(ROZPRAWY MATEMATYCZNE) CCCLXXIV. On general Franklin systems. GEGHAM GEVORKYAN and ANNA KAMONT

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1 P O L S K A A K A D E M I A N A U K, I N S T Y T U T M A T E M A T Y C Z N Y D I S S E R T A T I O N E S M A T H E M A T I C A E (ROZPRAWY MATEMATYCZNE) KOMITET REDAKCYJNY ANDRZ EJ BIA LYNICKI- BI RULA, BOGDAN BOJARSKI, ZBIGNIEW CIESIELSKI, JERZY LOŚ, ZBIGN IEW SEMADENI, JERZY ZABCZ YK redaktor, WIES LAW ŻELAZKO zastȩpca redaktora CCCLXXIV GEGHAM GEVORKYAN and ANNA KAMONT On general Franklin systems W A R S Z A W A 1998

2 Gegham Gevorkyan Erevan State University Department of Mathematics Alex Manoukian St Erevan, Armenia ggg@instmath.sci.am Anna Kamont Instytut Matematyczny PAN ul. Abrahama Sopot, Poland ak@impan.gda.pl Published by the Institute of Mathematics, Polish Academy of Sciences Typeset using TEX at the Institute Printed and bound by Publishing House of the Warsaw University of Technology ul. Polna 5, -644 Warszawa P R I N T E D I N P O L A N D c Copyright by Instytut Matematyczny PAN, Warszawa 1998 ISSN

3 C O N T E N T S 1. Introduction Notation Definition and properties of general Franklin systems Piecewise linear functions Franklin functions Sequences of partitions and Franklin functions Regularity of sequences of partitions Sequences of partitions and general Haar systems Technical lemmas Franklin series in L p, 1 < p < Franklin series in L p, < p 1, and H p, 1/2 < p The necessity of strong regularity in H p, 1/2 < p Haar and Franklin series with identical coefficients Characterization of the spaces BMO and Lip(α), < α < References Abstract We study general Franklin systems, i.e. systems of orthonormal piecewise linear functions corresponding to quasi-dyadic sequences of partitions of [, 1]. The following problems are treated: unconditionality of the general Franklin basis in L p, 1 < p <, and H p, 1/2 < p 1; equivalent conditions for the unconditional convergence of the Franklin series in L p for < p 1; relation between Haar and Franklin series with identical coefficients; characterization of the spaces BMO and Lip(α), < α < 1, in terms of the Fourier Franklin coefficients Mathematics Subject Classification: Primary 42C1. Key words and phrases: Franklin system, unconditional basis, unconditional convergence, L p spaces, real Hardy spaces, Hölder classes, BMO space. Part of this work was done while G. Gevorkyan was visiting the Mathematical Institute of the Polish Academy of Sciences in Sopot. Received

4 1. Introduction The classical Franklin system, introduced by Ph. Franklin in 1928 ([16]), is a complete orthonormal system of continuous, piecewise linear functions (with dyadic knots), obtained by means of Gram Schmidt orthogonalization of Schauder functions. Since then, it has been studied by many authors from different points of view. The basic properties of this system, including exponential estimates for the Franklin functions and L p -stability on dyadic blocks, have been obtained by Z. Ciesielski in [5] and [6]. These properties have turned out to be an important tool in further investigations of the Franklin system. It is known that this system is a basis in C[, 1] and L p for 1 p < ; moreover, the coefficients of a function in the Franklin basis give a linear isomorphism between the space of functions satysfying the Hölder condition in L p norm with exponent α, < α < 1+1/p, 1 p, and the appropriate sequence space ([5], [6]). The unconditionality of this basis in L p, 1 < p <, has been proved by S. V. Bochkarev in [1]. P. Wojtaszczyk has obtained a characterization of the BMO space in terms of the coefficients of a function in the Franklin basis and has proved that this system is an unconditional basis in the real Hardy space H 1 ([29]; see [8] for a simplified proof). The unconditionality of the Franklin basis in real Hardy spaces H p, 1/2 < p 1, has been obtained by P. Sjölin and J. Strömberg ([27]); they have also proved that for this range of p, the H p quasi-norm of f H p is equivalent to the L p quasi-norm of the square function of the Franklin series with coefficients a n = (f, f n ). Z. Ciesielski and Sun-Yung A. Chang have proved that f H 1 iff its Fourier Franklin series is unconditionally convergent in L 1 (cf. [4]). The equivalence of the Franklin system with the Haar system and higher order orthonormal spline systems in L p and H p spaces has also beeen studied (see [7], [11], [26], [27]), and results concerning the boundedness of the translation operator are known as well (see for example [1], [17], [25]). One of the authors of this paper has studied the unconditional convergence of Franklin series in L p for < p 1 ([18] [21]). He has proved that the unconditional convergence of Franklin series in L p is equivalent to each of the following conditions: (i) the square function of the series is in L p, and (ii) the maximal function of the series is in L p. Moreover, the Franklin series converges unconditionally in L p iff the Haar series with identical coefficients converges unconditionally in L p. Analogous results concerning the convergence of Franklin series in Lorentz spaces can be found in the recent paper [22]. It should be mentioned that Franklin s construction has been later generalized to higher order spline functions, periodic splines, splines on R and for functions of several

5 6 G. Gevorkyan and A. Kamont variables and defined on smooth compact manifolds; here one should mention names like Z. Ciesielski, J. Domsta, T. Figiel, P. Oswald, J. Strömberg. In the present paper we study some of the above problems for general Franklin systems corresponding to quasi-dyadic sequences of partitions of the interval [, 1]. By a quasidyadic sequence of partitions we mean a sequence of partitions {P j : j } such that P = {, 1}, P j P j+1 and P j+1 is obtained form P j by adding 2 j new points (one new point between two consecutive points of P j ), and the corresponding Franklin system is a sequence of orthonormal piecewise linear functions with knots from the sequence of partitions P j (see Section 2.2 and Definition 2.1 in Section 2.3 for the precise formulation). It follows from [5] that if lim j P j = ( P j denotes the diameter of the partition P j ), then the corresponding Franklin system is a basis in C[, 1] and L p, 1 p <. The main results of the present paper are the following. We prove that if the sequence of partitions is weakly regular (see Definition 2.2 for the weak, strong and strong periodic regularity of a quasi-dyadic sequence of partitions), then the corresponding Franklin system is an unconditional basis in L p for 1 < p < (Theorem 3.1). Next we show that if the sequence of partitions is strongly regular, then the Franklin system is an unconditional basis in H p for 1/2 < p 1 (Theorem 4.2). Moreover, we prove that strong regularity of the sequence of partitions is a necessary condition for the corresponding Franklin system to be a basis in H p for 1/2 < p 1 (Theorems 5.1 and 5.3). The question of the unconditional convergence of the Franklin series in L p for < p 1 is studied as well. We prove (see Theorem 4.1) that the unconditional convergence of the Franklin series in L p is equivalent to each of the following conditions: (i) the square function of the series is in L p, and (ii) the maximal function of the series is in L p. For 1/2 < p 1, all these conditions are equivalent to the fact that the series under consideration is a Fourier Franklin series of some element of H p (see Theorem 4.2). Further, we compare the Franklin series and the Haar series with identical coefficients. We prove that, under suitable regularity of the sequence of partitions, the square functions of the Franklin and Haar series are equivalent in L p, < p < (Propositions 6.1, 6.2; the Haar system under consideration corresponds to the same sequence of partitions as the Franklin system for a detailed description see Section 2.4). As a consequence, under the assumption of strong periodic regularity of the sequence of partitions, we deduce that the Haar and Franklin systems are equivalent bases in L p, 1 < p <, while for < p 1 the Franklin series converges unconditionally in L p iff the Haar series with the same coefficients converges unconditionally in L p. Moreover, we get the boundedness of the associated translation operator in L p, 1 < p < (for both the Haar and Franklin systems), and in H p, 1/2 < p 1 (for the Franklin system) see Corollary 6.4. Finally, we obtain a characterization of the spaces BMO and Lip(α), < α < 1, in terms of the coefficients in the Franklin system corresponding to a strongly regular sequence of partitions Theorems 7.1 and 7.2; recall that the spaces BMO and Lip(α) are the dual spaces to H 1 and H p, 1/2 < p < 1, α = 1/p 1, respectively (cf. for example [13]). It should be mentioned that analogous properties of the Haar system corresponding to a quasi-dyadic sequence of partitions (i.e. the unconditionality of the Haar system in

6 On general Franklin systems 7 L p for 1 < p < and equivalence of the square and the maximal functions of a Haar series in L p for < p 1) follow from general results on martingale transforms (see [2], [3]). For the reader s convenience, we recall these properties of the Haar system in Section 2.4. The paper is organized as follows. In Section 2 we define the Franklin system corresponding to a quasi-dyadic sequence of partitions and summarize the properties of the Franklin and Haar systems needed for the purpose of this paper. In Section 3, we prove that the Franklin system corresponding to a weakly regular sequence of partitions is an unconditional basis in L p for 1 < p <. In Section 4, for the Franklin system corresponding to a strongly regular sequence of partitions, we discuss conditions equivalent to the unconditional convergence of the Franlin series in L p for < p 1, and the unconditionality of this system in H p, 1/2 < p 1, is proved. In Section 5 we prove that strong regularity of the quasi-dyadic sequence of partitions is a necessary condition for the corresponding Franklin system to be a basis in H p, 1/2 < p 1. In Section 6 the Franklin and Haar series with identical coefficients are discussed and results concerning the translation operators are formulated. Finally, in Section 7 we give a characterization of the spaces BMO and Lip(α) with < α < 1 in terms of the Fourier Franklin coefficients of a function again for the Franklin system corresponding to a strongly regular sequence of partitions Notation Function spaces and H p spaces. For the reader s convenience, we recall the definitions of the spaces we work with. By L p, < p <, we Ì denote the Lebesgue space of real-valued functions defined on 1 [, 1] for which f p = ( f(u) p du) 1/p <. If < p < 1, then f p is not a norm, but then the space L p is equipped with the metric (f, g) = f g p p. Recall that L p with this metric is a complete space. By C[, 1] we denote the space of continuous functions on [, 1], and for < α < 1, by Lip(α) C[, 1] we mean the subspace of functions satisfying the Hölder condition with exponent α. It is well known that Lip(α), with the norm f(x) f(y) (1.1) f Lip(α) = f + sup x,y 1 x y α, is a non-separable Banach space. We need also the BMO space, i.e. the space of functions of bounded mean oscillation. If f L 1, then f BMO iff (1.2) f BMO = (f, 1) + sup Γ ( 1 Γ Γ f(u) f Γ 2 du) 1/2 <. where the supremum is taken over all subintervals Γ [, 1] and f Γ = 1 Γ f(v)dv ( Γ Γ denotes the length of the inerval Γ); for equivalent definitions of the BMO space we refer to [13]. It is known that BMO is a non-separable Banach space. Next, we recall the definition of real Hardy spaces on [, 1], denoted by H p, 1/2 < p 1. We use the atomic definition, introduced in [12], and developed in [13]; for more details, we refer to [13]. Ì

7 8 G. Gevorkyan and A. Kamont First, recall the definition of p-atoms: a function a : [, 1] R is called a p-atom (1/2 < p 1) iff either Ì a = 1, or there is an interval Γ [, 1] such that suppa Γ, sup a Γ 1/p 1 and a(u)du = ; note that if a is a p-atom, then a p 1. For p = 1, a function f L 1 is said to belong to H 1 iff there are 1-atoms a j and real coefficients c j, j N, with j=1 c j <, such that f = j=1 c ja j. The norm in H 1 is defined as f H 1 = inf( j=1 c j ), where the infimum is taken over all atomic decompositions of f; H 1 with this norm is a Banach space. The space H p with 1/2 < p < 1 is defined as a subspace of the dual of Lip(α) with α = 1/p 1: f (Lip(α)) is said to belong to H p if it admits an atomic decomposition f = j=1 c ja j, where a j are p-atoms and the real coefficients c j satisfy j=1 c j p < ; it should be noted that this condition implies the convergence of the series j=1 c ja j in the norm of (Lip(α)). For f H p we put f H p = inf( n c n p ) 1/p, with the infimum taken over all atomic decompositions of f. For p < 1, H p is not a norm, but (f, g) = f g p H is a metric on p Hp, and H p with this metric is complete; thus, (H p, p Hp) is a Fréchet space. Moreover, a linear functional L on H p is continuous iff there is a constant C L such that Lf C L f H p for all f H p ; similarly, a linear operator T : H p H p is continuous iff it is bounded, i.e. there is a constant C T such that Tf H p C T f H p for all f H p. The spaces BMO and Lip(α) are identified with the duals of H 1 and H p, α = 1/p 1, respectively; cf. [13], Theorem B. More precisely: if g BMO, f H 1 and f = j=1 c ja j is an atomic decomposition of f, then the formula L g (f) = lim n n j=1 c j 1 g(u)a j (u)du defines a continuous linear functional on H 1, and each continuous linear functional on H 1 is of this form; moreover, the norm of L g in (H 1 ) is equivalent to g BMO. The dual of H p, 1/2 < p < 1, is identified with Lip(α), where α = 1/p 1: if g Lip(α), f H p and f = j=1 c ja j is an atomic decomposition of f, then the formula 1 L g (f) = c j g(u)a j (u)du j=1 defines a continuous linear functional on H p, and each continuous linear functional on H p is of this form; moreover, the norm of L g in (H p ) is equivalent to g Lip(α). To shorten the notation, if f H 1 and g BMO, or f H p and g Lip(α) with α = 1/p 1, we denote by (f, g) the value of the functional L g on f. In Sections 3 and 4, the unconditional convergence of Franklin series in spaces L p, 1 < p < and H p, 1/2 < p 1, is studied. The unconditional convergence of a series n=1 x n in a metric space (X, ) means that for each permutation σ of N, the series n=1 x σ(n) is convergent in (X, ). It is known that if (X, ) is a complete linear metric space, then the series n=1 x n is unconditionally convergent if and only if the series n=1 ε nx n converges in (X, ) for each choice of the coefficients ε n { 1, 1} (cf. for example [24], Theorem 1 in Chapter 1).

8 On general Franklin systems 9 Though H p is not a norm for 1/2 < p < 1, we use for the H p spaces the same terminology as for Banach spaces; in particular, by a basis in H p we mean a sequence of elements y n H p, n N, such that for each f H p there is a unique sequence of coefficients b n (f) such that f = n=1 b n(f)y n, with the series convergent in the metric p H p, and a basis is called unconditional if for each f Hp, the series n=1 b n(f)y n is unconditionally convergent in H p. Quasi-dyadic sequences of partitions. Let {P j : j } be a quasi-dyadic sequence of partitions of [, 1]. By this we mean that P = {, 1}, and P j = {t j,i : i 2 j }, P j P j+1 for j, = t j, <... < t j,2 j = 1, t j+1,2k = t j,k for all j and k =,...,2 j, i.e. P j+1 is obtained from P j by adding one point in each interval (t j,k 1, t j,k ), k = 1,...,2 j. For j and 1 k 2 j, we put I j,k = [t j,k 1, t j,k ], and I j,k = (t j,k 1, t j,k ) is the interior of I j,k. Moreover, we let I j = {I j,k : 1 k 2 j } and I = j I j. The elements of I j are called intervals of rank (or order) j. Maximal and square functions. For f L 1, M(f, ) denotes the Hardy Littlewood maximal function of f over [, 1], and M (f, ) is the maximal function corresponding to the sequence of quasi-dyadic partitions {P j : j }, i.e. M (f, x) = sup I x I I 1 I I f(u) du. It is well known that the operator M is of type (p, p) for p > 1 and of weak type (1, 1) (cf. for example Theorem in [28]). Clearly, M has the same properties. For a given sequence of quasi-dyadic partitions {P j : j }, the corresponding Franklin system, as introduced in Definition 2.1 below, is denoted by {f n : n }. For a sequence of real numbers (a n ) n, the square function P and the maximal function S of the Franklin series with coefficients (a n ) n are defined by the respective formulae ( P( ) = a 2 nf n ( ) 2) 1/2 m and S( ) = sup a n f n ( ). n= Moreover, for f L p, 1 p, or f H p, 1/2 < p 1, we denote by Pf and Sf the square function and the maximal function of the Franklin series with coefficients a n = (f, f n ), i.e. a n are the Fourier coefficients of f with respect to the Franklin system corresponding to {P j : j } (note that the f n s are Lipschitz functions, and therefore they define continuous linear functionals on H p, 1/2 < p 1). Abbreviations. To shorten the notation, we use the following abbreviations. For a, b R, we put a b = max(a, b), a b = min(a, b). We write A B if there are positive constants C 1, C 2 such that C 1 A B C 2 B. The letters C, C p, C γ,p etc. denote various constants, the value of which may vary from line to line; the subscripts indicate the parameters on which the particular constant depends. m n=

9 1 G. Gevorkyan and A. Kamont By χ A we denote the indicator of a set A [, 1], A c is the complement of A in [, 1] and A denotes the Lebesgue measure of A. 2. Definition and properties of general Franklin systems 2.1. Piecewise linear functions. We start with recalling some known facts concerning piecewise linear functions, which are needed for the purpose of this paper. Let π = {t i : i n} be a partition of [, 1], = t <... < t n = 1; for later convenience, we also put t 1 = and t n+1 = 1, λ i = t i t i 1 for i n + 1, ν i = λ i + λ i+1 for i n. 2 Let S π be the space of piecewise linear, continuous functions on [, 1] with knots π. Moreover, let N i, i n, be the B-splines of order 2 corresponding to the partition π, i.e. N i is the unique function from S π satisfying N i (t j ) = δ i,j. Note that suppn i = [t i 1, t i+1 ], n i= N i(t) = 1 for each t [, 1], N i 1 = ν i and (λ i + λ i+1 )/3 for i = j, λ (2.1) (N i, N j ) = i+1 /6 for i = j 1, λ i /6 for i = j + 1, for i j > 1. Moreover, any function f S π can be written in the form f = n i= a in i, where a i = f(t i ), so the functions {N i : i n} are a basis in S π. Let G π = [(N i, N j ) : i, j n] be the Gram matrix of the system {N i : i n}, and define Gπ 1 = A π = [a i,j : i, j n]. In Proposition 2.1 we list some estimates for a i,j, which are needed later on. Proposition 2.1. Let π={t i : i n} be a partition of [, 1], and let A π = [a i,j : i, j n] be the inverse of the Gram matrix G π defined above. Then the entries of the matrix A π satisfy the following conditions: (2.2) (2.3) (2.4) (2.5) (2.6) Moreover, 3 2 a i,i ν i 2 for i n, a i,j = a j,i and a i,j = ( 1) i+j a i,j for i, j n, 2 a i 1,j a i,j for < i j n, 2 a i+1,j a i,j for j i < n, a i,j 2 2 1, i j n. i j max i k j ν k (2.7) (2.8) a i,j ( 3 2 λ ) i + 2λ i+1 ai+1,j λ i+1 2 a i,j (λ i + λ i+1 ) for i < j, a i,j ( 2λ i λ i+1) ai 1,j λ i 2 a i,j (λ i + λ i+1 ) for j < i n. Proof. Properties (2.2) (2.6) can be found for example in [9] and [24], or they are straightforward consequences of estimates given there, so their proof is omitted.

10 To check (2.7), note that for i < j, This together with (2.3) gives On general Franklin systems 11 λ i a i 1,j + 2(λ i + λ i+1 )a i,j + λ i+1 a i+1,j =. 2(λ i + λ i+1 ) a i,j = λ i+1 a i+1,j + λ i a i 1,j λ i+1 a i+1,j. On the other hand, applying (2.4) we get 2(λ i + λ i+1 ) a i,j = λ i+1 a i+1,j + λ i a i 1,j λ i+1 a i+1,j λ i a i,j, which gives the remaining inequality in (2.7). Inequalities (2.8) are obtained analogously. It should be noted that, for fixed j, formulae (2.4), (2.5) and the fact that a i,j and a i+1,j have opposite signs follow just from the system of equations n i= a i,j(n i, N k ) = δ j,k, k =,...,n, and these properties are sufficient to get (2.7) and (2.8). We refer to this fact in the proof of Lemma 5.2. In the sequel, we need the L p -stability of the functions N i, which can be checked by straightforward calculation: Proposition 2.2. Let π be a partition of [, 1], f S π, f = n i= a in i. Then for all 1 p, ( ) 1/p 1 ( n ) 1/p ( n ) 1/p. a i p ν i f p a i p ν i p + 1 i= 2.2. Franklin functions. Let (π, π) be a pair of partitions of [, 1] such that π π and π is obtained from π by adding one knot τ, τ, 1. Then there is a unique, up to sign, function ϕ S π such that ϕ S π (in L 2 ) and ϕ 2 = 1; the sign of ϕ is chosen in such a way that ϕ(τ) >. The function ϕ is called the Franklin function corresponding to the pair of partitions (π, π). Let us formulate some properties of the Franklin function. Let π = {t i : i n}; as τ π, we have τ = t k for some < k < n. Then π = {t i : i n, i k}, and for convenience we denote by Ñi, i k, the B-splines corresponding to π. Observe that Ñ i = N i for i < k 1 and i > k + 1, Ñ k 1 = N k 1 + λ k+1 λ k N k and Ñ k+1 = N k+1 + N k. λ k + λ k+1 λ k + λ k+1 Define (2.9) w i = λ k+1 λ k a i,k 1 + a i,k a i,k+1, λ k + λ k+1 λ k + λ k+1 where A π = [a i,j : i, j n] is the inverse of the Gram matrix G π, and introduce the function n (2.1) g = w i N i. i= i=

11 12 G. Gevorkyan and A. Kamont Clearly, g S π, and it can be checked by straightforward calculation that (g, Ñi) = for all i k, whence g S π. Formula (2.9) and properties of a i,j from Proposition 2.1 imply the following: (2.11) (2.12) (2.13) w i = λ k+1 λ k a i,k 1 + a i,k + a i,k+1, λ k + λ k+1 λ k + λ k+1 w i = ( 1) k i w i, so in particular g(τ) = w k >, a k,k w k 3 2 a k,k, 1 2 a k+l,k+l w k+l 3 2 a k+l,k+l for l = ±1 (to check (2.13), note that for j k), which gives (2.14) 3 4 a j,k = λ k a j,k 1 + λ k+1 a j,k+1 2(λ k + λ k+1 ) 1 ν i w i 3 1 ν i for i = k 1, k, k + 1. Thus, we have ϕ = g/ g 2. Moreover, these formulae for w i (cf. (2.11), (2.14)), decay of a i,j from Proposition 2.1 (cf. (2.3) (2.6)) and L p stability of B-splines from Proposition 2.2 imply (2.15) 1 8 µ1 1/p g p 15µ 1 1/p for 1 p, where µ = 1/ν k 1 + 1/ν k + 1/ν k+1. As a consequence of formulae (2.9) (2.15) and Proposition 2.1, we get the following pointwise estimates for the Franklin function ϕ: Proposition 2.3 (Pointwise estimates for the Franklin function). Let (π, π) be a pair of partitions as above, and let ϕ be the Franklin function corresponding to (π, π). Define ξ i = ϕ(t i ), i.e. ϕ = n i= ξ in i, and µ = 1/ν k 1 + 1/ν k + 1/ν k+1. Then for i k 1, and for i k + 1, 1 12 µ1/2 1/p ϕ p 12µ 1/2 1/p for 1 p, ξ i = ( 1) i+k ξ i, i =,..., n, 1 µ 1/2 2 ν i ξ i 24 µ 1/2 ν i, i = k 1, k, k + 1; ξ i 1 ξ i 2, ξ i 48 2 i k µ 1/2 max i l k 1 ν l 96 i k 2 i k µ 1/2 t k t i 1, ξ i 1 ( 3 2 λ i 1 + 2λ i ) ξi λ i 2 ξ i 1 (λ i 1 + λ i ); ξ i+1 ξ i 2, ξ i 48 2 i k µ 1/2 max k+1 l i ν l 96 i k 2 i k µ 1/2 t i+1 t k, ξ i+1 ( 2λ i λ i+2) ξi λ i+1 2 ξ i+1 (λ i+1 + λ i+2 ). The pointwise estimates from Proposition 2.3 imply the following decay of norms of the Franklin function on the intervals from the partition π:

12 On general Franklin systems 13 Proposition 2.4 (Decay of norms of the Franklin function on intervals). Let (π, π) be a pair of partitions as above, and let ϕ be the Franklin function corresponding to (π, π). Then for i < k 1 we have and for i > k + 1, 3( 2 1) t i t i 1 ϕ(u) du t i+1 t i ϕ(u) du, 2 max ϕ(t) max ϕ(t), t i 1 t t i t i t t i+1 3( 2 1) t i+1 t i ϕ(u) du t i t i 1 ϕ(u) du, 2 max ϕ(t) max ϕ(t). t i t t i+1 t i 1 t t i Proof. Consider the case i < k 1. By Proposition 2.3, ξ i = ϕ(t i ) and ξ i+1 = ϕ(t i+1 ) have opposite signs and ξ i 1 2 ξ i+1, which implies and m i+1 = t i+1 ϕ(u) du = λ i+1 2 t i t i ξ i 2 + ξ i+1 2 ξ i + ξ i+1 m i = ϕ(u) du λ i ξ i 2. t i 1 Therefore, using the estimates from Proposition 2.3 we get ( 2 1)λ i+1 ξ i+1 m i+1 2( 2 1) λ i+1 ξ i+1 ( 2 1) 3λ i + 4λ i+1 3( 2 1). m i λ i ξ i λ i The bound for max ti 1 t t i ϕ(t) is a straightforward consequence of Proposition 2.3. The case i > k + 1 is treated analogously. Remark. The constants appearing in Propositions 2.3 and 2.4 are not sharp. Moreover, estimates analogous to those from Proposition 2.4 (i.e. with constants independent of the pair of partitions (π, π)) can be obtained for integrals of ϕ( ) p for 1 < p <. However, for < p < 1, estimates of this type do not hold Sequences of partitions and Franklin functions. Let {P j : j } be a quasi-dyadic sequence of partitions, with P j = {t j,i : i 2 j }. Define π 1 = P, and for n 2, n = 2 j + k with 1 k 2 j, (2.16) (2.17) with {} = {1} = [, 1], π n = P j {t j+1,2l 1 : 1 l k}, t n = t j+1,2k 1, {n} = [t j+1,2k 2, t j+1,2k ] = [t j,k 1, t j,k ], (2.18) {n } = [t j+1,2k 3, t j+1,2k 1 ], {n + } = [t j+1,2k 1, t j+1,2k+2 ], where for convenience we put t j, 1 = and t j,2j +1 = 1.

13 14 G. Gevorkyan and A. Kamont Note that π n is obtained from π n 1 (n 2) by adding exactly one point t n. The Franklin system corresponding to the quasi-dyadic sequence of partitions {P j : j } is defined as follows: Definition 2.1. Let {P j : j } be a quasi-dyadic sequence of partitions, and for n 1, let π n be as in (2.16). Then the Franklin system {f n : n } corresponding to {P j : j } is the following family of functions: f = 1, f 1 (t) = 2 3(t 1/2), and for n 2, f n is the Franklin function corresponding to (π n, π n 1 ). Note that this definition guarantees f n 2 = 1 and (f n, f m ) = for n m. For a partition π, denote by Q π the orthogonal (in L 2 ) projection onto S π. Note that Q π is simultaneously a continuous linear operator on L p, 2 p, and can be uniquely extended to a continous linear operator on L p, 1 p < 2, and H p, 1/2 < p 1; these extensions are denoted by Q π as well. Clearly, n Q πn f = (f, f i )f i. i= Next, we list the properties of the projections Q π which are needed for our purpose. For the proofs, we refer to [5] and [9]. Theorem 2.5. (i) For any partition π and f L p, 1 p, Moreover, for each f L 1 we have Q π f p 3 f p. Q π f( ) 64 M(f, ). (ii) Let {P j : j } be a quasi-dyadic sequence of partitions satisfying lim n π n =. Then for all f L p with 1 p <, or f C[, 1] for p = we have lim n f Q πn f p =. Consequently, the Franklin system {f n : n } corresponding to {P j : j } is a basis in L p, 1 p <, and C[, 1]. Moreover, if f L 1 and u is a weak Lebesgue point of f, then f(u) = lim n Q πn f(u) Regularity of sequences of partitions. Recall that for a quasi-dyadic sequence of partitions {P j : j }, I j,k = [t j,k 1, t j,k ], λ j,k = I j,k = t j,k t j,k 1, k = 1,...,2 j. When we pass from P j to P j+1, then the interval I j,k is split into two intervals I j+1,2k 1 and I j+1,2k with disjoint interiors, i.e. we have I j,k = I j+1,2k 1 I j+1,2k, λ j,k = λ j+1,2k 1 + λ j+1,2k. Now, we introduce the weak, strong and strong periodic regularity of a quasi-dyadic sequence of partitions. Definition 2.2. Let γ 1 and let {P j : j } be a quasi-dyadic sequence of partitions of [, 1].

14 On general Franklin systems 15 (i) We say that the sequence {P j : j } satisfies the weak regularity condition with parameter γ if for all j 1 and k = 1,...,2 j 1, 1 γ λ j,2k 1 λ j,2k γ. (ii) We say that the sequence {P j : j } satisfies the strong regularity condition with parameter γ if for all j and k = 1,...,2 j 1, 1 γ λ j,k+1 λ j,k γ. (iii) We say that the sequence {P j : j } satisfies the strong periodic regularity condition with parameter γ if for all j and k = 1,...,2 j, where by definition λ j,2 j +1 = λ j,1. 1 γ λ j,k+1 λ j,k γ, Clearly, the sequence P j = {k/2 j : k 2 j } of dyadic partitions satisfies the strong periodic regularity condition with γ = 1. Another example of a strongly periodically regular quasi-dyadic sequence of partitions is the sequence of Chebyshev knots on [, 1], i.e. with t j,k = (1 + cos((2 j k)π/2 j ) ) /2 = sin 2 (kπ/2 j+1 ). The best approximation by spline functions with these knots appears to be closely related with the Ditzian Totik modulus of smoothness with the step-weight function w(x) = x(1 x) and the best approximation by algebraic polynomials (see [23] for details and more examples). In the sequel, the following estimates for the length of the intervals I j,k are used frequently: Proposition 2.6. partitions of [, 1]. Let γ 1 and let {P j : j } be a quasi-dyadic sequence of (i) Let the sequence {P j : j } satisfy the weak regularity condition with parameter γ. Then for all j and k = 1,...,2 j, 1 I j,k I j+1,2k 1, I j+1,2k γ I j,k. Consequently, if for some j, k, m, l we have I m,l I j,k, then ( ) m j ( ) m j 1 γ I j,k I m,l I j,k. (ii) Let the sequence {P j : j } satisfy the strong regularity condition with parameter γ and α γ = log 2 γ. Then for all j and 1 k, l 2 j, γ 2 ( k l + 1) αγ I j,k I j,l γ 2 ( k l + 1) αγ I j,k. Proof. The inequalities from (i) are straightforward consequences of Definition 2.2(i). To check (ii), let 1 k, l 2 j, k l, and choose µ, µ j 1, such that 2 µ k l < 2 µ+1. Let a, b be such that I j,k I j µ,a and I j,l I j µ,b.

15 16 G. Gevorkyan and A. Kamont Then a b 2, and by strong regularity γ 2 I j µ,b I j µ,a γ 2 I j µ,b, so applying (i) we obtain ( ) µ ( ) µ γ γ I j,l I j µ,b γ 2 I j µ,a γ 2+µ I j,k, and we get (ii) by the choice of µ Sequences of partitions and general Haar systems. For a partition π = {t i : i m} of [, 1], let H π be the space of functions constant on each interval [t i 1, t i ), 1 i m, and continuous at 1. For a pair of partitions (π, π) such that π is obtained from π by adding one point, there is a unique (up to sign) function h H π with h H π and h 2 = 1; it is called the Haar function corresponding to (π, π). Now, let {P j : j } be a quasi-dyadic sequence of partitions of [, 1], and let the partitions π n, n 1, be as defined in (2.16). The Haar system {h n : n 1} corresponding to {P j : j } is defined as follows: h 1 = 1, and for n 2, h n is the Haar function corresponding to (π n, π n 1 ). It can be calculated that for n = 2 j + k, λ j+1,2k 1 for u [t j+1,2k 2, t j+1,2k 1 ), λ j+1,2k 1 λj,k (2.19) h n (u) = otherwise, λ j+1,2k 1 λ j+1,2k 1 λj,k for u [t j+1,2k 1, t j+1,2k ), and for 1 p, 1 (2.2) (λ j+1,2k 1 λ j+1,2k ) 1/p 1/2 h n p 2(λ j+1,2k 1 λ j+1,2k ) 1/p 1/2. 2 Therefore, if the sequence {P j : j } of partitions is weakly regular with parameter γ, then for 1 p, 1 (2.21) {n} 1/p 1/2 h n p γ {n} 1/p 1/2 2(1 + γ) and moreover, (2.22) 1 γ {n} 1/2 h n (u) γ {n} 1/2 on {n}. Consider a quasi-dyadic sequence {P j : j } of partitions and the corresponding Haar system {h n : n 1}. Note that for any sequence (a n ) n 1 of real coefficients, the sequences {Sm H : m 1} and {S2 H : j }, where S H j m = m n=1 a nh n, are martingales with respect to the σ-fields generated by the appropriate Haar functions. Clearly, if a n = (f, h n ) for all n N and some f L p, 1 p <, then the Sm s H are partial sums of the Fourier Haar series of f, and for f L 2, Sm H is the orthogonal projection of f onto the space spanned by h 1,...,h m. Therefore, the results concerning the unconditional convergence of the Haar series follow from known results from martingale theory. The properties of Haar series which are needed later on (cf. Section 6) are summarized in Propositions 2.7 and 2.8. To formulate these propositions, we introduce the following

16 On general Franklin systems 17 notation: for a sequence (a n ) n 1 of real numbers, P H and S H denote the square and maximal functions of the corresponding Haar series, i.e. ( P H ( ) = a 2 n h n( ) 2) 1/2, S H m ( ) = sup a n h n ( ). n=1 Moreover, for f L 1, we denote by P H f, S H f the functions defined by the above formulae with the coefficients a n = (f, h n ). If {P j : j } is a quasi-dyadic sequence of partitions of [, 1] such that P j as j, then the corresponding Haar system is a basis in L p for all 1 p <. Combining this with D. L. Burkholder s result concerning martingales (cf. [2], Theorem 9), and Doob s inequality for submartingales (cf. for example [14], Theorem 3.4 in Chapter VII), we have Proposition 2.7. Let {P j : j } be a quasi-dyadic sequence of partitions of [, 1] such that P j as j, and let {h n : n 1} be the corresponding Haar system. Then {h n : n 1} is a basis in L p for all 1 p <. This basis is unconditional in each L p for 1 < p <, and for each p, 1 < p <, and f L p, with implied constants depending on p only. m 1 f p P H f p S H f p, Under the additional assumption of weak regularity of the sequence of partitions under consideration, applying Theorem 5.1 from [3], we get Proposition 2.8. Let {P j : j } be a quasi-dyadic sequence of partitions of [, 1] satisfying the weak regularity condition with parameter γ, and let {h n : n 1} be the corresponding Haar system. Then, for each sequence (a n ) n 1 of real coefficients and p, < p 1, the following conditions are equivalent: (1) P H ( ) L p, (2) S H ( ) L p, (3) the series n=1 a nh n converges unconditionally in L p Technical lemmas. For later reference, we present the formulation of Propositions 2.3 and 2.4 for sequences of Franklin functions. Proposition 2.9. Let {P j : j } be a quasi-dyadic sequence of partitions satisfying the weak regularity condition with parameter γ and let {f n : n } be the corresponding Franklin system. For n 2, n = 2 j + k with 1 k 2 j, let t n and {n} be as in (2.17). Then there is a constant C γ, depending only on γ, such that (2.23) (2.24) (2.25) n=1 1 C γ {n} 1/p 1/2 f n p C γ {n} 1/p 1/2 for 1 p, 1 C γ {n} 1/2 f n (t n ) C γ {n} 1/2, f n (t j+1,2k 2 ) C γ {n} 1/2, f n (t j+1,2k ) C γ {n} 1/2 ;

17 18 G. Gevorkyan and A. Kamont for i 2k 2, (2.26) f n (t j+1,i ) = ( 1) 2k 1 i f n (t j+1,i ), f n (t j+1,i 1 ) 1 2 f n(t j+1,i ), (2.27) f n (t j+1,i ) C γ {n} 1/2 2 2k 1 i C γ {n} 1/2 2 k i/2, (2.28) f n (t j+1,i ) C γ 2k 1 i 2 2k 1 i {n} 1/2 t j+1,2k 1 t j+1,i 1, (2.29) t j+1,i f n (u) du t j+1,i t j+1,i 1 f n (u) du and for i 2k, i = 2l, (3 2 3) i 2k+2 f n 1 (2.3) (2.31) (2.32) (2.33) f n (t j+1,i ) = ( 1) k 1 i/2 f n (t j+1,i ), f n (t j+1,i+2 ) 1 2 f n(t j+1,i ), f n (t j+1,i ) C γ {n} 1/2 2 k i/2, k i/2 1 {n} 1/2 f n (t j+1,i ) C γ, 2 k i/2 t j+1,i+2 t j+1,2k 1 1 f n (u) du 3 t j+1,i f n (u) du 2 4 t j+1,i (3 2 3) k i/2 f n 1. Proof. By definition,f n is the Franklin function corresponding to (π n, π n 1 ). Denote by µ n the number µ from Proposition 2.3 chosen for π = π n and π = π n 1 ; then we have µ n 2 = 1 {n } + 1 {n} + 1 {n + }, t j+1,i with {n } and {n + } given by (2.18). Note that formulae (2.18) and the definition of weak regularity imply {n } {n} /() and {n + } {n} /(). Thus 2 {n} µ n 4γ + 6 {n}, and now Proposition 2.9 is a consequence of Propositions 2.3 and 2.4. Lemma 2.1. Let {P j : j } be a quasi-dyadic sequence of partitions. Let n = 2 j +k, m = 2 j + l with 1 k < l 2 j. Then there are two constants α, β, depending on n and m, such that f n (u) = αf m (u) for u t j+1,2k 2 and f n (u) = βf m (u) for u t j+1,2l. Proof. By definition, f n S πn, f n S πn 1 and f m S πm, f m S πm 1. Denote N n = {i : t j+1,i π n }, N m = {i : t j+1,i π m }, and let N n,i, i N n, and N m,i, i N m, be the corresponding B-splines. Thus, f n = i N n a i N n,i with a i = f n (t j+1,i ) and f m = i N m b i N m,i with b i = f m (t j+1,i ). Consider the functions f n, f m on [, t j+1,2k 2 ].

18 On general Franklin systems 19 Clearly, for i 2k 3 we have N n,i = N m,i on [, 1], and in addition N n,2k 2 = N m,2k 2 on [, t j+1,2k 2 ]; moreover, the functions N n,i = N m,i with i 2k 3 belong to both π n 1 and π m 1. Therefore, formula (2.1) for inner products of B-splines and the orthogonality conditions imply that both (a i ) i 2k 2 and (b i ) i 2k 2 satisfy the following system of equations: { 2x + x 1 =, λ j+1,i x i 1 + 2(λ j+1,i + λ j+1,i+1 )x i + λ j+1,i+1 x i+1 = for 1 i 2k 3. Since this is a system of 2k 2 equations with 2k 1 variables, the dimension of the space of its solutions is 1. Since both (a i ) i 2k 2 and (b i ) i 2k 2 are non-zero, this implies that there is a constant α such that a i = αb i for all i 2k 2. This property and the representation of f n and f m imply that f n = αf m on [, t j+1,2k 2 ]. The existence of a constant β such that f n = βf m on [t j+1,2l, 1] follows by analogous arguments. Lemma Let the quasi-dyadic sequence of partitions {P j : j } satisfy the strong regularity condition with parameter γ and let {f n : n } be the corresponding Franklin system. Let < p 1. Then there is a constant C γ,p such that for I I, (2.34) {n} p/2 f n (u) p du C γ,p I, and moreover, for all n, (2.35) I c {n} I 1 f n (u) p du C γ,p {n} 1 p/2. Proof. To prove (2.34), let I I j. Note that if {n} I then n = 2 j + m with j j and {n} = [t j,m 1, t j,m ] (cf. (2.17)). Let I = I j,k = [t j,k 1, t j,k] = [t j,2 j j (k 1), t j,2 j j k ]. Thus, {n} I means that 2 j j (k 1) < m 2 j j k, so we get for u I j,l with l > 2 j j k (cf. the decay of f n Proposition 2.9, formulae (2.3) and (2.31)) {n} p/2 f n (u) p C γ,p 2 p m l 2 j <n 2 j+1 {n} I 2 j j (k 1)<m 2 j j k C γ,p 2 p 2j j k l. As the sequence of partitions is strongly regular, applying the above inequality and Proposition 2.6(ii), we obtain 1 t j,k 2 j <n 2 j+1 {n} I j 2 {n} p/2 f n (u) p du C γ,p l=2 j j k+1 C γ,p I j,2 j j k I j,l 2 p 2j j k l 2 j l=2 j j k+1 ( ) j j γ C γ,p I, (l 2 j j k) αγ 2 p 2j j k l

19 2 G. Gevorkyan and A. Kamont which gives 1 t j,k n I {n} p/2 f n (u) p du By analogous arguments we obtain t j,k 1 {n} I 1 j=j t j,k C γ,p I 2 j <n 2 j+1 {n} I j=j ( γ {n} p/2 f n (u) p du C γ,p I, {n} p/2 f n (u) p du ) j j C γ,p I. which implies inequality (2.34). Inequality (2.35) follows from (2.34) (with I = {n}) and the fact that {n} f n (u) p du C γ {n} 1 p/2, which in turn is an immediate consequence of the estimate for the supremum norm of the Franklin function (cf. Proposition 2.9, formula (2.23)). Next, we formulate some technical estimates which are used frequently; their proofs are elementary and therefore the details are omitted. Proposition Let {P j : j } be a quasi-dyadic sequence of partitions satisfying the strong regularity condition with parameter γ and let {f n : n } be the corresponding Franklin system. For n 2, n = 2 j +k with 1 k 2 j, let {n} and t n = t j+1,2k 1 be as in (2.17). Then there is a constant C γ, depending only on γ, such that (2.36) 1 C γ {n} 1/2 f n (t j+1,2k 2 ), f n (t j+1,2k 1 ), f n (t j+1,2k ) C γ {n} 1/2. Moreover, let < α 1. Then there is a positive constant C γ,α, depending only on γ and α, such that for all n and A {n} with A α {n}, A f 2 n (u)du C γ,α. Proof. To get (2.36), apply the lower estimates for the values f n (t j+1,2k 2 ) and f n (t j+1,2k ) from Proposition 2.3 and strong regularity of the sequence of partitions. The remaining part of Proposition 2.12 is a straightforward consequence of (2.36). Proposition Let < α 1. Then there is a positive constant C α, depending only on α, such that for any interval [a, b], A [a, b] with A α(b a) and a function f linear on [a, b], max f(u) C α max f(u) and f(u) du C α f(u) du. u [a,b] u A [a,b] A

20 On general Franklin systems 21 Proposition Let the quasi-dyadic sequence {P j : j } of partitions satisfy the weak regularity condition with parameter γ, E [, 1] and B = {u [, 1] : M (χ E, u) > 1/(2γ + 2)}. Let I I, and let I, I + be the intervals in I obtained by splitting I. If I B then I E c 1 2γ I and I + E c 1 2γ I+. 3.FranklinseriesinL p,<p< The main result of this section is the following: Theorem 3.1. Let the quasi-dyadic sequence of partitions {P j : j } satisfy the weak regularity condition with parameter γ. Then the corresponding Franklin system is an unconditional basis in L p for all 1 < p <. For the proof of Theorem 3.1 we need some auxiliary results. We start with a technical lemma. Lemma 3.2. Let the quasi-dyadic sequence of partitions {P j : j } satisfy the weak regularity condition with parameter γ. Let I I and let ϕ be a function such that suppϕ I, sup ϕ 1 I, ϕ(u)du =. Moreover, let a n = (ϕ, f n ). Then there is a constant C γ, depending only on γ, such that a n f n (u) du C γ. I c n= Proof. First, observe that the conditions imposed on ϕ imply a = and a 1 3 (recall that f and f 1 do not depend on the specific sequence of partitions cf. Definition 2.1), so it is enough to consider the sum beginning with n = 2. Since I I, we have I = I j,k for some j and 1 k 2 j. To estimate Ì I c n= a nf n (u) du, we split it into several parts. First, consider j j= 2 j+1 n=2 j +1 a n f n 1. To simplify the notation, let I =I j+1,2k 1, I + = I j+1,2k and τ = t j+1,2k 1. Note that if n 2 j +k 1, then f n is linear on I, and for 2 j + k n 2 j+1 the function f n is linear on both subintervals I and I +. Therefore, denoting by ξ n, ζ n the value of the derivative of f n on (I ), (I + ) respectively (clearly, ξ n = ζ n for n 2 j + k 1) and using the properties of ϕ we get (3.1) a n = 1 ϕ(u)f n (u)du = I I ϕ(u) ( f n (u) f n (τ) ) du 1 ( ) ξ n u τ du + ζ n u τ du I I I + = ξ n I 2 + ζ n I I

21 22 G. Gevorkyan and A. Kamont To estimate ξ n, let n = 2 j +l with 1 l 2 j, j j, and let j be the unique interval of order j + 1 containing I ; as j I j+1, we have j = I j+1,kj for some k j ; moreover, f n is linear on j. Applying the pointwise estimates for f n from Proposition 2.9 (inequalities (2.27), (2.31)) we get which gives sup f n (u) C γ 2 l kj/2 {n} 1/2, u j (3.2) ξ n C γ 2 l kj/2 {n} 1/2 1 j. Using the estimates for length of intervals from Proposition 2.6(i) we get ξ n C γ 2 l kj/2 {n} 1/2 ( γ and by similar arguments (note that I + j for j < j ), ζ n C γ 2 l kj/2 {n} 1/2 ( γ ) j j ) j j 1 I, 1 I +. These estimates for ξ n, ζ n and (3.1) give ( ) j j γ a n C γ 2 l kj/2 {n} 1/2. As f n 1 C γ {n} 1/2 (cf. Proposition 2.9, (2.23)), the last inequality gives 2 j+1 n=2 j +1 which implies (3.3) a n f n 1 C γ ( γ j 2 j+1 j= n=2 j +1 ) j j 2 j l=1 a n f n 1 C γ j j= ( ) j j γ 2 l kj/2 C γ, ( ) j j γ C γ. Now, let n = 2 j +l with 1 l 2 j and j > j. Then, by the properties of ϕ we have (3.4) a n ϕ(u) f n (u) du 1 f n (u) du. I I Recall that I = [t j,k 1, t j,k] = [t j+1,2 j+1 j (k 1), t j+1,2 j+1 j k ]. Consider n such that t n I, i.e. {n} I, or equivalently 2 j+1 j (k 1) < 2l 1 < 2 j+1 j k. (It should be noted that we cannot use Lemma 2.11, which has been obtained for strongly regular partitions and < p 1; now we obtain an analogous estimate for weakly regular partitions, but for p = 1 only.) Then Ì I f n(u) du f n 1, and using the decay of the integrals of f n from Proposition 2.9 (cf. (2.29), (2.33)) we obtain I c f n (u) du = t j+1,2 j+1 j (k 1) I f n (u) du + 1 t j+1,2 j+1 j k f n (u) du j+1 j (k 1) 2l (θ 2 + θ l 2j j k ) f n 1,

22 On general Franklin systems 23 where θ = 1/(3( 2 1)) < 1. Moreover, for these n s we have (cf. formula (2.23) in Proposition 2.9 and Proposition 2.6(i)) ( ) j j γ f n 2 1 C γ {n} C γ I. By (3.4) we have a n I 1 f n 1, so applying the last two inequalities we get a n f n (u) du C γ (θ 2j+1 j(k 1) 2l+2 + θ l 2j j k ) f n 2 1 I I c 2 j <n 2 j+1 t n I which means that (3.5) 2 j <n 2 j+1 t n I 2 j <n 2 j+1 t n I C γ ( γ a n ) j j θ l, l N I c f n (u) du C γ ( ) j j γ. Now, consider n = 2 j + l with 2l 1 < 2 j+1 j (k 1), i.e. t n < t j+1,2 j+1 j (k 1). Let J j be the interval from I j with left end coinciding with the left end of I. Since J j I, Proposition 2.6(i) implies that J j (γ/()) j j I. Applying for these n s the estimates for the integral and pointwise decay of f n from Proposition 2.9 (cf. (2.33) and (2.31)), we get (3.6) I f n (u) du J j f n (u) du J j f n (t j+1,2 j+1 j (k 1) ) ( ) j j γ C γ 2 l 2j j(k 1) {n} 1/2 I. Ì Clearly, f I c n (u) du f n 1, so the above inequlities, (3.4) and the estimate for f n 1 (cf. (2.23)) give ( ) j j γ ( ) j j γ a n f n (u) du C γ 2 l C γ. 2 j <n 2 j+1 I c l N t n<t j+1,2 j+1 j (k 1) By analogous arguments we check that a n 2 j <n 2 j+1 t n>t j+1,2 j+1 j k The last two inequalities and (3.5) give 2 j+1 n=2 j +1 I c f n (u) du C γ I c a n f n (u) du C γ ( ) j j γ. ( ) j j γ,

23 24 G. Gevorkyan and A. Kamont and summing over j > j we get n=2 j +1 This and (3.3) complete the proof. I c a n f n (u) du C γ As a consequence of Lemma 3.2 we get j=j ( ) j j γ C γ. Lemma 3.3. Let the quasi-dyadic sequence of partitions {P j : j } satisfy the weak regularity condition with parameter γ. Let T I be a subset such that I Ĩ = for all I, Ĩ T, I Ĩ, and let B = I T I. Let ψ be a function such that sup ψ b, suppψ B, I T ψ =, where b is some nonnnegative number. Then there is a constant C γ, depending only on γ, such that for any function ψ satisfying the above conditions, a n f n (u) du C γ b B, where a n = (ψ, f n ). B c n= Proof. Note that B c I T Ic ; put ψ I = ψχ I and a I,n = (ψ I, f n ). Since the functions ψ I satisfy the hypothesis of Lemma 3.2, we get a I,n f n (u) du a I,n f n (u) du C γ b I, B c n= and summing over I T we obtain a n f n (u) du n= I T B c B c n= I c n= I a I,n f n (u) du C γ b I = C γ b B. I T Theorem 3.4. Let the quasi-dyadic sequence of partitions {P j : j } satisfy the weak regularity condition with parameter γ. Let 1 < p < 2 and f L p, f = n= a nf n. Let ε = (ε n ) n with ε n { 1, 1} and T ε f = ε n a n f n. n= Then there is a constant C γ,p, depending only on γ and p, such that for each f L p and each sequence ε, T ε f p C γ,p f p. Proof. Let f L p, 1 < p < 2. For x [, 1], define Φ(x) = sup 1 f(u)du I. I:x I I First, observe that Φ(x) M(f, x). Since for p > 1 the operator M(f, ) is of type (p, p), this inequality implies Φ p M(f, ) p C p f p. I

24 For m Z let On general Franklin systems 25 B m = {x [, 1] : Φ(x) > 2 m }. Note that the set B m is a sum of some intervals from I, and let T m be the set of maximal intervals from I contained in B m ; thus we have B m = I T m I, with I Ĩ = for I, Ĩ T m, I Ĩ. Note that B m+1 B m ; moreover, for each pair of different intervals from I, either their interiors are disjoint or one is included in the other, which implies that for each interval I T m+1 there is a unique J T m such that I J. It is also clear that (3.7) 2 mp B m C p Φ p p C p f p p. Now, let m Z f(x) for x B m, F m (x) = 1 f(u)du for x I, I T m. I I We check that there is a constant C γ such that F m C γ 2 m for all m. Indeed, if x B m, then Φ(x) 2 m, which means that 1 f(u)du J 2m J for all J I with x J, and this implies f(x) 2 m a.e. on B m. On the other hand, if I T m is an interval of rank j, I is the unique interval of rank j 1 containing I and I is the other interval of rank j contained in I, then by maximality of I, neither I nor I is included in B m, which implies 1 I f(u)du 2m and 1 I f(u)du 2m. I I These inequalities and weak regularity of the sequence of partitions give 1 f(u)du I 1 ( ) f(u)du + f(u)du 2 m I + I (2)2 m. I I I I I Thus, for all m Z we have F m (2)2 m. Define ψ m = F m+1 F m ; then (3.8) ψ m 3(2)2 m and suppψ m B m. Let us prove that the function ψ m and set B m satisfy the Ì assumptions of Lemma 3.3 with constant b m = 3(2)2 m. It remains to check that ψ I m(u)du = for all I T m, but this follows from the fact that for I T m the set I B m+1 can be written as the union of some intervals from T m+1, and from the formulae for F m and F m+1 ; the technical details are omitted. Moreover, the function ψ m is constant on the intervals I T m+1, which together with the previous property implies that (ψ m, ψ m ) = for m m. Note that by (3.7) and (3.8) we have f = m= ψ m, with the series convergent in L p. Thus, putting a m,n = (ψ m, f n ), we obtain a n = m= a m,n and ε n a n =

25 26 G. Gevorkyan and A. Kamont m= ε na m,n. For l Z let { E l = u [, 1] : X l = {u [, 1] : Y l = { u [, 1] : ε n a n f n (u) > 2 l}, n= ( ε n n= m l 1 a m,n )f n (u) > 2 l 1}, ( ε n a m,n )f n (u) > 2 l 1}. n= m l Note that E l X l Y l. Let us estimate X l and Y l. First, using Chebyshev s inequality, estimates (3.8) for ψ m and the orthogonality of the functions ψ m, we obtain { ( X l = u [, 1] : ε n a m,n )f n (u) 2 > 2 2l 2} 1 2 2l 2 n= ( n= = 1 2 2l 2 C γ 2 2l m l 2 m l 1 m l 1 a m,n ) 2 m l 1 2 ψ m = l 2 ψ m 2 2 m l 1 ψ m 2 B m C γ 2 2l m l 1 2 2m B m. On the other hand, the functions ψ m and the sets B m satisfy the assumptions of Lemma 3.3, so using this lemma we get Y l B l + {u Bl c : a m,n f n (u) > 2 l 1} n= m l B l l 1 a m,n f n (u) du Thus, we have B l l 1 C γ 2 l 2 m B m. m l E l C γ ( 1 2 2l Bl c n= m l m l Bm c n= m l 1 a m,n f n (u) du 2 2m B m l m l Using this estimate and (3.7) we obtain (recall that 1 < p < 2) T ε f p p C p 2 lp E l l Z ( C γ,p 2 l(p 2) 2 2m B m + l Z m l l Z ) 2 m B m. 2 ) l(p 1) 2 m B m m l

26 On general Franklin systems 27 ( C γ,p m Z2 2m B m l m2 l(p 2) + m Z2 m B m l m C γ,p 2 mp B m C γ,p f p p. m Z 2 l(p 1)) Proof of Theorem 3.1. Recall that for each p, 1 p <, the system {f n : n } is a basis in L p. As an orthonormal system, it is an unconditional basis in L 2. Its unconditionality in L p for 1 < p < 2 follows from Theorem 3.4, and then the unconditionality in L p for 2 < p < is obtained by a duality argument. As a consequence of Theorem 3.4, using a well-known argument based on Khinchin s inequality and the maximal inequality from Theorem 2.5(i), we obtain Corollary 3.5. Let the quasi-dyadic sequence of partitions {P j : j } satisfy the weak regularity condition with parameter γ and 1 < p <. Then for f L p we have f p Pf p Sf p, with implied constants depending only on p and γ. Moreover, for a real sequence (a n ) n, the following conditions are equivalent: (1) The series n= a nf n is unconditionally convergent in L p. (2) There is f L p such that a n = (f, f n ) for all n. (3) P( ) = ( n= a2 n f2 n ( ))1/2 L p. (4) S( ) = sup m m n= a nf n ( ) L p. 4.FranklinseriesinL p,<p 1,andH p,1/2<p 1 Let {P j : j } be a quasi-dyadic sequence of partitions and let {f n : n } be the corresponding Franklin system. Let < p < and let a = (a n ) n be a given sequence of real numbers. Consider the following conditions: (A) P( ) = ( n= a2 n f2 n ( ))1/2 L p. (B) The series n= a nf n converges unconditionally in L p. (C) S( ) = sup m m n= a nf n ( ) L p. For 1 < p <, we have already proved the equivalence of (A) (C) under the assumption of weak regularity of the sequence of partitions under consideration cf. Corollary 3.5. In this section, we study the relations of the above conditions for < p 1. Under the assumption of strong regularity of the sequence of partitions, we prove the following: Theorem 4.1. Let the quasi-dyadic sequence of partitions {P j : j } satisfy the strong regularity condition with parameter γ and let {f n : n } be the corresponding Franklin system. Then, for each p, < p 1, conditions (A), (B) and (C) are equivalent. Moreover, we study the convergence of the Franklin series in H p, 1/2 < p 1. We obtain the following result: