Discrete Operators in Canonical Domains

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1 Discrete Operators in Canonical Domains VLADIMIR VASILYEV Belgoro National Research University Chair of Differential Equations Stuencheskaya 14/1, Belgoro RUSSIA Abstract: In this paper, we consier a certain class of iscrete pseuo-ifferential operators in a sharp convex cone an escribe their invertibility conitions in L -spaces. For this purpose we introuce a concept of perioic wave factorization for elliptic symbol an show its applicability for the stuying. Key Wors: Discrete operator, Multiimensional perioic Riemann problem, Perioic wave factorization, Invertibility 1 Introuction A classical pseuo-ifferential operator in Eucliean space IR m is efine by the formula [1,, 3, 4] (Au)(x) = Ã(x, ξ)ei(ξ y)ũ(ξ)ξy, IR m IR m where the sign over a function enotes its iscrete Fourier transform ũ(ξ) = u(x)e ix ξ x. IR m 1.1 Multiimensional Fourier series an symbols Given function u of a iscrete variable x Z m we efine its iscrete Fourier transform by the series (F u )(ξ) ũ (ξ) = e i x ξ u( x), x Z m ξ = [ π, π] m, where partial sums are taken over cubes Q N = { x Z m : x = ( x 1,, x m ), max x k N}. 1 k m Let D IR m be a sharp convex cone, D D Z m, an L (D ) be a space of functions of iscrete variable efine on D, an A( x) be a given function of a iscrete variable x Z m. We consier the following types of operators (A u )( x) = e T i(ỹ x) ξã(ξ)ũ (ξ)ξ, (1) m ỹ D x D, an introuce the function à (ξ) = e i x ξ A( x), ξ. x Z m Definition 1 The function Ã(ξ) is calle a symbol of the operator A, an this symbol is calle an elliptic symbol if Ã(ξ) 0, ξ. Our main goal is escribing a perioic variant of wave factorization for an elliptic symbol [9] an showing its usability for stuying invertibility for the operator A. 1. Discrete projection operators Let us enote P D projection operator on D, P D : L (Z m ) L (D ) so that for arbitrary function u L (Z m ) { u ( x), x D (P D u )( x) = 0, x / D Perioic Cauchy kernel If we consier a half-space case, then the Fourier image of the operator P D is evaluate [10, 11, 1] an we ll emonstrate it in the following Example If D = IR m + then 1 4πi lim π + π (F P D u )(ξ, ξ m ) = u (ξ, η m ) cot ξ m η m + iτ η m. E-ISSN: Volume 16, 017

2 1.. Perioic Bochner kernel If D is a sharp convex cone C a + = { x Z m : x = ( x 1,, x m ), x m > a x, x = ( x 1,, x m 1 ), a > 0} then we introuce the function B (z) = x D e i x z, z = ξ+iτ, ξ, τ C a +, an efine the operator (B u)(ξ) = lim B (z η)u (η)η. Lemma 3 For arbitrary u L (Z m ) the following property F P D u = B F u hols. Proof: Let χ + ( x) be an inicator of the set D. Thus (P D u )( x) = χ + ( x) u ( x). Further, since the function χ + ( x) is not summable, we can t apply irectly a convolution property of the Fourier transform. We choose the function e i x τ so the prouct χ + ( x)e i x τ will be summable for some amissible τ. Taking into account a forthcoming passing to a limit uner τ 0+ we have F (χ + ( x)e i x τ ) = B (z). Thus we can use the Fourier transform obtaining convolution of functions B (z) an ũ (ξ). It is left passing to a limit. Multiimensional perioic Riemann bounary value problem.1 A half-space case an a perioic oneimensional Riemann bounary value problem [10, 11, 1] For D = IR m + we will remin some author s constructions for iscrete equations in a half-space. We have B (z) = cot z, z = (ξ, ξ m + iτ), ξ = (ξ 1,, ξ m 1 ), τ > 0. Thus (see example 1) we use a perioic oneimensional Riemann problem with a parameter ξ 1 which is the following. Fining a pair of functions Φ ± (ξ, ξ m ) which are bounary values of holomorphic in half-strips Π ± = {z C : z = ξ m ± iτ, τ > 0} such that these are satisfie a linear relation Φ + (ξ)(ξ, ξ m ) = G(ξ, ξ m )Φ (ξ)(ξ, ξ m ) + g(ξ), ξ, for almost all ξ 1, where G(ξ), g(ξ) are given perioic functions.. Essential multiimensional case Let D be a conjugate cone for D i.e. D= {x IR m : x y > 0, y D}, an T ( D) C m be a set of the type + i D. For IR m such a omain of multiimensional complex space is calle a raial tube omain over the cone D [7, 8, 9]. Let us efine the subspace A( ) L ( ) consisting of functions which amit a holomorphic continuation into T ( D) an satisfy the following conition sup ũ (ξ + iτ) ξ < +. () τ D In other wors, the space A( ) L ( ) consists of bounary values of holomorphic in T ( D) functions. Let us enote B( ) = L ( ) A( ), so that B( ) is a irect complement of A( ) in L ( )...1 A jump problem We formulate the problem by the following way: fining a pair of functions Φ ±, Φ + A( ), Φ B( ), such that Φ + (ξ) Φ (ξ) = g(ξ), ξ, (3) where g(ξ) L ( ) is given. Lemma 4 The operator B : L ( ) A( ) is a boune projector. A function u L (D ) iff its Fourier transform ũ A( ). Proof: Accoring to stanar properties of the iscrete Fourier transform F, we have F (χ + ( x)u ( x)) = lim B (z η)ũ (η)η, E-ISSN: Volume 16, 017

3 where χ + ( x) is an inicator of the set D. It implies a bouneness of the operator B. The secon assertion follows from holomorphic properties of the kernel B (z). In other wors for arbitrary function v A( ) we have v(z) = B (z η)v(η)η, z T ( D). It is an analogue of the Cauchy integral formula. Theorem 5 The jump problem has unique solution for arbitrary right-han sie from L ( ). Proof: Inee it is equivalent to one-to-one representation of the space L (D ) as a irect sum of two subspaces. If we ll enote χ + (x), χ (x) inicators of iscrete sets D, Z m \D respectively then the following representation u ( x) = χ + ( x)u ( x) + χ ( x)u ( x) is unique an hols for arbitrary function u L (Z m ). After applying the iscrete Fourier transform we have F u = F (χ + u ) + F (χ u ), where F (χ + u ) A( ) accoring to lemma, an thus F (χ u ) = F u F (χ + u ) B( ) because F u L ( ). Example 6 If m = an C + is the first quarant in a plane then a solution of the jump problem is given by formulas Φ + (ξ) = 1 (4πi) lim π π π π cot ξ 1 + iτ 1 t 1 cot ξ + iτ t g(t 1, t )t 1 t Φ (ξ) = Φ + (ξ) g(ξ), τ = (τ 1, τ ) C +... A general statement It looks as follows. Fining a pair of functions Φ ±, Φ + A( ), Φ B( ), such that Φ + (ξ) = G(ξ)Φ (ξ) + g(ξ), ξ, (4) where G(ξ), g(ξ) are given perioic functions. If G(ξ) 1 we have the jump problem 3). Like classical stuies [5, 6] we want to use a special representation for an elliptic symbol to solve the problem (4)...3 Associate singular integral equation We can easily obtain so-calle characteristic singular integral equation associate with multiimensional perioic Riemann bounary value problem (4). Let us enote Q D = I P D an consier so-calle paire operator compose by two operators A (1), A() of the following type A (1) P D + A () Q D : L (Z m ) L (Z m ) (5) One can easily obtain the following Property 7 The invertibility of the operator (1) in the space L (D ) is equivalent to invertibility of the operator (5) in the space L (Z m ) with A (1) = A, A () = I. The Fourier image for the operator (5) is the following operator ũ (ξ) ((A (1) (ξ)b +A () (ξ)(i B )ũ )(ξ) (6) If D = IR m + then the operator (6) is a oneimensional singular integral operator with perioic Cauchy kernel an a parameter ξ [10, 11, 1]. 3 Perioic wave factorization Definition 8 Perioic wave factorization for elliptic symbol Ã(ξ) is calle its representation in the form à (ξ) = à (ξ)ã=(ξ) where the factors A ±1 (ξ), A±1 = (ξ) amit boune holomorphic continuation into omains T (± D). 3.1 Sufficient conitions We ll give here certain sufficient conitions for an existence of the perioic wave factorization for an elliptic symbol. Theorem 9 Let an elliptic symbol à (ξ) C( ) be a such that π π supp F 1 (ln Ã(ξ)) D ( D ), (7) arg Ã(, ξ k, ) = 0, k = 1,, m. (8) Then the symbol à (ξ) amits the wave factorization. E-ISSN: Volume 16, 017

4 Proof: If we start from equality à (ξ) = à (ξ)ã=(ξ) then by logarithm we obtain ln Ã(ξ) = ln à (ξ) + ln Ã=(ξ) an we have a special kin of a jump problem. Namely if we will enote by A 1 ( ) a subspace of the space L ( ) consisting of functions which amit a holomorphic continuation into T ( D) an satisfy the conition () for τ D. So eviently we speak on a possibility of ecomposition of the function ln Ã(ξ) into two summans one of which belongs to the space A( ) an the secon one belongs to the space A 1 ( ). Let us enote F 1 (ln Ã(ξ)) v(x). If supp v D ( D ) then we have the unique representation v = χ + v + χ v where χ ± is an inicator of the iscrete set ±D. Further passing to the Fourier transform an potentiating we obtain the require factorization. Remark 10 The conition (7) is not necessary but we have no an algorithm for constructing a perioic wave factorization. For D = IR m + a such algorithm exists always (see [1]). 3. Factorization an inex There is one point in previous consierations from proof of the Theorem for which one nees an explanation. Inee the function ln Ã(ξ) is efine correctly because the conition (8) provies an absence of bifurcation points. That s why one can call this factorization with vanishing inex. 4 Invertibility of iscrete operators Lemma 11 If f B( ), g A 1 ( ) then f g B( ). Proof: Accoring to properties of iscrete Fourier transform F we have (F (f g))( x) ((F f) (F g))( x) f 1 ( x ỹ)g 1 (ỹ) = f 1 ( x ỹ)g 1 (ỹ), ỹ Z m ỹ D where f 1 = F 1 f, g 1 = F 1 g an accoring to lemma supp g 1 D. Further since we have supp f 1 Z m \ ( D ) then for x D, ỹ D we have x ỹ D so that f 1 ( x ỹ) = 0 for such x, ỹ. Thus supp f 1 g 1 Z m \ D. Theorem 1 If the elliptic symbol à (ξ) C( ) amits perioic wave factorization then the operator A is invertible in the space L (D ). Proof: We will remin that accoring to the property 1 an invertibility of the operator A in the space L (D ) is equivalent to an invertibility of the operator A P D + IQ D in the space L (Z m ). It is easily concluing the last invertibility is equivalent to solving the Riemann problem (4) for arbitrary righthan sie g(ξ) L (Z m ) with G(ξ) à 1 (ξ). If we have the perioic wave factorization for the symbol Ã(ξ) then à (ξ)φ + (ξ) = à 1 = (ξ)φ (ξ) + à (ξ)g(ξ), (9) ξ, an we have a jump problem. The first summan à (ξ)φ + (ξ) A( ) accoring to a holomorphic property, an the secon one à 1 = (ξ)φ (ξ) B( ) accoring to the lemma 3. Taking into account the theorem we conclue that the Riemann problem (9) has a unique solution for arbitrary g(ξ) L ( ). Conclusion These iscrete consierations can be transferre on more general situations an operators. It will be a subject of forthcoming papers of the author. Acknowlegements: The research was partially supporte by Russian Founation for Basic Research an government of Lipetsk region, Grant rcenter-a References: [1] M. E. Taylor, Pseuoifferential Operators, Princeton Univ. Press, Princeton 1981 [] F. Treves, Introuction to Pseuoifferential Operators an Fourier Integral Operators, Springer, New York 1980 [3] G. Eskin, Bounary Value Problems for Elliptic Pseuoifferential Equations, AMS, Provience 1981 E-ISSN: Volume 16, 017

5 [4] M. A. Shubin, Pseuoifferential Operators an Spectral Theory, Springer, Berlin Heielberg 001 [5] F. D. Gakhov, Bounary Value Problems, Dover Publications, New York 1981 [6] N. I. Muskhelishvili, Singular Integral Equations, North Hollan, Amsteram 1976 [7] S. Bochner an W. T. Martin, Several Complex Variables, Princeton Univ. Press, Princeton, 1948 [8] V. S. Vlaimirov, Methos of the Theory of Functions of Many Complex Variables, Dover Publications, New York (007) [9] V. B. Vasil ev, Wave Factorization of Elliptic Symbols: Theory an Applications, Kluwer Acaemic Publishers, Dorrect Boston Lonon 000 [10] A. V. Vasilyev an V. B. Vasilyev, Discrete singular operators an equations in a half-space, Azerb. J. Math. 3, 013, pp [11] A. V. Vasilyev an V. B. Vasilyev, Discrete singular integrals in a half-space. In: Mityushev, V., Ruzhansky, M. (es.) Current Trens in Analysis an Its Applications. Proc. 9th ISAAC Congress, Kraków, Polan, 013, pp Birkhäuser, Basel 015. Series: Trens in Mathematics. Research Perspectives. [1] A. V. Vasilyev an V. B. Vasilyev, Perioic Riemann problem an iscrete convolution equations, Differential Equat. 51, 015, pp E-ISSN: Volume 16, 017