ISSN: 1402-1757 ISBN 978-91-7439-XXX-X Se i listan och fyll i siffror där kryssen är
LICENTIATE T H E SI S Aigerim Kopezhanova Relations Between Functions from some Lorentz fromsome BetweenCoefficients KopezhanovaFunctions Relations Fourier Type Aigerim their of Summability and Spaces
Department of Mathematics
ISSN: 1402-1757 ISBN 978-91-7439-170-1 Relations between Functions from some
Luleå University of Technology 2010 Lorentz Type Spaces and Summability of their Fourier Coefficients
Aigerim Kopezhanova
Relations between Functions from some Lorentz Type Spaces and Summability of their Fourier Coefficients
Aigerim Kopezhanova
Department of Mathematics Lule˚a University of Technology SE-971 87 Lule˚a, Sweden [email protected] Key words: Lorentz spaces, summability of Fourier series, inequalities, orthonormal bounded systems, regular systems, quasi-monotone functions, generalised monotone sequences.
Printed by Universitetstryckeriet, Luleå 2010
ISSN: 1402-1757 ISBN 978-91-7439-170-1 Luleå 2010 www.ltu.se Abstract
This Licentiate Thesis is devoted to the study of summability of the Fourier coefficients for functions from some Lorentz type spaces and contains three papers (papers A - C) together with an introduction, which put these papers into a general frame. Let Λp(ω),p>0, denote the Lorentz spaces equipped with the (quasi) norm 1 1 p ∗ p dt f Λp(ω) := (f (t)ω(t)) 0 t for a function f on [0,1] and with ω positive and equipped with some addi- tional growth properties. In paper A some relations between this quantity and some corresponding sums of Fourier coefficients are proved for the case with a general orthonormal bounded system. Under certain circumstances even two-sided estimates are obtained. In paper B we study relations between summability of Fourier coefficients and integrability of the corresponding functions for the generalized spaces Λp(ω) in the case of a regular system. For example, all trigonometrical systems, the Walsh system and Prise’s system are special cases of regular systems. Some new inequalities of Hardy-Littlewood-P´olya type with respect to a regular system for the generalized Lorentz spaces Λp(ω) are obtained. It is also proved that the obtained results are in a sense sharp. The following inequalities are well-known: ∞ p c f ≤ kp−2|a |p ≤ c tf p , for 1 iii iv Preface This Licentiate thesis contains the following papers: [A] A. Kopezhanova and L.-E. Persson, On summability of the Fourier coefficients in bounded orthonormal systems for functions from some Lorentz type spaces, Eurasian Mathematical Journal, 1 (2010), No. 2, 76-85. [B] A. Kopezhanova, E. Nursultanov and L.-E. Persson, Relations between summability of the Fourier coefficients in regular systems and functions from some Lorentz type spaces, Proc. Razmadze Math. Inst., 152 (2010), 73-88. [C] A. Kopezhanova, Some new results concerning the Fourier coefficients in Lorentz type spaces, Research report 5, Department of Mathematics, Lule˚a University of Technology, (15 pages), 2010. These publications are put to a more general frame in an introduction, which also serves as a basic overview of the field. Some of the results in the papers [A]-[C] are strongly influenced by the following paper: [D] A. Kopezhanova, E. D. Nursultanov and L.-E. Persson, On summabi- lity of the Fourier coefficients for functions from some Lorentz type spaces, Research report 8, Department of Mathematics, Lule˚a University of Tech- nology, (27 pages), 2009. v vi Acknowledgement I thank my main supervisors Prof. Lars-Erik Persson (Department of Ma- thematics, Lule˚a University of Technology, Sweden) and Prof. Erlan Nur- sultanov (Eurasian National University, Kazakhstan) for guiding me in this fascinating field of mathematics, for giving encouragement and support in times of need and for always showing a generous attitude towards their stu- dents. I also thank Prof. Ryskul Oinarov (Eurasian National University, Kaza- khstan), Prof. Lech Maligranda (Department of Mathematics, Lule˚a Univer- sity of Technology, Sweden), Prof. Nazerke Tleukhanova (Eurasian Natio- nal University, Kazakhstan), Dr. Niklas Grip (Department of Mathematics, Lule˚a University of Technology, Sweden) and Dr. Kuanush Bekmaganbe- tov (Moscow State University (Kazakh branch), Kazakhstan) for generously sharing their valuable knowledge in the field with me and for their constant support during my studies. Moreover, I thank PhD students John Fabricius (Department of Mathematics, Lule˚a University of Technology, Sweden) and Yulia Koroleva (Moscow State University, Russia) for helping me with many practical things. Furthermore, I thank everyone at the Department of Mathematics at Lule˚a University of Technology for their friendly attitude to me and for the inspiring atmosphere. This research has been done within the frame of the general agreement between Eurasian National University in Astana, Kazakhstan, and Lule˚a University of Technology in Sweden concerning research and PhD education in mathematics. We thank both these universities for financial support, which made this cooperation possible. Finally, I thank my family for all support and just being there. vii Introduction Let f be a measurable function on a measure space (Ω,μ), where μ is an additive positive measure. The distribution function m(σ, f) and the nonincreasing rearrangement f ∗ of a function f are defined as follows: m(σ, f):=μ {x ∈ Ω:|f(x)| >σ} , f ∗(t) := inf {σ : m(σ, f) ≤ t} . Let 1 ≤ p ≤∞and 0 Note that for the case p = q the Lpq spaces coincide with the usual Lp– spaces equipped will the norms f Lp (quasinorms for 0 ix x are called the Fourier coefficients of the function f with respect to the system { }∞ Φ= ϕn n=1. In 1926 the following inequalities by Hardy-Littlewood with respect to the trigonometrical system were established (see e.g. [15], [17], [18] and [55]): If 2 ≤ p<∞, then ∞ p ≤ p−2| |p f Lp[0,1] c1 k ak . (0.3) k=1 If 1 Here and in the sequel ci,i∈ Z+, denote positive constants dependent on p but not dependent an the involved function f. The constants need not to be the same in different occurances. In 1931 these inequalities were established for the uniform orthonormal bounded system by P´olya ( see e.g. [20], [40] and [55]): { }∞ Theorem 0.1. Let ϕk k=1 be the orthonormal system on [a, b] such that |ϕk(x)|≤M for all k ∈ N and x ∈ [a, b]. Then the following estimates hold: 1) 1 ∞ p 2−p p p−2 p |ak| k ≤ c1M f p, k=1 where 1 1 ∞ p p p−2 |ak| k < ∞ k=1 ∈ p n and the function f L [a, b], is given by the formula f = limn→∞ k=1 akϕk. p−2 p−2 Remark 0.1. The weight k can be replaced by uk in the case when the { }∞ sequence uk k=1 consists of positive numbers uk satisfing −2 ≤ uk B/z uk>z for all z>0 (see [15]). xi Many mathematicians have been interested in this type of questions. Therefore a lot of different results on the Hardy-Littlewood-P´olya type in- equalities have been obtained. In this introduction we will present some of the most important of these results. In 1937 Marcinkiewicz and Zygmund [35] obtained an estimate for the norm in Lq[0, 1] of the sum of the series ∞ ≤ { } n=1 cnϕn(x) under the condition that ϕn(x) ∞ Mn, where Mn is a monotone increasing sequence. A generalizations of the Marcinkiewicz - Zygmund theorem and of several related results were established in [25]. The proofs were based on some ele- mentary numerical inequalities. Several analogous estimates for Orlicz classes ϕ(L) were proved in the case of orthonormal systems of bounded functions. Also necessary and sufficient conditions were derived for imbedding into ϕ(L) of classes of functions with a given majorant of best approximations in Lp in the trigonometric system. In [16] some analogues of the Hardy-Littlewood inequalities with respect to the Haar and Walsh systems were obtained. Moreover, in [37] the theorem of Hardy - Littlewood for Walsh series in a more general setting was proved. The properties of a class of periodic multiplicative orthonormal systems on [0,1] were studied in [53]. Some theorems concerning the quantitative properties of these systems were proved. In particular, completeness of the systems was discussed together with an application concerning the represen- tation of a given function in the form of a Fourier series. In [7] the author considered different conditions on the coefficients of dou- ble Fourier series which lead to the Fourier series of functions f ∈ Lp(0, 2π). It was proved that some of these results cannot be improved. In [36] the author extended to two dimensions results obtained by Hardy and Littlewood, Stechkin and himself to the case with cosine, sine and Walsh series with monotone coefficients. Let smn(x, y) be the rectangular partial sums of a double cosine series whose coefficients {ajk} satisfy ajk ≥ 0, ajk ≥ aj+1,k,ajk ≥ aj,k+1, and ajk + aj+1,k+1 ≥ aj+1,k + aj,k+1. Denote by 1 p p p−2 p−2 Sp = |ajk| (j +1) (k +1) . j,k | | ≤ ≥ → It was shown that supm,n smn p c1Sp,p 1. If, in addition, ajk 0 as max(j, k) →∞, and if the sum f of the double cosine series belongs to 1 p p p−2 p ≤ L for some p>1, then j,k ajk(jk) c1 f p,p>1. For the case ≥ p q ≥ p =1, it was proved that if ajk 0, then supp,q s2 ,2 1 c2S1. Similar results were proved for double sine and Walsh series. xii In [23] the author dealt with a generalization of the well-known Hardy- Littlewood-P´olya inequality to dimensions n ≥ 2. In particular, in this con- nection it was proved that |u(x)|p p p |∇u|p dx ≤ dx, (0.5) k+2 k−p+2 Rn |x| |n − k − 2| Rn |x| where p is a real number ≥ 2,x∈ Rn, |x| is the Euclidean norm of x in Rn, and u belongs to a proper function space. It was also included a construction of a counterexample that, for certain exponents and consequently in some spaces, such an extension is impossible. Inequalities of the form (0.5) are of extremely great independent interest. Weighted forms of (0.5) are usually called Hardy type inequalities and a lot of information can be found in the recent books [30] and [31] and the references given there. In [9] the author discussed classes of periodic functions of m variables that are of piecewise monotonic type. In particular, for such functions, the connections between the property of belonging to Lp, 1 2m for (m+1) Theorem 0.2. Let {ϕn} be an orthonormal system of functions on [0, 1] ∗ − 1 1 − 1 ≤ ··· ≤ 2 2 q such that ϕn ∞ M, n =1, 2, . Then cn Cn (log n) f 2q, ··· { ∗ } for every n =2, 3, , where cn is a nonincreasing rearrangement of the Fourier coefficients {cn} of f. xiii In the second theorem a trigonometric polynomial was constructed such that for ”most” of its coefficients the opposite estimate (with different con- stant) holds. This example showed that the Hausdorff-Young theorem and the Riesz theorem are not valid for f ∈ L2q. The construction of the poly- nomial was based on an interesting generalization of the well-known Rudin- Shapiro polynomials. In [48] the author considered trigonometric cosine and sine series with monotone coefficients. The conditions for the sums of these series to belong to the Lorentz spaces Λp(ω) were found. Under some additional conditions on the function ω the quasinorms of the sums of the series were estimated above and below by expressions involving the coefficients. In [24] some new estimates of the norms of functions in Lorentz spaces Lq,r (q ≥ 2,r>0) were obtained, which are analogous to the estimates given by V. I. Kolyada [25] in the case of Lebesgue spaces. Also some estimates which generalized a well-known theorem of J. Marcinkiewicz and A. Zygmund [35] were obtained. In [47] the expansions of functions in the space Lp with respect to systems similar to orthogonal ones were studied. The estimates for the coefficients and sufficient conditions on them under which the corresponding expansions converge in Lp were found. These results are analogues of the well-known Hausdorff-Young-Riesz and Hardy-Littlewood-P´olya theorems in the theory of trigonometric and orthogonal series. It was shown that the resulting esti- mates are more exact than the classical ones even in the case of orthogonal systems. In [34] the author studied the convergence of Fourier series in the Lorentz 1 spaces Λp( ψ ), where ψ satisfied certain growth properties. In particular, it was proved that the Fourier series of any f ∈ Λ ( 1 )convergestof in the p ψ T ψ(τ) norm of Lorentz spaces Λψψ,p where ψ(t)ψ(t)= t τ dτ. In [1] an interpolation theorem for a class of network spaces was proved. In terms of Fourier-Haar coefficients the authors obtained a test for a function to belong to the network space Np,q(M), where 1 In [13] the convergence results for trigonometric series in Lp-spaces on one-dimensional and n-dimension torus were studied. The sufficient condi- tions for these results to hold as well as criteria were derived for series with general monotone coefficients. Moreover, a Hardy-Littlewood type theorem was obtained for multiple series. Finally, several corollaries, in particular, concerning u−convergence were presented. (One can find the definition of u−convergence e.g. in [13]). The aim of [32] was to generalize two fundamental theorems of Boas. One of them dealt with conditional integrability; another proved pth power integrability. Both theorems considered Fourier series with nonnegative coef- ficients and classical power weight xγ. The weight functions in our theorems aremoregeneral.Theyhaveβ-power-monotone properties. In [12] the sufficient and necessary conditions on the Fourier coefficients n ∞ for functions to belong to the Lebesgue spaces Lp for 1 n Theorem 0.3. Let 1 − ∞ n p 2 ∈ n p ∞ f Lp iff am mj < m=1 j=1 xv for the following type of series (N = {1, 2, .., n},B⊆ N) ∞ am cos mjxj sin mjxj. m=1 j∈B j∈N−B n A non-negative sequence a = {am}, m =(m1, ..., mn) ∈ N ,n≥ 1, satisfies the GM n-condition if n am → 0as|m|≡ mj →∞ j=1 and ∞ n ∞ (n) ∗ ak1,...,ki−1,mi,ki+1,...,kn |Δ am|≤C ak + + mi m=k i=1 mi=ki+1 ⎞ ∞ ∞ ∞ a 1 n a + k ,...,mi,...,mj ,...,k + ... + m1,...,mn ⎠ , mimj m1 ···mn 1≤i n (n) ≡ j j − Δ Δ and Δ am = am am1,...,mj−1,mj +1,mj+1,...,mn . j=1 It was introduced earlier a continuous scale of monotonicity for sequences (classes Mα,α≥ 0), where, for example, M0 is the set of all nonnegative vanishing sequences, M1 is the class of all nonincreasing sequences, tending to zero, etc. In addition, several results obtained for trigonometric series with monotone convex coefficients onto more general classes were proved. The aim of [10] was to generalize the well-known Hardy-Littlewood theorem for trigonometric series, whose coefficients belong to the classes Mα, where ∈ 1 ∈ 1 1 ∈ α 2 , 1 . It was proved that if α 2 , 1 , α The Hardy-Littlewood-P´olya inequalities were early generalized to hold also for Lorentz spaces by Stein [50] and for the more general Lorentz spaces Λp(ω) the following was proved in 1974 by L.E. Persson, when ∞ 2πikx + Φ= e k=−∞ is a trigonometrical system (see [43]-[44]): ∞ Theorem 0.4. ∞ 2πikt + Let 0 1 ∞ 1 p (a∗ ω(n))p ≤ c f . (0.6) n n 1 Λp(ω) n=1 − 1 − b) If there exists a positive number δ>0 satisfying that: ω(t)t 2 δ is an increasing function of t and ω(t)t−1+δ is a decreasing function of t, then 1 ∞ 1 p f ≤ c (a∗ ω(n))p , (0.7) Λp(ω) 2 n n n=1 { ∗ }∞ {| |}∞ where an n=1 is the nonincreasing rearrangement of the sequence an n=1 of Fourier coefficients of f on the system Φ. { }∞ We say that the orthonormal system Φ = ϕk(x) k=1 is regular if there exists a constant B such that 1) for every segment e from [0, 1] and k ∈ N it yields that ϕk(x)dx ≤ B min(|e|, 1/k), e 2) for every segment w from N and t ∈ (0, 1] we have that ∗ ϕk(·) (t) ≤ B min(|w|, 1/t), k∈w · ∗ where k∈wϕk( ) (t) as usual denotes the nonincreasing rerrangement of function k∈w ϕk(x). This concept was introduced and studied by E. D. Nursultanov [38]. For example, all trigonometrical systems, the Walsh sistem and Prise’s system are regular ones. The papers [38]-[39] by E. D. Nursultanov dealt with the deriving of the following Hardy-Littlewood type inequalities of Fourier coefficients with res- pect to the regular system: xvii 1. If 1 Short description of some of the obtained results: Let Λp(ω),p>0, denote the generalized Lorentz spaces equipped with the (quasi) norm 1 1 p ∗ p dt f Λp(ω) := (f (t)ω(t)) 0 t for a function f on [0,1] and with ω positive and equipped with some addi- tional growth properties. In paper A (see also [29] and c.f. also [27]) some xviii relations between this quantity and some corresponding sums of Fourier coef- ficients are proved for the case with a general orthonormal bounded system. This results generalize the results of L.-E. Persson [43], [44]. In paper B (see also [28] and c.f. also [27]) we study relations between summability of Fourier coefficients and integrability of the corresponding functions for the generalized spaces Λp(ω) in the case of a regular system. For example, all trigonometrical systems, the Walsh system and Prise’s sys- tem are special cases of regular systems. Some new inequalities of Hardy- Littlewood-P´olya type with respect to a regular system for the generalized Lorentz spaces Λp(ω) are obtained. It is also proved that the obtained results are in a sense sharp. In the case when the sequences of coefficients with re- spect to a regular system are generalized monotone then relation is obtained, which is analogous to the well-known theorem of Hardy and Littlewood. The following inequalities are well-known: ∞ p c f ≤ kp−2|a |p ≤ c tf p , for 1 Remark 0.2. Note that to work with weights which will be monotone after multiplication by a power function are well-known in the literature. For ex- ample, in an implicit form such functions were used already in [3] and the classes Q(α0,b0),α0,b0 ∈ R, consisting of all weights ω such that for some α b real numbers a α ≤ α ≤ ω(t1)t1 C0ω(t2)t2 ,t1 t2. xix It is obvious that each weight λ(t) in the class Q(α0,b0,C0,B0) can be replaced by an equivalent weight ω(t) in the class Q(α0,α1). All results in this thesis are formulated in situations corresponding to a class of type Q(α0,α1) but, according to this remark and our methods of proofs, they could as well have been formulated for weights in a more class of type Q(α0,α1,C0,C1). Recently, such classes of weights were also used in the context of Hardy type inequalities for variable Lp−spaces and Stein type inequalities, see [46]. Finally, it should be mentioned that such classes of quasi-monotone func- tions are not only useful in various situations but also of independent interest, see e.g. the new review article [45] by L.-E. Persson and N. Samko and the references given there. Remark 0.3. Some questions which are related to some Lorentz spaces Λp(ω) were considered in [6], [21] and [22]. In particular, necessary and sufficient conditions on convexity and concavity, lower and upper estimates and type and cotype of weighted Lorentz spaces Λp(ω) with 1 ≤ p<∞ and a decreasing weight w were presented in [22].In[21] the authors studied order covexity and concavity of quasi-Banach Lorentz spaces Λp(ω), where 0 Bibliography [1] T. U. Aubakirov and E. D. Nursultanov, The Hardy-Littlewood Theorem for Fourier-Haar Series, Mat. Zametki, 73 (2003), No. 3, 340-347. [In Russian]; translation in Math. Notes, 73 (2003), No. 3-4, 314–320. [2] G. J. Bao, T. X. Li and Yu. M. Xing, Two-weighted Hardy-Littlewood in- equality for A−harmonic tensors, Chinese Ann. Math. Ser., 26 (2005), No. 1, 113–120. [In Chinese] [3] N. K. Bari and S. B. Stechkin, Best approximations and differential prop- erties of two conjugate functions, Trudy Mosk. Matem. Obva, 5 1956, 483–522. [In Russian] [4] J. Bergh and J. L¨ofstr¨om, Interpolation spaces. An Introduction. Grund- lehren der Mathematischen Wissenschaften, Springer Verlag, Berlin- New York, No. 223, (1976). [5] S. V. Bochkarev, Estimates for the Fourier coefficients of functions in Lorents spaces, Dokl. Akad. Nauk, 360 (1998), No. 6, 730–733. [In Rus- sian] [6] M. Cwikel,´ A. Kami´nska, L. Maligranda and L. Pick, Are generalized Lorentz ”spaces” really spaces?, Proc. Amer. Math. Soc., 132 (2004), No. 12, 3615–3625. [7] M. I. Dyachenko, Convergence of double trigonometric series and Fourier series with monotone coefficients, Mat. Sb. (N.S.), 129(171) (1986), No. 1, 55–72. [In Russian] [8] M. I. Dyachenko, Norms of Dirichlet kernels and of some other trigono- metric polynomials in Lp spaces, Mat. Sb., 184 (1993), No. 3, 3-20. [In Russian]; translation in Russian Acad. Sci. Sb. Math., 78 (1994), No. 2, 267–282. xxi xxii BIBLIOGRAPHY [9] M. I. Dyachenko, Piecewise monotone functions of several variables and a theorem of Hardy and Littlewood, Izv. Akad. Nauk SSSR Ser. Mat., 55 (1991), No. 6, 1156–1170. [In Russian]; translation in Math. USSR-Izv., 39 (1992), No. 3, 1113–1128. [10] M. Dyachenko, The Hardy-Littlewood theorem for trigonometric series with generalized monotone coefficients, Izv. Vyssh. Uchebn. Zaved. Mat., (2008), No. 5, 38–47. [In Russian]; translation in Russian Math. (Iz. VUZ) 52 (2008), No. 5, 32–40. [11] M. I. Dyachenko and E. D. Nursultanov, Hardy-Littlewood theorem for trigonometric series with α-monotone coefficients, Mat. Sb., 20 (2009), No. 11, 45–60. [In Russian] [12] M. Dyachenko and S. Tikhonov, A Hardy-Littlewood theorem for mul- tiple series, J. Math. Anal. Appl., 339 (2008), No. 1, 503–510. [13] M. Dyachenko and S. Tikhonov, Convergence of trigonometric series with general monotone coefficients, C. R. Math. Acad. Sci. Paris 345 (2007), No. 3, 123–126. [14] M. Dyachenko and S. Tikhonov, Integrability and continuity of functions represented by trigonometric series: coefficients criteria, Studia Math., 193 (2009), No. 3, 285–306. [15] R. Edvards, Fourier series with a modern exposition, Mir, Moscow, Vol. 2, 1985. [In Russian] [16] B. I. Golubov, Best approximations of functions in the Lp metric by Haar and Walsh polynomials, Mat. Sb. (N.S.), 87(129) (1972), No. 2, 254– 274. [17] G. H. Hardy and J. E. Littlewood, Some new properties of Fourier con- stants, Math. Ann., 97 (1926), 159–209. [18] G. H. Hardy, J. E. Littlewood and G. P´olya, Inequalities 2d ed., Cam- bridge, at the University Press, 1952. [19] K. Hu, On Hardy-Littlewood-P´olya inequality and its applications, Chi- nese J. Contemp. Math., 15 (1994), No. 4, 325–330. [20] S. Kaczmarz and G. Steingauz, Theory of orthogonal series, Fiz.-Mat. Lit., Moscow, 1958. [In Russian] BIBLIOGRAPHY xxiii [21] A. Kami´nska and L. Maligranda, Order convexity and concavity in Lorentz spaces Λp(ω), 0 [22] A. Kami´nska, L. Maligranda and L.-E. Persson, Convexity, concavity, type and cotype Lorentz spaces, Indag. Math., 9 (1998), No. 3, 367– 382. [23] P. Khajeh-Khalili, Generalization of a Hardy-Littlewood-P´olya inequa- lity, J. Approx. Theory , 66 (1991), No. 2, 115–124. [24] S. A. Kirillov, Norm estimates of functions in Lorentz spaces, Acta Sci. Math. (Szeged), 65 (1999), No. 1-2, 189–201. [25] V. I. Kolyada, Some generalizations of the Hardy-Littlewood-P´olya the- orem, Mat. Zametki, 51 (1992), No. 3, 24–34. [In Russian]; translation in Math. Notes, 51 (1992), No. 3-4, 235–244. [26] A. Kopezhanova, Some new results concerning the Fourier coefficients in Lorentz type spaces, Research report 5, Department of Mathematics, Lule˚a University of Technology, (15 pages), 2010. [27] A. N. Kopezhanova, E. D. Nursultanov and L.-E. Persson, On summa- bility of the Fourier coefficients for functions from some Lorentz type spaces, Research Report 8, Department of Mathematics, Lule˚a Univer- sity of Technology, (27 pages), 2009. [28] A. Kopezhanova, E. Nursultanov and L.-E. Persson, Relations between summability of the Fourier coefficients in regular systems and functions from some Lorentz type spaces, Proc. Razmadze Math. Inst., 152 (2010), 73–88. [29] A. Kopezhanova and L.-E. Persson, On summability of the Fourier coeffi- cients in bounded orthonormal systems for functions from some Lorentz type spaces, Eurasian Mathematical Journal., 1(2010), No. 2, 76–85. [30] A. Kufner, L. Maligranda and L.-E. Persson, The Hardy Inequality. About its History and some Related Results, Vydavatelsky Servis Pub- lishing House, Pilsen, 2007. [31] A. Kufner and L. -E. Persson, Weighted Inequalities of Hardy Type, World Scientific Publ., New Jersey/ London/ Singapore/ Hong Kong, 2003. xxiv BIBLIOGRAPHY [32] L. Leindler, On integrability of functions defined by Fourier series, Acta Sci. Math. (Szeged), 74 (2008), No. 3-4, 565–580. [33] G. G. Lorentz, Some new functional spaces, Ann. of Math. (2), 51 (1950), 37–55. [34] S. F. Lukomskii, Convergence of Fourier series in Lorentz spaces, East J. Approx., 9 (2003), No. 2, 229–238. [35] J. Marcinkiewicz and A. Zygmund, Some theorems on orthogonal sys- tems, Fund. Math., 28 (1937), 309–335. [36] F. M´oricz, On double cosine, sine, and Walsh series with monotone co- efficients, Proc. Amer. Math. Soc., 109 (1990), No. 2, 417–425. [37] F. M´oricz, On Walsh series with coefficients tending monotonically to zero. Acta Math. Acad. Sci. Hungar., 38 (1981), No. 1–4, 183-189. [38] E. D. Nursultanov, On the coefficients of multiple Fourier series from Lp-spaces, Izv. Ross. Akad. Nauk Ser. Mat., 64(2000), No.1, 95–122. [In Russian]; translation in Izv. Math., 64 (2000), No. 1, 93–120. [39] E. D. Nursultanov, Network spaces and inequalities of Hardy-Littlewood type, Mat. Sb., 189 (1998), No. 3, 83–102. [In Russian] [40] R. E. Paley, Some theorems on orthonormal functions, Studia Math., 3 (1931), 226–238. [41] L. -E. Persson, An exact description of Lorentz spaces, Acta Sci. Math. (Szeged), 46 (1983), No. 1-4, 177–195. [42] L. -E. Persson, Interpolation with a parameter function, Math. Scand, 59 (1986), No. 2, 199–222. [43] L. -E. Persson, Relation between regularity of periodic functions and their Fourier series, Ph.D thesis, Dept. of Math., Ume˚a University, (1974). [44] L. -E. Persson, Relation between summability of functions and Fourier series, Acta Math. Acad. Sci. Hungar., 27 (1976), No. 3-4, 267–280. [45] L -E. Persson and N. Samko, Quasi-monotone weight functions and their applications, American Institute of Physics, to appear 2010. [46] L. -E. Persson and S. Samko, Some new Stein and Hardy type inequali- ties, J. Math. Sci., 169(2010), No. 1, 113–129. BIBLIOGRAPHY xxv [47] T. V. Rodionov, Analogues of the Hausdorff-Young and Hardy-Little- wood theorems, Izv. Ross. Akad. Nauk Ser. Mat., 65 (2001), No. 3, 175–192. [In Russian]; translation in Izv. Math., 65 (2001), No. 3, 589– 606. [48] B. V. Simonov, On trigonometric series with monotone coefficients in Lorentz spaces, Izv. Vyssh. Uchebn. Zaved. Mat., (1998), No. 5, 59–67. [In Russian]; translation in Russian Math. (Iz. VUZ), 42 (1998), No. 5, 57–65. [49] B. Simonov and S. Tikhonov, Norm inequalities in multidimensional Lorentz spaces, Math. Scand., 103 (2008), No. 2, 278–294. [50] E. M. Stein, Interpolation of linear operators, Trans. Amer. Math. Soc., 83 (1956), 482–492. [51] S. Tikhonov, On the integrability of trigonometric series, Mat. Zametki, 78 (2005), No. 3, 476–480. [In Russian]; translation in Math. Notes, 78 (2005), No. 3-4, 437–442. [52] S. Tikhonov, Trigonometric series with general monotone coefficients. J. Math. Anal. Appl., 326 (2007), No. 1, 721–735. [53] M. F. Timan and K. Tukhliev, Properties of certain orthonormal sys- tems, Izv. Vyssh. Uchebn. Zaved. Mat., (1983), No. 9, 65–73. [In Rus- sian] [54] F. Weisz, Hardy-Littlewood inequalities for Ciesielski – Fourier series, Anal. Math., 31 (2005), No. 3, 217–233. [55] A. Zygmund, Trigonometric series, Cambridge University Press, London – New York, Vols. I, II, 1968. Paper A EURASIAN MATHEMATICAL JOURNAL ISSN 2077-9879 Volume 1, Number 2 (2010), 76 – 85 ON SUMMABILITY OF THE FOURIER COEFFICIENTS IN BOUNDED ORTHONORMAL SYSTEMS FOR FUNCTIONS FROM SOME LORENTZ TYPE SPACES A.N. Kopezhanova and L.-E. Persson Communicated by V. Guliyev Key words: summability of Fourier series, inequalities, orthonormal bounded systems, Lorentz spaces. AMS Mathematics Subject Classification: 46E30, 42A16. Abstract. We denote by Λβ(λ),β>0, the Lorentz space equipped with the (quasi) norm 1 1 β β ∗ 1 dt f Λβ (λ) := f (t)tλ 0 t t for a function f on [0,1] and with λ positive and equipped with some addi- tional growth properties. Some estimates of this quantity and some corre- sponding sums of Fourier coefficients are proved for the case with a general orthonormal bounded system. 1 Introduction Let f be a measurable function on a measure space (Ω,μ), where μ is an additive positive measure. The distribution function m(σ, f) and the nonincreasing rearrangement f ∗ of a function f are defined as follows: m(σ, f):=μ {x ∈ Ω:|f(x)| >σ} , f ∗(t) := inf {σ : m(σ, f) ≤ t} . Let 1≤ p ≤∞and 0 1 ∞ q q − ∗ p 1 q ∞ f Lpq := t (f (t)) dt < . (1) 0 1 Note that for the case p = q the Lpq spaces coincide with the usual Lp spaces equipped with the norms f Lp (quasinorms for 0 Let the function f be periodic with period 1 and integrable on [0, 1] and { }∞ let Φ = ϕn n=1 be an orthonormal system on [0, 1]. The numbers 1 an = an(f)= f(x)ϕn(x)dx, n ∈ N, 0 are called the Fourier coefficients of the function f with respect to the system { }∞ Φ= ϕn n=1. We also remark that there are many relations between summability of Fourier coefficients and integrability of the corresponding functions e.g. the following two-sided ones for the trigonometrical system: ∞ p ≤ p−2| |p ≤ ∞ f Lp[0,1] c1 k ak , if 2 p< , (2) k=1 ∞ p ≥ p−2| |p ≤ f Lp[0,1] c2 k ak , if 1 ∞ Theorem 1. ∞ 2πikt + Let 0 <β< and Φ= e k=−∞ be a trigonometrical system. a) If there exists a positive number δ>0 satisfying such conditions like: 2 − − 1 − λ(t)t δ is an increasing function of t and λ(t)t ( 2 δ) is a decreasing function of t, then 1 ∞ 1 β (a∗ λ(n))β ≤ c f . n n 1 Λβ (λ) n=1 b) If there exists a positive number δ>0 satisfying such conditions like: − 1 − − λ(t)t 2 δ is an increasing function of t and λ(t)t 1+δ is a decreasing function of t, then 1 ∞ 1 β f ≤ c (a∗ λ(n))β , Λβ (λ) 2 n n n=1 { ∗ }∞ {| |}∞ where an n=1 is the nonincreasing rearrangement of the sequence ak k=1 of Fourier coefficients of f with respect to the system Φ. The aim of this paper is to derive some analogues of the inequalities (2) { }∞ and (3) both in the case of bounded orthonormal systems Φ = ϕn n=1 and for generalized Lorentz spaces of type Λβ(λ). Conventions. The letter c (c1,c2,etc.) means a constant which does not dependent on the involved functions and it can be different in different oc- curences. Moreover, for C, D > 0 the notation C ≈ D means that there exist positive constants a1 and a2 such that a1D ≤ C ≤ a2D. 2 The main result Let δ>0andλ(t) be a nonnegative function on [1, ∞) . We define the following classes (see also [3]): A = {λ(t):λ(t)t−δ is an increasing function and δ − 1 − λ(t)t ( 2 δ)is a decreasing function , − 1 − B = {λ(t):λ(t)t 2 δ is an increasing function and δ λ(t)t−1+δ is a decreasing function . Then the classes A and B are defined as follows: A = ∪δ>0Aδ,B= ∪δ>0Bδ. { }∞ In the sequel we denote by Φ= ϕn n=1 a bounded orthonormal system, i.e. |ϕn(t)|≤M, t ∈ [0, 1] ,n∈ N. Our result reads: 3 Theorem 2. Let 0 <β≤∞, and assume that the orthonormal system { }∞ Φ= ϕk k=1 is bounded. (a) If λ(t) belongs to the class A, then 1 ∞ 1 β (a∗ λ(n))β ≤ c f , (4) n n 1 Λβ (λ) n=1 { ∗ }∞ { }∞ where an n=1 is the nonincreasing rearrangement of the sequence ak k=1 of Fourier coefficients of f with respect to the system Φ. a.e. ∞ (b) If λ(t) belongs to the class B and f = n=1 anϕn, then 1 ∞ 1 β f ≤ c (a∗ λ(n)β . (5) Λβ (λ) 2 n n n=1 Here the constants c1 and c2 don’t depend on f. Proof. (a) Let λ(t) belongs to the class A. This means that there exists δ>0 − − 1 − such that: λ(t)t δ is an increasing function and λ(t)t ( 2 δ) is a decreasing function. Suppose that the function f satisfies the condition: 1 1 1 β dt β f ∗(t)tλ < ∞. 0 t t Let f = f0+f1, where f0 and f1 are defined later on. By using the inequalities ≤ a l2∞ c1 f L21 , ≤ a l∞ c2 f L1 and ∗ ≤ ∗ ∗ an(f) a n (f0)+a n (f1),n=1, 2, ..., [ 2 ] [ 2 ] ∗ we estimate an(f) from above as follows: 1 2 1 ∗ ≤ ∗ 2 n 2 ∗ ≤ an(f) a n (f0)+ a n (f1) [ 2 ] n 2 [ 2 ] 1 1 ∗ 1 − 1 ∗ ≤ 2 c3 f0 (t)dt + 1 t f1 (t)dt . 0 n 2 0