ON ISOMORPHISM OF TWO BASES IN MORREY-LEBESGUE TYPE SPACES

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1 Sahand Communications in Mathematical Analysis (SCMA) Vol. 4 No. 1 (2016), ON ISOMORPHISM OF TWO BASES IN MORREY-LEBESGUE TYPE SPACES FATIMA A. GULIYEVA 1 AND RUBABA H. ABDULLAYEVA 2 Abstract. Double system of exponents with complex-valued coefficients is considered. Under some conditions on the coefficients, we prove that if this system forms a basis for the Morrey-Lebesgue type space on [ π, π], then it is isomorphic to the classical system of exponents in this space. (1.1) 1. Introduction Consider the double system of exponents { A (t) e int ; B (t) e ikt} n Z, k N, with complex-valued coefficients A (t) = A (t) e iα(t), B (t) = B (t) e iβ(t) on the interval [ π, π], where N is the set of natural numbers and Z = {0} N. The system (1.1) is a generalization of the following double sine and cosine system (1.2) 1 {cos (nt γ (t)) ; sin (nt γ (t))} n Z, where γ : [ π, π] C is a complex-valued function in general. The study of basis properties (such as completeness, minimality, basicity) of the systems of type (1.1) and (1.2) in the space L p ( π, π), 1 p < (L ( π, π) C [ π, π]), dates back to the classical works by Paley- Wiener [31] and N. Levinson [22] who considered the perturbed systems of exponents. A lot of works have appeared in this field since then. A 2010 Mathematics Subject Classification. 46E30, 30E25. Key words and phrases. Morrey-Lebesgue type space, System of exponents, Isomorphism, Basicity. Received: 29 February 2016, Accepted: 11 June Corresponding author. 79

2 80 F.A. GULIYEVA AND R.H. ABDULLAYEVA well known Kadets-1/4 theorem [19] also refers to this range of issues. Criterion for the basicity of a system of exponents (1.3) { e i(nαsignn)t} n Z, for L p ( π, π) 1 < p <, where Z is the set of all integers, was first found by A.M. Sedletsky [38]. Similar result was obtained by E.I. Moiseyev [24] who used the method of boundary value problems. Note that the single versions of these systems are the cosine system of (1.4) 1 {cos (n α) t} n N, and the sine system of (1.5) { sin (n α) t} n N, that arise when solving equations of mixed type by the Fourier method (see, e.g., [25 27, 33, 34]). The basis properties of the systems (1.4) and (1.5) in L p (0, π) were fully studied by E.I. Moiseyev [24,28] in case where a R is a real parameter. G.G. Devdariani [14,15] extended these results to the case of complex parameter. Riesz basicity in L 2 ( π, π) for the system (1.2), when γ : [ π, π] C is a Hölder function, was studied by A.N. Barmenkov [1]. One of the most effective methods for treating basis properties of systems like (1.1)-(1.5) is the method of boundary value problems of the theory of analytic functions which dates back to A.V. Bitsadze [11]. This method was successfully used by many authors [1 4, 11, 14, 15, 24 28, 33, 34]. B.T. Bilalov [2 5] considered the most general case, namely, the systems of the form (1.1), and using the results concerning basis properties of (1.1), found a basicity criterion for the completeness and minimality of the sine system of the form (1.6) {sin (nt γ (t))} n N, in L p (0, π), 1 < p <, where γ : [0, π] C is a piecewise continuous function. Similar results for the system (1.6) were obtained earlier in [13]. The study of basis properties of systems (1.1)-(1.6) in different spaces of functions is still ongoing. The weighted case of the L p space was considered in [6,29,35], while [30,36,39] treated the case of Sobolev spaces. In [8, 9], the basicity of the system (1.3) was studied for generalized Lebesgue spaces.

3 ON ISOMORPHISM OF TWO BASES IN MORREY-LEBESGUE TYPE SPACES 81 It should be noted that in recent years interest in the study of various problems of analysis in Morrey-type spaces increased to a great extent. There is obviously a need to study the approximation properties of systems like (1.1)-(1.6) in Morrey-type spaces. Some questions of the approximation theory were studied in [17, 18, 21]. In [10], the basicity of classical exponential systems in Morrey-Lebesgue type spaces was treated. In this paper, we consider a system of exponents (1.1) and prove that if it forms a basis for Morrey-Lebesgue type space M p, α = M p, α ( π, π), then it is isomorphic to the classical system of exponents in this space. To do so, we use the method of boundary value problems. A similar result was obtained earlier in [7]. 2. Necessary Information We will need some facts about the theory of Morrey-type spaces. Let Γ be a rectifiable Jordan curve in the complex plane C. By M Γ we denote the linear Lebesgue measure of the set M Γ. All the constants throughout this paper (can be different in different places) will be denoted by c. The expression f (x) g (x), x M, means δ > 0 : δ f (x) g (x) δ 1, x M. Similar meaning is intended by the expression f (x) g (x), x a. By Morrey-Lebesgue space L p, α (Γ), 0 α 1, p 1, we mean the normed space of all measurable functions f ( ) on Γ with the finite norm L p, α (Γ) : f L p, α (Γ) = sup B ( B α 1 Γ Γ B Γ )1 / p f (ξ) p dξ <. L p, α (Γ) is a Banach space with L p, 1 (Γ) = L p (Γ), L p, 0 (Γ) = L (Γ). The inclusion L p, α 1 (Γ) L p, α 2 (Γ) is valid for 0 α 1 α 2 1. Thus, L p, α (Γ) L 1 (Γ), α [0, 1], p 1. For Γ = [ π, π] we will use the notation L p, α ( π, π) = L p, α. More details on Morrey-type spaces can be found in [12,20,23,32,37, 40]. Let ω = {z C : z <} be the unit circle on C and ω = γ be its circumference. Define the Morrey-Hardy space H p, α of analytic functions f (z) inside ω equipped with the following norm

4 82 F.A. GULIYEVA AND R.H. ABDULLAYEVA f H p, α = sup f (re n ) L p, α. 0<r<1 In [10], the following theorem was proved. Theorem 2.1. The function f ( ) belongs to H p, α only when the nontangential boundary values f belong to L p, α, 1 < p <, 0 < α 1, and the Cauchy formula holds. f (z) = 1 2πi γ f (τ) dτ, τ z Denote by L p, α the linear subspace of L p, α functions, whose shifts are continuous in L p, α, i.e. f ( δ) f ( ) L p, α 0 as δ 0. Consider the closure of L p, α in L p, α and denote it by M p, α. The following theorem was also proved in [10]. Theorem 2.2. Functions infinitely differentiable on [0, 2π] are dense in the space M p, α, 1 p <, 0 < α 1. In the sequel, we will extensively use the following lemma. Lemma 2.3. If f L, g M p,α, 1 < p <, 0 < α 1, then fg M p,α. Consider the following singular operator (Sf) (τ) = 1 f (ξ) dξ 2πi γ ξ τ, τ γ. Using the results of [18, 20, 37], it is easy to prove the following. Theorem 2.4. Singular operator S acts boundedly in M p, α (γ), 1 < p <, 0 < α 1. Consider the space H p, α α. Denote by Lp, the subspace of Lp, α, generated by the restrictions of the functions from H p, α to γ. From the results mentioned above it immediately follows that the spaces H p, α and L p, α are isomorphic and f ( ) = (Jf) ( ), where f H p, α, f are nontangential boundary values of f on γ, and J performs the corresponding isomorphism. Let M p, α = M p, α L p, α. It is clear that M p, α is a subspace of M p, α with respect to the norm L p, α. Let MH p, α = J 1 ( M p α ). The latter is a subspace of H p, α α. Let f Hp, and f be its boundary values. It is absolutely clear that the norm in f H p, α can be defined also as f H p, α = f L p, α. Similar to the classical case, we define the Morrey-Hardy class outside ω. Let D = C\ ω ( ω = ω γ). We say the analytic function f in D

5 ON ISOMORPHISM OF TWO BASES IN MORREY-LEBESGUE TYPE SPACES 83 has finite order k at infinity, if its Laurent series in the neighborhood of the point at infinity has the following form (2.1) f (z) = k n= a n z n, k <, a k 0. Thus, for k > 0 the function f (z) has a pole of order k; for k = 0 it is bounded; and in case of k < 0 it has a zero of order ( k). Let f (z) = f 0 (z) f 1 (z), where f 0 (z) is the principal, and f 1 (z) is the regular part of decomposition (2.1) of the function f (z). Consequently, if k 0, then f 0 (z) = 0. If k > 0, then f 0 (z) is a polynomial of degree k. We say that the function f (z) belongs to the class m H ( p, α, if it has the order at infinity less or equal to m, i.e. k m and f 1 ) 1 z H p, α. The class m MH p, α is defined in a very similar manner to the case of MH p, α. In other words, mmh p, α is the subspace of mh p, α -functions whose shifts are continuous on the unit circle s circumference with respect to the norm L p, α (γ). We will also use the following result of [10]. Theorem 2.5 ( [10]). The system of exponents { e int} forms a basis n Z for M p, α, 1 < p <, 0 < α 1. We will need the Sokhotski-Plemelj formula for the boundary values of a Cauchy type integral (2.2) Φ (z) = 1 f (τ) dτ 2πi γ τ z, where f ( e it) L 1 ( π, π). Then the boundary values Φ ± (τ), τ γ, satisfy the following Sokhotski-Plemelj expression (2.3) Φ ± (τ) = ± 1 f (τ) (Sf) (τ), a.e.. τ γ, 2 where S ( ) is a singular integral (Sf) (τ) = 1 f (ξ) dξ 2πi γ ξ τ, τ γ. Let s show the validity of the following direct decomposition (2.4) L p, α = H p, α 1 Hp, α, with 1 < p <, 0 < α < 1.

6 84 F.A. GULIYEVA AND R.H. ABDULLAYEVA In fact, let f L p, α. Then it is clear that f L p. Consider the Cauchy type integral (2.2). Then, by Theorem 2.1, we have Φ (z) H p, α for z < 1, and Φ (z) 1 H p, α for z > 1. From the Sokhotski-Plemelj formulas (2.3) it follows that (2.5) f (τ) = Φ (τ) Φ (τ), a.e. τ γ. Denote by 1 L p, α the subspace of Lp, α, generated by the restrictions of the functions from 1 H p, α to γ. It is not difficult to see that L p, α 1 Lp, α = {0}. Then from (2.4) we immediately derive the direct decomposition (2.6) L p, α = L p, α 1 Lp, α Identifying H p, α Lp, α and 1L p, α 1 H p, α, we obtain the decomposition (2.4). Applying Theorem 2.4 to the expression (2.5), we obtain in a similar. way that the direct decomposition M p, α = MH p, α Let P ± be projectors 1 MHp, α P : M p, α M p, α, P : M p, α 1 M p, α, also holds. generated by the decomposition (2.4) (or (2.6). Denote by T ± : M p, α M p, α the multiplication operators defined as follows T f = Af, T f = Bf, f L p, α. Let A ±1 ; B ±1 L ( π, π). Suppose that the system (1.1) forms a basis for M p, α. Take g M p, α and expand it in terms of this basis g (t) = A (t) g n e int B (t) g n e int. As A ± L, B ± L, it follows from Lemma 2.3 that the series f (t) = g n e int, and f (t) = g n e int, represent some functions from M p, α. Suppose

7 ON ISOMORPHISM OF TWO BASES IN MORREY-LEBESGUE TYPE SPACES 85 f (t) = n= g n e int, t [ π, π]. It s clear that f M p, α. Let us show that the inclusions f M p, α, f 1 M p, α, hold. In fact, we have π f (arg ξ) ξ n dξ = i f (t) e i(n1)t dt γ = i π π g k e i(kn1)t dt π = 0, n Z. Then from the theorem of Privalov [16] it follows that f (t) are the boundary values of the function F H 1 with (2.7) F (z) = 1 2πi γ We have F (z) = 1 2πi γ = 1 2πi = f (arg ξ) dξ, z < 1. ξ z g n g n γ e int e it z deit ξ n dξ ξ z g n z n, z < 1. As f M p, α, it follows from (2.7) that F MH p, α. In a similar way we can prove that F 1 MH p, α, where F (z) = g n z n, z > 1. Consider the operator T = T P T P. We have T f = T P f T P f = T f T f = A ( ) f B ( ) f = g.

8 86 F.A. GULIYEVA AND R.H. ABDULLAYEVA Then, the equation (2.8) T f = g, g M p, α, has a solution in M p, α for all g M p, α, i.e. R T = M p, α, where R T is { the range of the operator T. Let f KerT. Expand f in the basis e int } n Z : We have f (t) = n= f n e int. 0 = T f = A (t) f n e int B (t) f n e int. As the system (1.1) forms a basis for M p, α, this gives us f n = 0, n Z, i.e. KerT = {0}. By Banach theorem, from T L (M p, α ) it follows that T 1 L (M p, α ), and this in turn implies the correct solvability of the equation (2.8). Now, on the contrary, let the equation (2.8) be correctly solvable in M p, α. Take an arbitrary g M p, α and let f = T 1 g. Expand f in the basis { e int} n Z in M p, α : We have f (t) = n= f n e int, t [ π, π]. and therefore P f = f n e int ; P f = f n e int, T f = A (t) f n e int B (t) f n e int = g (t), i.e. every element of M p, α can be expanded with respect to the system (1.1) in M p, α. Let s show that this expansion is unique. Let Suppose A (t) f n e int B (t) f n e int = 0.

9 ON ISOMORPHISM OF TWO BASES IN MORREY-LEBESGUE TYPE SPACES 87 f (t) = n= f n e int. It s clear that f M p, α. We have T f = A (t) f n e int B (t) f n e int = 0 f = T 1 0 = 0 f n = 0, n Z. Thus, we have proved the following theorem. Theorem 2.6. Let A ±1, B ±1 L ( π, π). The system (1.1) forms a basis for M p, α only when the equation (2.8) is correctly solvable in M p, α, 1 < p <, 0 < α 1. Now let s prove the following main theorem. Theorem 2.7. Let A ±1, B ±1 L ( π, π). If the system (1.1) forms a basis for M p, α, 1 < p <, 0 < α 1, then it is isomorphic to the classical system of exponents { e int} n Z in M p, α, with the isomorphism given by means of the operator T 0 : ( (2.9) (T 0 f) (t) = A (t) f; e inx ) e int ( B (t) f; e inx ) e int, where (f; g) = 1 π f (t) g (t)dt. 2π π Proof. Let the system (1.1) forms a basis for M p, α. Take an arbitrary f M p, α. As the system { e int} n Z forms a basis for M p, α, it is clear that the series and f (t) = f (t) = ( f; e inx ) e int, ( f; e inx ) e int, converge in M p, α. Moreover, the following inequality holds f ± p, α c f p, α.

10 88 F.A. GULIYEVA AND R.H. ABDULLAYEVA Then from (2.9) it directly follows that T 0 L (M p, α ). Let s show that KerT 0 = {0}. Let f KerT 0, i.e. ( T 0 f = A (t) f; e inx ) e int ( B (t) f; e inx ) e int = 0. From the basicity of the system (1.1) for M p, α it follows that ( f; e inx) = 0, n Z f = 0, because the system { e inx} forms a basis for n Z M p, α. Consequently, KerT 0 = {0}. Now let s show that R T0 = M p, α. Let g M p, α be an arbitrary element. By Theorem 2.1, f M p, α : T f = g. On the other hand, it is not difficult to see that T 0 = T, and as a result, R T = M p, α. Then it follows from the Banach theorem that T 0 is an automorphism in M p, α [. It s clear that T 0 e inx ] [ = A (t) e int, n Z and T 0 e inx ] = B (t) e int, n N. The theorem is proved. This theorem has the following immediate corollary. Corollary 2.8. If the perturbed system of exponents { e i(nαsignn)t} n Z, forms a basis for the space M p, α, 1 < p <, 0 < α 1, then it is isomorphic to the classical system of exponents { e int} in this space. n Z Acknowledgment. The authors express their deep gratitude to Prof. B.T.Bilalov for his constant guidance and valuable advice. References 1. A.N. Barmenkov and A.Y. Kazmin, Completeness of a system of functions of special type. In the book: theory of mappings, some its generalization and applications, Kiev, Naukova Dimka 1982, p B.T. Bilalov, On uniform convergence of series in a system of sines, Diff. func., 1988, Vol. 24, No 1, p B.T. Bilalov, Basicity of some systems of functions, Diff. func., 1989, Vol. 25, p B.T. Bilalov, Basicity of some systems of exponents, cosines and sines, Diff. func., 1990, Vol. 26, No 1, p B.T. Bilalov, Basis problems of some systems of exponents, cosines and sines, Sibirskiy matem. zhurnal, 2004, Vol. 45, No 2, p B.T. Bilalov, On completeness of exponent system with complex coefficients in weight spaces, Trans. of NAS of Azerbaijan, XXV, No 7, 2005, p B.T. Bilalov, On isomorphism of two bases in L p, Fundam. Prikl. Mat., 1:4 (1995), B.T. Bilalov and Z.G. Guseynov, Basicity of a system of exponents with a piecewise linear phase in variable spaces, Mediterr. J. Math. Vol. 9, No 3 (2012),

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