The space indicated in the title is introduced and studiedyesBelgorod State Universit

Soldatov, A. P.

DSpace at Belgorod State University

The Hardy Space of Solutions to First-Order Elliptic SystemsA. P. Soldatov“ *On a domain D c  C  in the plane, consider the first- order elliptic system^ - A ^  = 0dy ox ( 1 )with coefficient A e  C /:;/. Ellipticity means that the matrix A  has no real eigenvalues. It is well known that the solutions U = (Uu U2, . . Ut) to this system are real analytic on the domain D.Suppose that the boundary r  = dD of the domain is a piecewise Lyapunov contour; i.e., its connected com­ponents are homeomorphic to the circle and can be rep­resented as unions of finitely many Lyapunov arcs, which can intersect only in endpoints. Recall that an arc L  is Lyapunov if  it admits a smooth parameterization of class C1' M for some 0 < |i < 1. Take a sequence of con­tours T„ c  D, where « = 1 ,2 , . . . ,  converging to T  in the sense that there exists a homeomorphism a n: V —> r„with piecewise continuous derivative a„ such that a n(t) - 1 —> 0 as n —> <x> in the sup-norm and the deriva­tives a n are uniformly bounded.The Hardy space Hp(D), where 1 < p < °°, of solu­tions to elliptic systems is defined by the condition that the norm\U\ = sup I U\ LP(Tn) (2)must be finite. The domain D can be finite or infinite; in the latter case, the solutions u to system (1) are assumed to be bounded at infinity. The I f - norm of a function U(i) on T„ is related to the number function \U(I)\, where |-| denotes some fixed norm on W. The space №can also be defined for p = 1, but we do not consider this case.For analytic functions, this definition generalizes the classical Hardy class on the unit disk and is due to V.I. Smirnov, M.A. Lavrent’ev, and M.V. Keldysh (see [1]). This class is also denoted by EP{D).Of special interest are systems of the form (1) for whichT^?. = 0— J  W  idy ax (3)where the spectrum a(J) of the matrix J  e  C /x/ is con­tained in the upper half-plane. System (3) is a natural generalization of the Cauchy-Riemann equation; it was introduced by Douglis in [2]. The basic properties of analytic functions are inherent in the solutions to this system [3]; for this reason, they are called hyperana- lytic functions, or functions analytic in the sense of Douglis (or, briefly, /-analytic functions).In the general case, the solutions of system (1) can be uniquely expressed in terms of those of (3). Indeed, if an invertible matrix B  reduces A to a block-diagonalform J  = diag(./h J 2) (i.e., B 'AB = J) and the eigenval­ues of Jk are contained in the upper half-plane, then wehave the representation U = B(j), where (|) = (c^, (j>2), of a solution to system (1) in terms of the / r  analytic func­tions §k. This allows us to consider only system (3) in what follows.If a /-analytic function (|) is continuous on a closed simple domain D = D u  T, then the following Cauchy formula is valid [3]:2 (Kz) = d t j ^ ( t ) ,  tn i Je D, (4)a Belgorod State University, ul. Pobedy 85,Belgorod, 308015 Russia b Research Institute of Applied Mathematics and Automation, Kabardino-Balkar Research Center, Russian Academy of Sciences, Nalchik, Russia e-mail: soldatov@bsu.edu.ruwhere (x, + Lx2), = x, 1 + x2J  for Xj e R. the expression MX, + dx2)j = MX,) 1 + (dx2).f has similar meaning for the matrix differential, and the contour r  = dl) has positive orientation with respect to D. Replacing the boundary value (|)+ by an arbitrary density cp, we obtain a general­ized Cauchy-type integral, which we denote by (/(p)(z).THE HARDY SPACE OF SOLUTIONS TO FIRST-ORDER ELLIPTIC SYSTEMSThe definition of Hardy spaces can be extended to the weight case in a natural way. Let F  be a finite set of points 1 6  F. For a family X = (X-, x e F) of real num­bers (of weight order), we denote the class of continu-Xous functions on r \ F  with asymptotics 0 (l)\t-% \  as t —> x by Q jT , F). A scalar function p e Cx is said to be weight (of order X) if it does not vanish and p 1 e C , . We indicate the dependence on X by a subscript: p = p, .In a similar way, weight functions on the domain D with respect to the corresponding class C% = C,( I ) , F) are defined. In this case, of main interest are /-analytic (I x Z)-matrix weight functions R(z) = Rx(z) satisfying system (3) and subject to the additional constraint R(z)J = JR(z) for z e I). This constraint means that the operation § R§ of multiplication by a weight func­tion does not lead out from the class of /-analytic vector functions ((). Weight functions RJz) mentioned above always exist. For example, if the domain I) is finite and T is a simple contour, then we can setRx(z) = ] ^ [ ( z - x ) ^ .IE FHere, in the notation of (4), the multiplier (z, -  x)" = exp[aln(z -  i)/1 is understood as a function of a matrix for the branch of the logarithm ln(z -  x) continuous on I).Let Lpx ( r ,  F) be the Banach space on the boundaryr  = })D such that L{ = LP for X = and the weight transformation (p —> p v(p implements an isomorphism between LP on Lp _ 1 /p. Note that the class C, + 0( r ,  F) =l J C l  + o is densely contained in Lpx ( r ,  F).£> 0In a similar way, we define a space Lpx (I), F) on the2domain I) by the condition L{ = LP with k  = —  and aPfamily of weight Hardy spaces H- (I), F) by the condi­tion H i = HP with X = and the weight transforma­tion (() —> Rv(j). Obviously, for k  = 0. the space L[\ on Y and on D is an LP-space with respect to the measures d\\. = p i(t)ds, where / e V, and d\\. = p_2(z)dxdy, where z = x  + iy e D, respectively.The following theorem gives all basic properties of the Hardy spaces.Theorem 1. (a) Let (|) e H px (D) be a solution to sys­tem (1). Then, there exist almost everywhere angular limit values (|)+(0, which determine a function (|)+ eZ ^(r). The space Hj- (D) is Banach, the norm |(()|0 ' is equivalent to (2), and there are continuous dense embeddings Cx+0( D , F) c  H p% (I), F ) c i J  (I), F).(b) I f§  e H p% (D), then the restriction of§ to each sub- domain D(, c  I) bounded by a piecewise Lyapunov con­tour belongs to H pk (D0). Conversely, if  such subdomainsDj, D2, are disjoint, D\ u  D 2 u  ... u  Dm = D ,and (() is a solution to system (I) on the domain Dbelonging to H p% (Dj)for j  = 1 ,2 ,. . . ,  m, then (|) e Hp{D)\ moreover, the norm on H {(D ) is equivalent to |(()| =j(c) The Cauchy-type integral k p determines a bounded linear operator L{ ( r ,  F) —> H i  (D , F) fo r  -1  < X < 0, and, fo r  almost all t0 e T, the Sokhotskii- Plemelj formula( / t p ) +( r 0) =  9 ( ^ o )  +  ^ J ( ^ - ^ o ) / 1^ / 9 ( 0rholds. Here, the integral is singular; it is understood in the sense o f the Cauchy principal value.The classes C%+0 and L{ (D) in assertion (b) of The­orem 1 refer to /-analytic functions. Assertions (a) and (b) imply, in particular, that the space H{ does not depend on the choice of the sequence T„ in (2) and can be defined as the completion of the class C/ + 0 >n thenorm |(()| = <|)+| , . This scheme was implemented in [4]for usual analytic functions. The above spaces for ana­lytic functions were studied in [5] under more general assumptions about the boundary of the domain D and the weight. Theorem 1 (b) makes it possible to naturallyextend the definition of H{ (D) to domains D  with any piecewise Lyapunov boundary. The proofs of asser­tions (a) and (b) of Theorem 1 are based on assertion (c), which is proved in precisely the same way as in [6].The analogue of N.I. Muskhelishvili’s theorem about a representation of (|) by the integral Rp with a real vector function (p, which was proved in [3] for the Holder classes, remains valid for the Hardy classes.Recall that a bounded operator N: X —> Y between Banach spaces X  and Y is said to be Fredholm if its ker­nel kerA/ is finite-dimensional, the image Im A' is closed, and there exists a finite-dimensional subspace F0 c 7  such that Y = Y0 © ImA' [7]. It is convenient to call this subspace the coimage of the operator N  and denote it by coImA'. although it is not uniquely deter­mined by N. The difference indA' between the dimen­sions of the kernel and the coimage is called the index of the operator N.SOLDATOVTheorem 2. (a) I f  the domain D is bounded by a Lyapunov contour r ,  then the operator I: L{ (T, F) —>H j- (D , F) (-1 < X< 0) is Fredholm, and its index equals l(s -  2), where s is the number o f connected components o f the contour. Moreover, k e r/ c  C+0(T) = CM (r),H> 0and there exists a coimage co lm / c  C+0(Z)).(b) Suppose that the matrix J  is triangular, the domain D is bounded by a piecewise Lyapunov contour T without cusp points, and the set F  contains all corner points o f the contour.Then, the operator I: LPK (T, F) —> H PK (D , F) ( -^  <X<  0) is Fredholm, and its index equals 1(2 -  s). More­over, the elements o f the kernel are constant on the con­nected components o fT , and the operator has a coim­age whose elements are constant on D.More precisely, the kernel and the coimage men­tioned in assertion (b) are described as follows. If the domain D is infinite, then ker/ consists of all locally constant functions, and we can set colm / = 0 .  If the domain D is finite, then ker/consists of all locally con­stant functions vanishing on the exterior component of the contour T, and coIm /= ((q\ c, e  R 'j. By the exterior component we understand the connected component of T enclosing the domain D.There is another approach to defining weight spaces, which is based on the consideration of homogeneity with respect to dilations. Suppose that F  consists of one point, which is convenient to take for the origin. Let uscover I\{0} by smooth curves r ,, T2, ... so that Y„ = 2T „ ^  j ^  -  2 j starting with some number n> m .Setting Y„ = T„ for n < m , we can associate each func­tion (p e L l  ( r ,  0) with the sequence of functions~ . . |cp(2~nt), n > m  ~ , . t P ~ .<P»(0 = i LP(T„),[cp(0> n < m ,and the space Lpx (T, 0) can be described by using the equivalent norm\ i ipThen, fo r  ф e H{ (D , 0),1ф1 = ьр(гпу (5)A similar fact is valid for the weight Hardy space.Theorem 3. Suppose that the domain D is repre­sented as the union o f domains /),, D2, ... such thatD„ = 2 nDn — |  ^  -  kl -  2 1 fo r  n > m  and the bound­aries dD„ converge to some contour t  in the same sense as in (2).фл(г) =ф(2 nz), n > m  ^ ( z ) ,  n < m ,ф „(г)е H p(D n),and the space Hj- (D , 0) can be described by using the equivalent normE l ? .V  nUp(6)This theorem shows that the family of spaces intro­duced above monotonically decreases with respect to each of the parameters p  and X-L in the sense of an embedding of Banach spaces. Note that the contours dDn in the theorem can be chosen to be smooth. Thus, the weight Hardy space can be introduced on the basis of the definition of the spaces №  on domains with smooth boundary. Applying translations, we can extend the above definition to the case of F = {x} for any point x 5* 0. In the general case, the domain D  can be repre­sented as the union of domains Dx with z e F. where each I)x is bounded by a contour r T smooth outside x.The space H i  (D, F) can be defined by the condition (() e H- (I)x, x) for all x e  F. According to Theorem 1 (b), this definition does not depend on the choice of Dx.On the basis of Theorem 2, we can transfer the results obtained in [6], including those on the Rie- mann-Hilbert problem with piecewise continuous matrix coefficient G, from weight Holder spaces to Hardy spaces. Consider the simplest case (G = 1) of the Schwarz problemКеф+ = / (7)with real right-hand s id e /e  L \  (Г, F). By the Fredholm property and the index of the problem we mean those of the [R-linear operator H- —> L \  of its boundary condi­tion.The criterion for the Fredholm property, which is given below, is stated in terms of the end symbol, i.e., the family of entire functions xx( "Q of a complex vari­able £ (where x e  f )  defined as follows. We associate each pair of different unit vectors a = a x + ia2 and b = bi + ib2 with the analytic function со of two variables £ and u, where Irm/ Ф 0. defined byю(a, b \u ,z )  = -argai + ua2 Y b1 + ub2ü \  +  UÜ, 2b1 + ub2< к.THE HARDY SPACE OF SOLUTIONS TO FIRST-ORDER ELLIPTIC SYSTEMSSince the spectra of the matrices /  and 7 are contained in the upper and lower half-planes, respectively, in which co is analytic as a function of u, it follows that the values of this function at the above matrices can be defined. Thus, we can define an entire function of the variable C, by the formulah(a, b; 0  = det[e’I‘^ co(a, b; J, Q-e-KKw(a,b; 7, 0 1 - (8)It is easy to show that, on each strip A., < Re£ < k 2 of finite width, the function h has finitely many zeros. Therefore, the projection of the zero set of this function onto the real axis is a discrete subset of R. which we denote by A (a, b).If a = -b , then co = 1 and, therefore, the function h(a, -a; Q  coincides with sin'rt^ up to a constant mul­tiplier. Thus,À (a, -a )  = Z .In the scalar case (I = 1), in which /  = v e C, defini­tion (8) takes the formh (a ,b \  Ç) = 2i\q\ sin0Ç, q =b1 + v b 2 ai + \ a 2where 0 = arg q , 0 < 0 < 2n. Obviously, in this case, we haveA (a, b) = \ k e  Z (10)Theorem 4. Suppose that the domain D is bounded by a piecewise Lyapunov contour without cusp points and the set F  contains all corner point o f this contour.Then, problem (7) of class H- (D , F) is Fredholm if  and only ifAT, x e F, ( 1 1 )and its index k  is given by k  = 1(2 -  s) + (kx).Twhere s denotes the number o f connected compo­nents in T.Note that, in the scalar case (I = 1, /  = v), the set \  is determined by (10), where the quantity 0 = 0T has the geometric meaning of the angle of the sector to which the curvilinear sector Sx is mapped under the affine transformation x + iy —> x  + vv. I n the special case of analytic functions (v = i), 0T coincides with the angle of the sector Sx itself. In the scalar case under consider-0 A-(9) ation, condition (11) reduces to g Z, and y x(t) =+ 1, where the brackets denote the integer partConsider problem (7). In a small neighborhood of a corner point x e T, the set Sx = D n  {\z -  x| < r } is a cur­vilinear sector; we denote its lateral sides by r T±0 (it is assumed that the counterclockwise rotation inside the sector is from r T + 0 to r x 0). Let q-L±(l he the unit tan­gent vectors to the lateral sides r T± 0 at the point x. Sup­pose that these sides are not tangent to each other, i.e., x is not a cusp point of the contour T. Then, qx (l^ q -  + (l. and we can consider the function jct(£) = h(qx + Q, qx 0; Q, which we call the end symbol of the problem at the point x. For the corresponding set A(qT + 0, qx ,,) we use the short notation Let yj. t) be an entire function constant on the intervals from the complement to A and such that, for t e  A,., the jump y j i  -  0) -  yj. / + 0) equals the number of zeros of the function jct(£) on the line Re£ = t with multiplicities taken into account. This function monotonically decreases and is defined up to an additive constant, which is determined by the condi­tion %(-0) = 0.of a number. If the contour T  is smooth, then qx + (l = -q x o at x e F, and, according to (9), the situation is the same as in the scalar case.So far, the boundary r  of the domain D was assumed to be finite. Suppose that the curve T  is unbounded, i.e., <=o e T, and it is a piecewise Lyapunov contour on the Riemann sphere C u = » .  The convergence of the con­tours T„ c  D to T is understood in the same sense as above with the additional requirement that 00 e T„ for all 1 1 (in particular, the functions a„(7) must be unbounded, i.e., a„( oo^J = 0°). The Hardy space FP’(D) under these assumptions is defined by the same condi­tion (2). In particular, for the half-plane {z = x  + iy, y > 0}, we can define T„ to be the lines y = en, where en —> 0. For analytic functions, we obtain the classical Hardy space [8].Let us define weight spaces in the case under con­sideration. We include the infinite point in F  and define the class C JT , F) as above with the additional- Xrequirement that (p(t) = 0(l)\t\ " as t For this class, we define the weight scalar functions p(7) = p,(7) and the /-analytic (I x Z)-matrix functions R(z) = R, (z.) as in the finite case. Using these functions, we define the weight spaces as above, with the only difference thatL{(Y ,F ) = U(Y) and H { (T, F) = W (T )  for the weight order k  taking the value at the finite points x e fand ^ at the point x = and, similarly, Lpx (D, F) =SOLDATOV2LP(D) for the weight order A. taking the value - -  at the 2finite points x e  F and -  at the point x = For the half­plane, the weight Hardy space for /-analytic functions was introduced and studied in [9].Theorems 1 and 2 remain valid in the case under consideration, with the only difference that the condi­tions -1  < X<  0 and < X<  0 for x = are replacedby 0 < < 1 and () < ^ . respectively. An analogueof Theorem 3 is valid as well. Suppose that the set F  comprises only one point x = i.e., the unboundedcurve T is smooth. Let us cover Y  by smooth curves Tmwhere n < 1, so that f  „ = 2_T n — j  ^  -  N -  2 j  startingwith some number n > m. Let Dn be domains defined in a similar way for D. Then, the space Lvx ( r ,  «>) can bespecified by norm (5), and the space H{ (D, in The­orem 3 can be specified by (6). Thus, the families of spaces under consideration, as well as in the case of finite contours, monotonically decrease with respect to each of the parameters p  and ~kx, where x e  f .Theorem 4 also extends to the unbounded case; we must only enumerate the arcs r T±0 with endpoint x = <x> and define the corresponding unit tangent vec­tors qx± o to these arcs at this endpoint. Recall that, for x ^  °o, the lateral sides r T±0 of the curvilinear sector ,S'T = I) n  {|z — x| < e }, where 8 > 0 is sufficiently small, with vertex x are enumerated as follows. When the boundary <)SZ is traversed through the point x from r x 0 to r T + 0, the sector Sx remains on the left. The arc r T±0can be specified by the parametric equation z -  x = rqz ± 0exp[ih(r)\, where 0 < r  < e and the real function h(r) is continuous and vanishes at r  = 0. The case of x = «. can be handled similarly. Namely, for sufficiently small R > 0, the lateral sides of the curvilinear “sector” S,_„ = D n  {|z| > R] are “arcs” r T±0 with endpoint x = which can be specified by the parametric equation z = rq-L±llcxp\ih(r)\ on the half-axis r > R ,  where the real function h(r) is continuous and tends to zero as r  —> These arcs can be enumerated in the same way as for finite vertices x.ACKNOWLEDGMENTSThis work was supported in part by the Russian Foundation for Basic Research, project no. 07-01- 00299.REFERENCES1. G. M. Goluzin, The Geometric Theory of Functions of a Complex Variable (Nauka, Moscow, 1972) [in Russian],2. A. A. Douglis, Common. Pure Appl. Math. 6, 259-289 (1953).3. A. P. Soldatov, Izv. Akad. Nauk SSSR, Ser. Mat. 55 (5), 1070-1100(1991).4. A. P. Soldatov, Differ. Equations 38, 855-864 (2002) [Differ. Uravn. 38, 809-817 (2002)].5. V. I. Vlasov and A. V. Rachkov, Dokl. Akad. Nauk 328, 281-284 (1993).6. A. P. Soldatov and A. V. Aleksandrov, Differ. Uravn. 27 (1), 3-8 (1991).7. R. Palais, Seminar on the Atiyah-Singer Index Theorem (Princeton Univ. Press, Princeton, 1965; Mir, Moscow, 1970).8. K. Hoffman, Banach Spaces of Analytic Functions (Prentice-Hall, Englewood Cliffs, N. J., 1962; Inostran- naya Literatura, Moscow, 1963).9. A. P. Soldatov, Izv. Akad. Nauk, Ser. Mat. 70 (6) (2006).

The Hardy space of solutions to first-order elliptic systems

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The Hardy space of solutions to first-order elliptic systems

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