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0 With the exception of the special case s = 0, this is the maximal range of parameters s d for which Bp,q,1(I ) is a separable quasi-Banach space and contains the Haar systems ˜ d d s d H and H . Moreover, for this parameter range Bp,q,1(I ) admits a characterization in terms of best Lp-approximations with piecewise constant functions on dyadic partitions of Id which is used in the proofs. Our main result for the Haar wavelet system Hd is the following theorem. Theorem 1 Assume (2). a) If 1 ≤ p < ∞ then the Haar wavelet system Hd is an unconditional Schauder basis s d for Bp,q,1(I ) for all parameters 0 s d Theorem 2 If 0 s d Consequently, for these parameters Bp,q,1(I ) does not have a Schauder basis at all which is a stronger statement than proving that the Haar wavelet system Hd fails to be a Schauder basis. For the parameters 0 d s d where according to Theorem 1 H is not a Schauder basis in Bp,q,1(I ), we have the s d d d s d ′ continuous embedding Bp,q,1(I ) ⊂ L1(I ), and thus L∞(I ) ⊂ Bp,q,1(I ) . We do not s d know if the spaces Bp,q,1(I ) satisfying (3) possess (unconditional) Schauder bases at all. s d ˜ d As Schauder basis in Besov spaces Bp,q,1(I ), the tensor-product Haar system H behaves the same as Hd for 1 ≤ p < ∞ but fails completely for the parameter arnge p< 1. This is the essence of Theorem 5 in Section 5.3. 2 Let us comment on known results that motivated this study. For d > 1, the two Haar systems H˜ d and Hd have formally been introduced in the 1970-ies, see [3, 6], the system Hd implicitly appeared already in [25]. In the univariate case d = 1, part a) of Theorem 1 has essentially been established by Triebel [25] and Ropela [21] under the restriction q ≥ 1. Extensions to d ≥ 1 are due to Ciesielski [3] and Triebel. The results s Rd of Triebel who worked in the framework of distributional Besov spaces Bp,q( ) and s d Bp,q(I ) are summarized in [29, Section 2]. Starting from [26], Triebel considered the parameter values 0 This is the essence of Theorem 2.13 (i) (d = 1) and Theorem 2.26 (i) (d > 1) in [29]. Note that for the range 0 < p,q ≤∞ we have s d s d Bp,q(I )= Bp,q,1(I ) ⇐⇒ max(0,d(1/p − 1)) s d s d i.e., for the parameters in (4), the scales Bp,q(I ) and Bp,q,1(I ) coincide up to equivalent quasi-norms. Consequently, with the exception of the range 1 ≤ s< 1/p for 0 ditions on the parameters p,q,s,τ for which the map f → (f, χId )L2 = Id f dx extends s,τ Rd to a bounded linear functional on Besov-Morrey-Campanato-type spaces Bp,q ( ). d R s d Independently, for 0 3 Theorems 1 and 3. The necessity of these conditions and Theorem 2 are dealt with in Section 4, where we construct specific counterexamples consisting of piecewise constant functions. Similar in spirit examples have already been used in [16]. We conclude in Section 5 with some remarks on higher-order spline systems, analogous results for the tensor-product Haar system H˜ d, and the exceptional cases s = 0 and q = ∞. 2. Definitions and auxiliary results 2.1. Haar systems Recall first the definition of the univariate L∞ normalized Haar functions. By χΩ we denote the characteristic function of a Lebesgue measurable set Ω ⊂ Rd, and by −k −k −k ∆k,i := [(i − 1)2 , i2 ) the univariate dyadic intervals of length 2 , k ∈ Z+, i ∈ Z. Then the univariate Haar system H = {hm}m∈N on I := [0, 1] is given by h1 = χI , and k−1 k−1 N h2 +i = χ∆k,2i−1 − χ∆k,2i , i =1,..., 2 , k ∈ , Throughout the paper, we work with L∞-normalized Haar functions. The Haar functions hm with m ≥ 2 can also be indexed by their supports, and identified with the shifts and dilates of a single function, the Haar wavelet h0 := χ[0,1/2) − χ[1/2,1). Indeed, k−1 k−1 k−1 N h∆k−1,i := h2 +i = h0(2 · −i), i =1,..., 2 , k ∈ . The above introduced enumeration of the Haar functions hm is the natural ordering used in the literature, however, one can also define H as the union of dyadic blocks ∞ k−1 N H = ∪k=0Hk, H0 = {h1}, Hk = {h∆k−1,i : i =1,..., 2 }, k ∈ , and allow for arbitrary orderings within each block Hk. Below, we will work with the multivariate counterparts of the spaces 2k 2k Sk = span({hm}m=1) = span({χ∆k,i }i=1), k =0, 1,..., of piecewise constant functions with respect to the uniform dyadic partition Tk = {∆k,i : i =1,..., 2k} of step-size 2−k on the unit interval I. The Haar wavelet system d ∞ d H = ∪k=0Hk (5) on the d-dimensional cube Id, d > 1, is defined in a blockwise fashion as follows. Let d −k d d the partition Tk be the set of all dyadic cubes of side-length 2 in I . Each cube in Tk is the d-fold product of univariate ∆k,i, i.e., d k d i Tk = {∆k, := ∆k,i1 × ... × ∆k,id : i =(i1,...,id) ∈{1,..., 2 } }. d d The set of all piecewise constant functions on Tk is denoted by Sk . With each ∆k−1,i ∈ d k−1 d N d d d Tk−1, i ∈ {1,..., 2 } , k ∈ , we associate the set Hk,i ⊂ Sk of 2 − 1 multivariate Haar functions with support ∆k−1,i, given by all possible tensor products ψk,i1 ⊗ ψk,i2 ⊗ ... ⊗ ψk,id , ψk,i = h∆k−1,i or χ∆k−1,i 4 d where at least one of the ψ equals h − . The blocks H appearing in (5) are given k,il ∆k 1,il k d as follows: The block H0 is exceptional, and consists of the single constant function d d d χId . The block H1 coincides with H1,1 and consists of 2 − 1 Haar functions, where 1 = (1,..., 1). For general k ≥ 2, the block d d H := ∪ d H i k ∆k−1,i∈Tk−1 k, consists of (2d − 1)2(k−1)d Haar functions of level k. It is obvious that d k d Sk = span(∪l=0Hl ), d d and that H is a complete orthogonal system in L2(I ). Since each Haar function in Hd has support on a d-dimensional dyadic cube, we sometimes call this system isotropic, in contrast to the anisotropic tensor-product Haar system H˜ d defined in (1), where the supports of the tensor-product Haar functions h˜ ∈ H˜ d are d-dimensional dyadic rectangles. Note that H˜ d can also be organized into ˜ d ˜ d blocks Hk , where for k ≥ 1 the block Hk consists of the tensor-product Haar functions ˜ d d ˜ d d h ∈ Sk orthogonal to Sk−1, and H0 = H0 = {χId }. Obviously, we have ˜ d d Z span(Hk ) = span(Hk ), k ∈ +, such that H˜ d represents a level-wise transformation of Hd vice versa. The basis proper- ˜ d s d ties of H in the spaces Bp,q,1(I ) are exhaustively dealt with in Section 5.3, see Theorem 5. It turns out that for 0 2.2. Function spaces s d The Besov function spaces Bp,q,1(I ) are traditionally defined for s> 0, 0 < p,q ≤∞, d as the set of all f ∈ Lp(I ) for which the quasi-norm (kfkq + kt−s−1/qω(t, f) kq )1/q, 0 d ω(t, f)p := sup k∆yfkLp(Iy ), t> 0, 0<|y|≤t stands for the first-order Lp modulus of smoothness, and d d d Rd ∆yf(x) := f(x + y) − f(x), x ∈ Iy := {z ∈ I : z + y ∈ I }, y ∈ , 5 denotes the first-order forward difference. Here and throughout the remainder of the paper, we adopt the following notational convention: If the domain is Id, we omit s the domain in the notation for spaces and quasi-norms, e.g., we write Bp,q,1 instead of s d d Bp,q,1(I ), and k·kLp instead of k·kLp(I ). An exception are the formulation of theorems. Also, by c,C we denote generic positive constants that may change from line to line, and, unless stated otherwise, depend on p,q,s only. The notation A ≈ B is used if cA ≤ B ≤ CA holds for two such constants c,C. s The space Bp,q,1 is a quasi-Banach space equipped with a γ-quasi-norm, where γ = s min(p, q, 1), meaning that k·kBp,q,1 is homogeneous and satisfies γ γ γ kf + gk s ≤kfk s + kgk s . Bp,q,1 Bp,q,1 Bp,q,1 Similarly, Lp is a quasi-Banach space equipped with a γ-quasi-norm if we set γ = γp := min(p, 1). s If 0 s kfkBp,q,1 ≈kfkLp , f ∈ Lp, s< 0. s In other words, Bp,q,1 = Lp for all s< 0 (this holds also for s = 0 and q = ∞). s To conclude this short discussion of the definition and properties of Bp,q,1-spaces, let us motivate our basic assumption (2) on the parameter range adopted in this paper. The case s = 0 is in some sense exceptional and often not considered at all, we return to it in Section 5.2. Since our main concern is the Schauder basis property of the countable Haar d ˜ d s systems H and H , we can also neglect all parameters for which Bp,q,1 is non-separable or does not contain piecewise constant functions on dyadic partitions. The separability requirement excludes the spaces with p = ∞ or q = ∞. Since, with constants also depending on k, we have 1/p ω(t, h)p ≈ t , t → 0, d d s for any Haar function h ∈ Hk , k = 1, 2,..., we see that H is not contained in Bp,q,1 whenever s ≥ 1/p, 0 ks −1 s Rd Rd Z Z kfkBp,q( ) := k(k2 F φkFfkLp( ))k∈ + kℓq( +) 6 d is finite. By exchanging the order of taking Lp(R ) and ℓq(Z+) quasi-norms in (??), s Rd one defines the Triebel-Lizorkin spaces Fp,q( ). Spaces on domains are defined by restriction. In particular, for the domain Id s s Rd Bp,q = {f : ∃ g ∈ Bp,q( ) such that f = g|Id } and s s Rd kfkBp,q := inf kgkBp,q( ). g: f=g|Id This definition, and many equivalent ones, are surveyed in [27] and [29, Chapter 1], with references to earlier papers. In particular, the equivalence (4) is mentioned in [29, Section 1.1]. 2.3. Piecewise constant Lp-approximation s We start with introducing an equivalent quasi-norm in Bp,q,1 which is based on ap- proximation techniques using piecewise constant approximation on dyadic partitions. Let Ek(f)p := inf kf − skLp , k =0, 1,..., d s∈Sk d denote the best approximations to f ∈ Lp with respect to Sk . Lemma 1 Let 0 ∞ 1/q q ks q s kfkAp,q,1 := kfkLp + (2 Ek(f)p) (6) ! Xk=0 s provides an equivalent quasi-norm on Bp,q,1. This result follows from the direct and inverse inequalities relating best approximations Ek(f)p and moduli of smoothness ω(t, f)p which have many authors. In the univariate case d = 1, see e.g. Ul’yanov [30], Golubov [11] for 1 ≤ p < ∞, and [24, Section 2] for 0 7 We refer to [15] for d = 1, and to [7, Theorem 7.4] for d > 1. A local version of the associated embedding inequality will be used in Section 3. At the heart of the counterexamples constructed in Section 4 for p ≤ 1 is a simple observation about best Lp approximation by constants which we formulate as Lemma 2 Let (Ω, A,µ) be a finite measure space, and let the function f ∈ Lp(Ω) := ′ Lp(Ω, A,µ), 0 < p ≤ 1, equal a constant ξ0 on a measurable set Ω ∈ A of measure ′ 1 µ(Ω ) ≥ 2 µ(Ω). Then kf − ξ0kLp(Ω) = inf kf − ξkLp(Ω), ξ∈R i.e., best approximation by constants in Lp(Ω) is achieved by setting ξ = ξ0. Proof. Indeed, under the above assumptions and by the inequality |a + b|p ≤|a|p + |b|p we have p p ′ p kf − ξk = |f(x) − ξ| dµ(x)+ µ(Ω\Ω )|ξ − ξ0| Lp(Ω) ′ ZΩ p p ≥ (|f(x) − ξ| + |ξ − ξ0| ) dµ(x) ′ ZΩ p p ≥ |f(x) − ξ0| dµ(x)= kf − ξ0k ′ Lp(Ω) ZΩ for any ξ ∈ R, with equality for ξ = ξ0. This gives the statement. ✷ Note that the equivalence (up to constants depending on parameters but not on f) between Lp quasi-norms and best approximations by constants holds also for p ≥ 1 and under weaker assumptions on the relative measure of Ω′ (e.g., µ(Ω′)/µ(Ω) ≥ δ > 0 would suffice). We will apply this lemma to the Lebesgue measure on dyadic cubes in Id and special examples of dyadic step functions constructed below. Extensions to higher degree polynomial and spline approximation are possible as well (see the proof of the lemma on p. 535 in [16] for d = 1). 2.4. Schauder basis property A sequence (fm)m∈N of elements of a quasi-Banach space X is called a Schauder basis in X if every f ∈ X possesses a unique series representation ∞ f = cmfm m=1 X converging in X. If every rearrangement of (fm)m∈N is a Schauder basis in X then this system is called unconditional Schauder basis. Below we will rely on the following criterion whose proof for Banach spaces can easily be extended to the quasi-Banach space case. 8 Lemma 3 The sequence (fm)m∈N of elements of a quasi-Banach space X is a Schauder basis in the quasi-Banach space X if and only if its span is dense in X, and there exists a sequence (λm)m∈N of continuous linear functionals on X such that 1, m = n, λ (f )= m, n ∈ N, n m 0, m =6 n, and the associated partial sum operators n Sn(x) := λm(x)fm, n ∈ N, x ∈ X, m=1 X are uniformly bounded operators in X. In order for (fm)m∈N to be unconditional, all operators SJ (x) := λm(x)fm, x ∈ X, m∈J X where J is an arbitrary finite subset of N, must be uniformly bounded operators in X. An immediate consequence of Lemma 3 is that, in order to possess a Schauder basis at all, X must have a sufficiently rich dual space X′ of continuous linear functionals. If Hd is a Schauder basis in a quasi-Banach space X of functions defined on Id then Sd = span(Hd) must be a dense subset of X by the density condition in Lemma s 3. This is satisfied for all X = Bp,q,1 with parameters satisfying (2). Moreover, since d d d S ⊂ L∞ ⊂ L2 and H is an orthogonal system in L2, any dyadic step function g ∈ S has a unique Haar expansion given by kd g = λh(g)h, λh(g) := 2 gh dx. (8) d d I hX∈H Z d Since for g ∈ S only finitely many coefficients λh(g) do not vanish, the summation in (8) is finite, and there are no convergence issues. Thus, for the Schauder basis property of d H in X to hold, the coefficient functionals λh(g) in (8) must be extendable to elements in X′, and the level k partial sum operators k Pkg = λh(g)h, k =0, 1,..., (9) l=0 h∈Hd X Xl must form a sequence of uniformly bounded linear operators in X. Due to the local d support properties of the Haar functions within each block Hl and the assumed ordering of the Haar wavelet system it often suffices to deal with this subsequence of partial sum operators. The statements in Theorem 1 b) about the failure of the Schauder basis d s property of H in Bp,q,1 will be shown by either relying on Theorem 2 or proving that the operators Pk are not uniformly bounded. 9 Whenever X is continuously embedded into L1, the level k partial sum operators Pk d extend to bounded projections with range Sk , and with constant values on the dyadic d cubes in Tk explicitly given by averaging. This comes in handy when computing Pkf for concrete functions f. Indeed, the constant values taken by Pkf on dyadic cubes in d Tk are given by kd d Pkf(x)= av∆(f) := 2 f dx, x ∈ ∆, ∆ ∈ Tk ,, k =0, 1,..., (10) Z∆ d if f ∈ L1(I ). Note that the coefficient functionals λh of the Haar expansion are finite linear combinations of functionals as defined in (10), vice versa. Finally, for X = L2 ⊂ L1 d the level k partial sum operator Pk realizes the orthoprojection onto Sk . 3. Proofs: Sufficient conditions 3.1. The case s = d(1/p − 1) d d(1/p−1) The positive results on the Schauder basis property of H in Bp,q,1 stated in Theorem 1 b) for the parameter range d − 1 d(1/p−1) d(1/p−1) Bp,q,1 ⊂ Bp,p,1 ⊂ L1. This ensures that the Haar coefficient functionals λh defined in (8) are continuous on s d d s Bp,q,1. Moreover, S = span(H ) is dense in Bp,q,1. Due to Lemma 1 and Lemma 3, it is therefore sufficient to establish the inequality d(1/p−1) kPgk d(1/p−1) ≤ Ckgk d(1/p−1) , g ∈ Bp,q,1 , (12) Ap,q,1 Ap,q,1 for any partial sum operator P of the Haar expansion (8), with a constant C independent of g and P , if the parameters satisfy (11). According to our ordering convention for Hd, any partial sum operator P can be ¯ d d written, for some k =0, 1,... and some subset Hk+1 ⊂ Hk+1, in the form d Pg = Pkg + λh(g)h ∈ Sk+1. (13) h∈H¯ d Xk+1 ¯ d For Hk+1 = ∅, we get P = Pk as partial case. 10 The first step for establishing (12) is the proof of the inequality kPgkp ≤ C2kd(p−1) kgkp , (14) Lp L1(∆) ∆∈T d Xk with the explicit constant C =2d. By (10), we have p kP gkp = 2−kd 2kd g dx ≤ 2kd(p−1) kgkp . k Lp L1(∆) ∆ ∆∈T d ∆∈T d Xk Z Xk ¯ d d The remaining h ∈ Hk+1 can be grouped by their support cubes ∆ ∈ Tk . Each such d d group may hold up to 2 − 1 Haar functions with the same supp(h) = ∆ ∈ Tk . By the definition of the Haar coefficient functionals λh(g) for each term λh(g)h associated with such a group we obtain the estimate kλ (g)hkp = |λ (g)|pkhkp ≤ 2kdpkgkp · 2−kd =2kd(p−1)kgkp . h Lp h Lp(∆) L1(∆) L1(∆) Thus, using the p-quasi-norm triangle inequality for Lp, p p p kPgkLp ≤ kPkgkLp + kλh(g)hkLp h∈H¯ d Xk+1 ≤ 2kd(p−1) kgkp + (2d − 1)2kd(p−1)kgkp L1(∆) L1(∆) ∆∈T d ∆∈T d Xk Xk = 2d2kd(p−1) kgkp , L1(∆) ∆∈T d Xk we obtain (14). Now we apply the embedding inequality associated with (7), with the appropriate coordinate transformation, locally on ∆ to the terms kgkp . This gives L1(∆) ∞ p kd(1−p) p ld(1−p) p kgkL (∆) ≤ C 2 kgkL (∆) + 2 El(g)p,∆ 1 p ! Xl=k d for each ∆ ∈ Tk , where El(g)p,∆ := inf kg − skLp(∆), l = k,k +1,..., d s∈Sl denotes the local best Lp approximation by dyadic step functions restricted to cubes ∆ d from Tk . Since kgkp = kgkp , E (g)p = E (g)p , l = k,k +1,..., Lp Lp(∆) l p l p,∆ ∆∈T d ∆∈T d Xk Xk 11 after substitution into (14), we arrive at the estimate ∞ p p kd(p−1) ld(1−p) p kPgkLp ≤ C kgkLp +2 2 El(g)p (15) ! Xl=k for the Lp quasi-norm of any partial sum Pg. d(1/p−1) With the auxiliary estimate (15) at hand, we turn now to the estimate of the Ap,q,1 d quasi-norm of g − Pg. Since Pg ∈ Sk+1, we have El(g − Pg)p = El(g)p, l > k, while for l ≤ k the trivial bound d El(g − Pg)p ≤kg − PgkLp(I ) will suffice. This gives ∞ q q ld(1/p−1) q − kg − Pgk d(1/p 1) = kg − PgkLp + (2 El(g − Pg)p) Ap,q,1 Xl=0 ∞ kd(1/p−1)q q ld(1/p−1) q ≤ C 2 kg − PgkLp + (2 El(g)p) , (16) ! l=Xk+1 uniformly for all P and g ∈ Sd. Recall that 0 kg − skkLp = Ek(g)p, and estimate with (15) and Psk = Pksk = sk as follows: p p p kg − PgkLp ≤ kg − skkLp + kP (g − sk)kLp ∞ p p kd(p−1) ld(1−p) p ≤ kg − skkLp + C kg − skkL (Id) +2 2 El(g − sk)p p ! l=Xk+1 ∞ kd(p−1) ld(1−p) p ≤ C2 2 El(g)p. (17) Xl=k Now we can finish the proof of (12). According to (17) we get for the first term in the right-hand side of (16) ∞ q/p ∞ kd(1/p−1)q q ld(1−p) p ld(1/p−1) q 2 kg − PgkLp ≤ C 2 El(g)p ≤ C (2 El(f)p) , ! Xl=k Xl=k 12 where the inequality ∞ γ ∞ γ al ≤ al , al ≥ 0, 0 <γ ≤ 1, ! Xl=0 Xl=0 ld(1−p) p has been used with γ = q/p ≤ 1, al = 2 El(g)p for l ≥ k, and al = 0 for l ∞ q ls q q d(1/p−1) kg − Pgk d(1/p−1) ≤ C (2 El(g)p) ≤ Ckgk d(1/p−1) , g ∈ Bp,q,1 , (18) Ap,q,1 Ap,q,1 Xl=k s for the parameters (11). Since the Ap,q,1-quasi-norm is a q-quasi-norm for the parameters in (11), (18) is equivalent with (12). This proves the Schauder basis property for Hd in d(1/p−1) Bp,q,1 for this parameter range. Under the condition (11), unconditionality of Hd does not hold, see Section 4 for a counterexample. The parameter range for which Theorem 1 asserts that Hd is an s unconditional Schauder basis in Bp,q,1 is dealt with in the next subsection. For use in the next subsection, we mention the following by-product of the consider- ations leading to (17). Consider the k-th block Qkg := (Pk − Pk−1)g = λh(g)h, k =1, 2,..., h∈Hd Xk of the Haar expansion (8). Using a standard compactness argument and the dyadic dilation- and shift-invariance of the Haar wavelet system Hd, we have the equivalence of quasi-norms k γ hkp ≈ 2−kd |γ |p, 0 p −kd p k γhhkLp ≈ 2 |γh| , 0 p p −kd p kQkgkLp = k λh(g)hkLp ≈ 2 |λh(g)| , 0 p kd p kd p p |λh(g)| ≈ 2 kQkgkLp ≤ C2 (kg − PkgkLp + kg − Pk−1gkLp ), h∈Hd Xk−1 13 k =1, 2,..., for arbitrary g ∈ L1 and 0 ∞ p kdp ld(1−p) p |λh(g)| ≤ C2 2 El(g)p, k =1, 2,..., (20) h∈Hd l=k−1 Xk−1 X d(1/p−1) d−1 for the k-th block of Haar coefficients of arbitrary g ∈ Bp,p,1 , d max(d(1/p − 1), 0) d s and establish the unconditional Schauder basis property of H in Bp,q,1 by proving a slightly stronger statement. Theorem 3 For the parameter range (21), the mapping Λ: g 7−→ Λg := (λh(g))h∈Hd , g ∈ L1, (22) s d provides an isomorphism between Bp,q,1(I ) and a weighted ℓq(ℓp) space, more precisely, we have q/p 1/q ∞ k(sp−d) p kgkBs ≈kΛgkℓ (ℓ ) := 2 |λh(g)| . (23) p,q,1 p q k=0 h∈Hd X Xk d s d Consequently, H is an unconditional Schauder basis in Bp,q,1(I ) for the parameter range (21). s Proof. For all parameters considered in (21) we have the continuous embedding Bp,q,1 ⊂ s L1. This ensures that Λ is well-defined on Bp,q,1. In the following, we will concentrate on the case (d − 1)/d s s kΛgkℓp(ℓq ) ≤ CkgkBp,q,1 , g ∈ Bp,q,1, (24) can be proved as follows. For (d − 1)/d p ak := |λh(g)| , k =0, 1,.... h∈Hd Xk 14 With this notation, we have ∞ kΛgkq = (2k(sp−d)a )q/p ℓp(ℓq ) k Xk=0 ∞ ∞ q/p q/p k(sp−d(1−p)) ld(1−p) p ≤ a0 + C 2 2 El(g)p ! Xk=0 Xl=k ∞ ∞ q/p q k(sp−d(1−p)) ld(1−p) p ≤ C kgkLp + 2 2 El(g)p . (25) ! Xk=0 Xl=k In the last estimation step, we substituted the estimate q q/p q q − |a0| = g(x) dx ≤kgkL1 ≤ Ckgk d(1/p 1) d A ZI p,p,1 q/p ∞ p ld(1−p) p ≤ C kgkLp + 2 El(g)p . ! Xl=0 which follows from the definition of a0 and the continuous embedding (7). Now recall that under the assumption (21) we have d(1/p − 1) ∞ 1/r ∞ 1/q r −kǫ lǫ q bl ≤ Cǫ,q/r2 (2 bl) , ǫ> 0, k =0, 1,..., (26) ! ! Xl=k Xl=k ld(1/p−1) valid for non-negative sequences (bl)l∈Z+ and all 0 q/p ∞ ld(1−p) p −kǫq l(ǫ+d(1/p−1)) q 2 El(g)p ≤ C2 (2 El(g)p) , k =0, 1,.... ! Xl=k Xl=k After substitution into (25), we arrive at ∞ ∞ kΛgkq ≤ C kgkq + 2kq(s−d(1/p−1)−ǫ) (2l(ǫ+d(1/p−1))E (g) )q ℓp(ℓq ) Lp l p k=0 l=k ! X∞ X k q l(ǫ+d(1/p−1)) q kq(s−d(1/p−1)−ǫ) = C kgkLp + (2 El(g)p) 2 l=0 l=0 ! X∞ l X q ls q kq(s−d(1/p−1)−ǫ) q s ≤ C kgkLp + (2 El(g)p) 2 ≤ CkgkA . ! p,q,1 Xl=0 Xk=0 This proves (24) for the range (d − 1)/d 15 For 1 ≤ p< ∞ we can use the inequality Z kg − PkgkLp ≤ CEk(g)p, g ∈ Lp, k ∈ +, which follows from the uniform boundedness of the projectors Pk in Lp (for details, see Section 5.2). As above, in conjunction with (??) this gives p kd p kd p ak = |λh(g)| ≤ C2 kQkgkLp ≤ C2 Ek−1(g)p, k =1, 2,..., h∈Hd Xk p and a0 ≤ kgkLp . Then (24) follows by simple substitution into the expression for the weighted ℓq(ℓp)-norm of Λg defined in (23). It remains to prove that Λ is surjective. Since the operator Λ is obviously injective on s any Besov space Bp,q,1 embedded into L1, surjectivity together with (24) automatically implies boundedness of the inverse mapping Λ−1 as a consequence of the open mapping theorem for F -spaces (all γ-quasi-normed Banach spaces and, in particular, the Besov s spaces Bp,q,1 are F -spaces). Let Γ=(γh)h∈Hd be an arbitrary sequence with finite weighted ℓq(ℓp)-quasi-norm: q/p 1/q ∞ kΓk := 2k(sp−d)|γ |p < ∞. (27) ℓq(ℓp) h k=0 d h∈Hk X X s To show the surjectivity of Λ, we need to find a g ∈ Bp,q,1 such that d λh(g)= γh, h ∈ H . (28) We will show this in all detail only for the case (d − 1)/d 1. Set k qk := γhh, pk := ql, k ∈ Z+. h∈Hd l=0 Xk X We first show that (pk)k∈Z+ is fundamental in L1 whenever s>d(1/p − 1) in (27). Indeed, by (19) and (26) we have 1/p ∞ m ∞ kp − p k ≤ kq k ≤ C 2−ld |γ | ≤ C 2−ld |γ |p m k L1 l L1 h h l=k+1 l=k+1 h∈Hd l=k h∈Hd X X Xl−1 X Xl q/p 1/q ∞ ≤ C2−k(s−d(1/p−1)) (2l(sp−d)|γ |p h l=k d h∈Hl X X −k(s−d(1/p−1)) ≤ C2 kΓkℓq(ℓp) 16 for arbitrary 1 ≤ k 1/p b =2−ld |γ |p , l ≥ k, l h h∈Hd Xl whereas the parameter r in (26) has been set to r = 1. The above bound on kpm − pkkL1 shows that (pk)k∈Z+ is fundamental in L1, and that it converges to a function g ∈ L1. Obviously, this implies the L1 convergence of the Haar series with coefficients γh to g, i.e., g = γhh, d hX∈H assuming the agreed upon blockwise ordering of the Haar functions. By orthogonality d ′ of H and λh ∈ L∞ = L1 for all Haar functions, we also must have (28). s It remains to show that the above g belongs to Bp,q,1. We use p p p Z Ek(g)p ≤kg − pkkLp ≤ kqlkLp , k ∈ +, Xl>k and, similarly, ∞ p p kgkLp ≤ kqlkLp . Xl=0 As above, together with Lemma 1, (19), and (26) with 0 <ǫ ∞ ∞ q/p q q ksp p kgk s ≤ Ckgk s ≤ C 2 kqlk Bp,q,1 Ap,q,1 Lp k=0 l=k ! ∞ X∞ X ∞ k(s−ǫ)q lǫq q q ≤ C 2 2 kqlkLp ≤ C kqlkLp k=0 l=k ! l=0 X X q/p X ∞ lsq −ld p q ≤ C 2 2 |γh| = CkΓk . ℓq(ℓp) l=0 h∈Hd X Xl This finishes the proof of Theorem 3 for (d − 1)/d ∞ ∞ 1/q −ks ls q kpm − pkkLp ≤ kqlkLp ≤ C2 (2 kqlkLp ) ! l=Xk+1 l=Xk+1 q/p ∞ ≤ C2−ks 2l(sp−d) |γ |p , 1 ≤ k ∞ Ek(g)p ≤ kqlkLp , kgkLp ≤ kqlkLp . Xl>k Xl=0 The rest of the proof is, up to obvious changes in the application of (26), the same as for p< 1. 4. Proofs: Necessary conditions 4.1. Proof of Theorem 2 s Let the parameters satisfy (3). For (d − 1)/d < p < 1 the Besov space Bp,q,1 is continuously embedded into L1 if and only if d(1/p − 1) s see (7), while for p ≥ 1 this embedding is obvious. In all these cases, the dual of Bp,q,1 is therefore infinite-dimensional. This proves the necessity of the condition in Theorem 2. For the sufficiency, we can concentrate on the range 0 be a dyadic cube ∆0 such that φ(χ∆0 ) =6 0. Without loss of generality, we may assume that d ∆0 = I , φ(χ∆0 )=1. d d d Each dyadic cube in Tk , k ≥ 0, is the disjoint union of 2 dyadic cubes from Tk+1. Therefore, using the linearity of φ, we can construct by induction a sequence of dyadic d cubes ∆k ∈ Tk such that −kd ∆0 ⊃ ∆1 ⊃ ... ⊃ ∆k ⊃ ..., φ(χ∆k ) ≥ 2 , k =0, 1,.... For a given sequence a =(ak)k∈Z+ , consider the function m d fm = alχ∆l ∈ Sm, m =0, 1,.... (29) Xl=0 The Lp and Besov space quasi-norms of fm can be computed exactly. Indeed, fm is d k ′ constant on all cubes ∆ ∈ Tk but ∆k, and equals the constant ξk := l=0 al on ∆k := ∆k\∆k+1, where P 1 µ(∆′ )=(1 − 2−d)µ(∆ )=(1 − 2−d)2−kd ≥ µ(∆ ), k k 2 k 18 and µ denotes the Lebesgue measure on Rd. Thus, m n p p −d −nd kfmkLp = (1 − 2 ) 2 al . (30) n=0 X Xl=0 ′ ′ Moreover, by Lemma 2 applied to Ω = ∆k and Ω = ∆ we have k m n p p p −d −nd Ek(fm)p = kfm −ξkkL (∆ ) = (1−2 ) 2 al , k =0, 1,...,m−1, (31) p k n=k+1 l=k+1 X X while obviously Ek(fm)p = 0 for k ≥ m. Now we choose the particular sequence kd −1 ak =2 (k + 1) , k =0, 1,..., in (29). Substituting into (30) and (31), we have for 0 19 4.2. The case s = d(1/p − 1) > 0, 0 < q ≤ p< 1 d d(1/p−1) In Section 3.1 we established that H has the Schauder basis property for Bp,q,1 if 0 < q ≤ p, (d − 1)/d md d −m d d fm =2 χ∆m ∈ Sm, ∆m := [0, 2 ) ∈ Tm, m =0, 1,.... Obviously, Ek(fm)p = 0 for k ≥ m, and using Lemma 2 we compute −md/p md −md(1/p−1) Ek(fm)p = kfmkLp =2 2 =2 , k =0, 1,...,m − 1. Thus, since m−1 q q q kd(1/p−1)q − − kfmk d(1/p 1) ≈kfmk d(1/p 1) = kfmkLp 1+ 2 ≈ 1, m =0, 1,..., Bp,q,1 Ap,q,1 k=0 ! X (33) d(1/p−1) the Bp,q,1 quasi-norms of fm are uniformly bounded for the indicated parameter range. The Haar coefficients of fm are easily computed: d 1, h ∈ H0 , λ (f )= 2(k−1)d, ∆ ⊂ supp(h), h ∈ Hd, k =1,...,m h m m k 0, otherwise. From this, it is immediate that λh(fm)h = fl − fl−1, l =0, 1, . . . , m, h∈Hd Xl where we have set f−1 = 0 for convenience. Consider now the function k 2k 2k l l ld g2k := λh(fm)h = (−1) fl = (−1) 2 χ∆l , l=0 h∈Hd l=0 l=0 X X2l X X where 2k ≤ m. This function is of the same type as the functions f2k considered in the l ld previous subsection but with a different coefficient sequence a = ((−1) 2 )l∈Z+ and a d specific nested sequence of dyadic cubes ∆l ∈ Tl . Using the formula (31), we compute −ld(1/p−1) El(g2k)p ≈ 2 , l =0,..., 2k − 1, El(g2k)p =0, l ≥ k, and conclude that 2k−1 q ld(1/p−1) q kg2kk d(1/p−1) ≥ c (2 El(g2k)p) ≥ ck, 2k ≤ m, Bp,q,1 Xl=0 20 grows unboundedly if k, m → ∞, independently of 0 4.3. The case s = d(1/p − 1) > 0, (d − 1)/d s = d(1/p − 1), (d − 1)/d the failure of the Schauder basis property claimed in Theorem 1 b) cannot be deduced s from Theorem 2. Since the Haar coefficient functionals λh are continuous on Bp,q,1, the finite-rank partial sum operators Pk defined in (9) represent bounded operators. s However, they are not uniformly bounded on Bp,q,1 for the parameter values in (34). To establish this fact, we need another type of examples which were introduced in ′ d [18] in a slightly different form. Fix a k ≥ 1 arbitrarily, and select a subset T ⊂ Tk d d−1 ′ ′ such that each dyadic cube ∆ ∈ Tk−1 contains exactly 2 cubes ∆ ∈ T (and thus d−1 d ′ ′ kd−1 d exactly 2 cubes from Tk \T ). This way T contains 2 cubes from Tk which will ′ kd−1 be enumerated in an arbitrary order, and denoted by ∆i, i = 1,..., 2 . For each kd−1 d ′ i =1,..., 2 , choose a dyadic cube ∆i ∈ Tk+i contained in ∆i, and set 2kd−1 (k+i)d −α kd−1 fk := biχ∆i , bi := 2 i , i =1,..., 2 , i=1 X where α< 1/q is fixed. It is clear by this construction that on each dyadic cube ∆ we have either fk = 0 ′ ′ 1 d on a subset Ω ⊂ ∆ of measure µ(Ω ) ≥ 2 µ(∆), or fk is constant on ∆. For ∆ ∈ Tl ′ d with l 2kd−1 2kd−1 p p −(k+i)d p −(k+i)d(1−p) −αp −kd(1−p) El(fk)p = kfkkLp = 2 bi = 2 i ≈ 2 . i=1 i=1 X X For l = k +1,...,k +2kd−1 − 1 we similarly have 2kd−1 2kd−1 p p −(k+i)d(1−p) −αp −ld(1−p) −αp El(fk)p = k biχ∆i kLp = 2 i ≈ 2 (l − k) , i=Xl+1−k i=Xl+1−k 21 kd−1 d(1/p−1) while El(fk)p = 0for l ≥ k+2 . Substitution into the Ap,q,1 quasi-norm expression gives k k+2kd−1−1 q −kd(1/p−1)q ld(1/p−1)q ld(1/p−1) −ld(1/p−1) −α q kfkk d(1/p−1) ≈ 2 2 + (2 2 (l − k) ) Bp,q,1 Xl=0 l=Xk+1 2kd−1 ≈ 1+ i−αq ≈ 2kd(1−αq), k =1, 2,..., i=1 X where αq < 1 has been used. The construction of T ′ also allows us to compute the best approximations of the partial sum Pkfk of the finite Haar expansion of fk. Indeed, according to (10-??), Pkfk is given by 2kd2−(k+i)db =2kdi−α, x ∈ ∆′ , i =1,..., 2kd−1, P f (x)= i i k k 0, x ∈ ∆′, ∆′ ∈ T d\T ′. k d Since Pkfk ∈ Sk we have El(Pkfk)p = 0 for l ≥ k. By the selection rule of the cubes ′ ′ d ′ d ′ ∆i ∈ T , on any cube ∆ ∈ Tl with l 2kd−1 2kd−1 p p −kd kd −α p −kd(1−p) −αp El(Pkfk)p = kPkfkkp = 2 (2 i ) =2 i i=1 i=1 X X for l =0,...,k − 1. Consequently, we have q/p 2kd−1 q kd(1/p−1)q q −αp kd(1/p−1)q kPkfkk d(1/p−1) ≈ 2 kPkfkkp ≈ i ≈ 2 , Bp,q,1 i=1 X since for the parameter range of interest α< 1/q < 1/p. Comparing the above estimates for the Besov quasi-norms of Pkfk and fk shows that kPkfkk d(1/p−1) Bp,q,1 kd(1/p−1/q) − − kPkkBd(1/p 1)→Bd(1/p 1) ≥ ≥ c2 , k ≥ 1, p,q,1 p,q,1 kfkk d(1/p−1) Bp,q,1 for values 0 5. Remarks and extensions 5.1. Higher-order spline systems The Schauder basis property of spline systems of order m > 1 in Lebesgue-Sobolev s spaces and in Besov spaces Bp,q,m defined by m-th order moduli of smoothness m d ωm(t, f)p := sup k∆y fkLp(Iy,m), t> 0, 0≤|y|≤t 22 m where ∆y denotes the m-th order forward difference operator with step-size y, and d d d Iy,m = {x ∈ I : x + my ∈ I }, has also attracted attention, see [1, 4, 5, 6] for 1 ≤ p ≤∞. Note that, with the exception of [6], in these papers the case d> 1 has been treated by the tensor product construction, i.e., with analogs of H˜ d, not the Haar wavelet system Hd. For 0 kP mgkp ≤ C2kd(p−1) kgkp , 0 1 we have m,d s F 6⊂ Bp,q,m, s ≥ m, 0 23 s 1 ≤ s < 1/p, 0 m ≤ s < m − 1+1/p, 0 s also for d> 1. In order for a spline system to belong to Bp,q,m for this parameter range, it is desirable that it consists of splines which are locally polynomials of exactly total degree m − 1, and globally belong to Cm−2 over well-shaped and refinable partitions. These are the spline functions that are maximally smooth in Lp, 0 5.2. The exceptional case s =0, 1 ≤ p< ∞ s With a few exceptions, the literature about Besov function spaces such as Bp,q,1 deals only with parameters s > 0 although the definition given in Section 2.2 formally leads to meaningful spaces in the case s = 0 as well. As our proof of Theorem 2 reveals, the statement s ′ (Bp,q,1) = {0} about the triviality of the dual space remains true also for s = 0 if we are in the range 0 d 0 d Theorem 4 The Haar wavelet system H is an unconditional Schauder basis in Bp,q,1(I ) if and only if 1 Ek(g)p ≈kg − PkgkLp , k =0, 1,..., (36) 24 for all g ∈ Lp, 1 ≤ p< ∞. Indeed, Ek(g)p ≤kg − PkgkLp ≤kg − skkLp + kPk(g − sk)kLp ≤ CEk(g)p d since Pksk = sk for the best approximating element sk ∈ Sk and since the partial sum operators Pk are uniformly bounded in Lp, see Lemma 3. Thus, if P is a partial sum d d operator for H of the form (13) then El(Pg)p = 0 for l>k since Pg ∈ Sk+1, and El(Pg) ≤kPg − PlPgkLp = kP (g − Pl)gkLp ≤ Ckg − PlgkLp ≤ CEl(g)p (37) for l =0,...,k, since P and Pl commute, and the P are uniformly bounded in Lp. This, together with Lemma 1 for s = 0, establishes the uniform boundedness of the operators P , and consequently the Schauder basis property of Hd in the assumed natural ordering, 0 in Bp,q,1 for all 1 ≤ p< ∞, 0 kPJ gkLp ≤ CkgkLp , PJ g := θhλh(g)h, (38) Xh∈J d for all g ∈ Lp and all finite subsets J of H . Since the projectors PJ and Pk commute, as in (37) this also implies Ek(PJ g) ≤ CEk(g), k =0, 1,..., g ∈ Lp. (39) 0 We use (38-39) in conjunction with Lemma 1, and bound the Bp,q,1 quasi-norm of PJ g as follows: ∞ q q q q kPJ gk 0 ≤ CkPJ gk 0 = C kPJ gkL + Ek(PJ g)p Bp,q,1 Ap,q,1 p k=0 ! ∞ X ≤ C kgkq + E (g)q = Ckgkq ≤ Ckgkq . Lp k p A0 B0 ! p,q,1 p,q,1 Xk=0 d 0 This proves the unconditionality of H in Bp,q,1 for 1 1, l < m, E (f ) = l m 1 0, l ≥ m, and ≈ 2k − l, l< 2k, E (g ) l 2k 1 =0, l ≥ 2k, 25 d where g2k = PJk fm for a specific choice of Jk ⊂ H and 2k ≤ m. Substitution into the 0 formulas for the equivalent A1,q,1 quasi-norms of these functions gives q q q q q+1 kfmk 0 ≈ m, kg2kk 0 = kPJk fmk 0 ≈ (2k − l) ≈ k . B1,q,1 B1,q,1 B1,q,1 l ∞ −ld 2 2 1/2 kfkLp ≈k( 2 λh(f) |h| ) kLp , f ∈ Lp, 1 ∞ q q q q kgk 0 ≈ kgk 0 ≈kgkL + kg − PkgkL Bp,q,1 Ap,q,1 p p Xk=0 1/2 q ∞ ∞ ≈ 2−ldλ (f)2|h|2 . h k=0 l=k h∈Hd l X X X Lp 2 d Recall that |h| = χ∆, where ∆ is the support cube of the Haar function h ∈ H . 0 We doubt that the above equivalent quasi-norm for Bp,q,1, 1 q/2 ∞ ∞ q −ld 2 kgkB0 ≈ 2 λh(g) , 0 ∞ 2 −kd 2 kgk 0 ≈ (k + 1)2 λh(g) , 0 26 ˜ d s 5.3. Tensor-product Haar system H as basis in Bp,q,1 The historically first constructions of Schauder bases for Banach spaces of smooth functions over higher-dimensional cubes and manifolds [1, 4, 5] exclusively used tensor products of univariate Schauder bases. Later, due to the desire to work with systems with better support localization and the need to cover quasi-Banach spaces as well, wavelet-type constructions became more popular. As a matter of fact, in more recent texts such as [29], the tensor-product construction is examined only with respect to function spaces of dominating mixed smoothness. This raises the following question: Can the Haar tensor-product system H˜ d be a s Schauder basis in any of the separable Besov function spaces Bp,q,1 it is contained in? Note that if we write ˜ d ∞ ˜ d H = ∪k=0Hk , ˜ d ˜ d ˜ d ˜ d d where Hk = {h ∈ Sk : h 6∈ Sk−1 for k ≥ 1 and H0 = H0 , and order by blocks, then the ˜ ˜ d set of partial sum operators P associated with H expansions contains {Pk}k∈Z+ as a subsequence. Consequently, such an ordering of H˜ d assumed, {P˜} is uniformly bounded s d in Bp,q,1 only if the Haar wavelet system H with the natural ordering is a Schauder basis in this space. When this is the case, one may hope that H˜ d is a Schauder basis as well. For 1 ≤ p < ∞ this is indeed the case: The properly ordered tensor-product Haar ˜ d s system H is a Schauder basis in Bp,q,s, 1 ≤ p < ∞, 0 ˜ −2 ˜ ˜ ˜ ˜ d Λh˜ : f → Λh˜ f := khkL2 fh dx h, h ∈ H , d ZI onto one-dimensional subspaces spanned by tensor-product Haar functions h˜ ∈ H˜ d can- s d ˜ d not be uniformly bounded in Bp,q,1(I ). As a consequence, H cannot be a Schauder basis s d ˜ d in Bp,q,1(I ) for 0 ˜ 2 −k+1 ˜ −kd khkkL2 =2 , fkhk dx =2 , d ZI and q k−1 −kd q ˜ q kΛ˜ fkk s = (2 · 2 ) khkk s , hk Bp,q,1 Bp,q,1 we arrive at q kΛ˜ fkk s −k(d−1)q −kq/p q hk Bp,q,1 2 2 k(1/p−1)(d−1)q kΛ˜ k s s ≥ ≈ =2 . hk Bp,q,1→Bp,q,1 q −kdq/p kfkk s 2 Bp,q,1 Since 0 To this end, recall from Lemma 1 and (36) that ∞ q q q ks q s kfk s ≈kfk s ≈kfk + (2 kf − PkfkLp ) , f ∈ B , (42) Bp,q,1 Ap,q,1 Lp p,q,1 Xk=0 for the parameters in (41). Using ∞ kf − PkkLp ≤ k(Pl+1 − Pl)fkLp Xl=k and (26), this further transforms to ∞ q ks q s kfk s ≈ (2 kPk − Pk−1fkLp ) , f ∈ B , (43) Bp,q,1 p,q,1 Xk=0 28 if 0 1/2 ˜ −1 ˜ 2 kgkLp ≈ (µ(supp(h)) λh˜(g)h) , g ∈ Lp, 1 q 1/2 ∞ q ks ˜ −1 ˜ 2 s kfkBs ≈ 2 (µ(supp(h)) λh˜ (g)h) , f ∈ Bp,q,1. p,q,1 k=0 h˜∈H˜ d k X X Lp 0 For s = 0, we obtain a similar characterization of the Bp,q,1 quasi-norm if we substitute ˜ d s (44) into (42). This shows that H is an unconditional Schauder basis in Bp,q,1 for the parameters in (41) if 1 ˜ 2 2k ˜ 2k ˜ Hk+1 = ∪i=1{h2k+i,n}n=1,...,2k+1 ∪ ∪i=1{hn,2k+i}n=1,...,2k (45) ˜ ′ ˜ ′′ Hk+1 Hk+1 | {z } | {z } ˜ ′ ˜ ′′ for k ≥ 0. The enumeration of the functions within the subblocks Hk+1 and Hk+1 in (45) is assumed lexicographic with respect to the index pairs (i, n). With this at hand, any partial sum operator P˜ with respect to H˜ 2 is the linear combination of a few projectors ˜ Qn1,n2 = Qn1 ⊗ Qn2 . Indeed, similar to (13), any partial sum operator P takes the form ˜ ˜ ˜ ˜ 2 Pg = Pkg + ∆Pg; ∆Pg := λh˜ (g)h ∈ Sk+1. h˜∈H˜¯ 2 Xk+1 29 ˜¯ 2 ˜ 2 for some k = 0, 1,... and some section Hk+1 of Hk+1 taken in the described order. ˜¯ 2 ˜ ′ Obviously, Pk = Q2k,2k . If the section Hk+1 is contained in Hk+1 then ˜ ∆P = (Q2k+i−1 − Q2k ) ⊗ Q2k+1 +(Q2k+i − Q2k+i−1) ⊗ Qn = Q2k+i−1,2k+1 − Q2k,2k+1 + Q2k+i,n − Q2k+i−1,n for some n =1,..., 2k+1 and i =1,..., 2k. Otherwise, we have ˜ ∆P = (Q2k+1 − Q2k ) ⊗ Q2k+1 + Q2k ⊗ (Q2k+i−1 − Q2k )+ Qn ⊗ (Q2k+i − Q2k+i−1) = Q2k+1,2k+1 − Q2k,2k+1 + Q2k+,2k+i−1 − Q2k,2k + Qn,2k+i − Qn,2k+i−1, for some n = 1,..., 2k and i = 1,..., 2k. Altogether, this shows that P˜ is always a linear combination of a few tensor-product projectors Qn1,n2 . Since tensor-products of uniformly L1 bounded operators are uniformly L1 bounded as well, it follows that ˜ kPgkL1 ≤ Ckgk, g ∈ L1, (46) d uniformly for all partial sum operators P˜. Thus, H˜ is a Schauder basis in L1 for d = 2. By induction, this holds for all d > 1. This is most probably known, and has been proved here only in the absence of a proper reference. ˜ d s For the parameters under consideration, the Schauder basis property of H in B1,q,1 follows now from (46) using the same arguments as in the proof of Theorem 4 in Section 5.2. ˜ d s Step 4. Finally, it is easy to see that H is not unconditional in B1,q,1 for any 0 ≤ s < 1, 0 ˜ d Jk := {hn ⊗ h2k+1 ⊗ ... ⊗ h2k+1}n=1,...,2k+1 ⊂ Hk+1, k =0, 1,.... This Jk is a finite section of H ⊗ h2k+1 ⊗ ... ⊗ h2k+1. Since H is not unconditional in L1(I), the subsets Jk cannot be uniformly unconditional in L1 either. This means that ′ ˜ d there is a sequence of subsets Jk ⊂ Jk ⊂ Hk+1, and a sequence of functions ˜ d d d d gk ∈ span(Hk+1) = span(Hk+1)= Sk+1 ⊖L2 Sk , k =0, 1,..., such that kP˜J′ gkkL k 1 →∞, k →∞, (47) kgkkL1 where P˜ ′ denotes the partial sum projector with respect to the subset J ′ ⊂ H˜ d . Jk k k+1 Consider now the Bs quasi-norms of g and P˜ ′ g . Since these functions belong 1,q,1 k Jk k to Sd but are L -orthogonal to Sd we have E (g ) = E (P˜ ′ g ) = 0 for l>k while k+1 2 k l k 1 l Jk k 1 for l ≤ k we have P g = P P˜ ′ g = 0 and according to (36) l k l Jk k E (g ) ≈kg k , E (P˜ ′ g ) ≈kP˜ ′ g k . l k 1 k 1 l Jk k 1 Jk k 1 30 s If we substitute this into the expressions for the A1,q,1 norms then using Lemma 1 we get kP˜ ′ gkkBs kP˜ ′ gkkAs kP˜ ′ g k Jk 1,q,1 Jk 1,q,1 Jk k L1 ˜ ′ s s kPJ kB1,q,1→B1,q,1 ≥ ≈ ≈ . k s s kgkkB1,q,1 kgkkA1,q,1 kgkkL1 Due to (47), this contradicts the unconditionality criterion formulated in Lemma 3. ✷ 5.4. The exceptional case q = ∞ s Since Bp,∞,1 is non-separable for 0 < s ≤ max(1, 1/p), it cannot possess Schauder bases. However, one can talk about the (unconditional) Schauder basis property of a system (fm)m∈N in a quasi-Banach space X with respect to its closure in X. In this case, d (fm)m∈N is called (unconditional) basis sequence in X. 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