldatov, A.P., Izv. Akad. Nauk Ser. Mat., 1992, vol. 56, no. 3, pp. 566-604.VASHCHENKO, SOLDATOV

The Hardy space of solutions of the generalized Beltrami system

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The Hardy space of solutions of the generalized Beltrami system

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