THE HAAR SYSTEM AS A SCHAUDER BASIS IN SPACES OF HARDY-SOBOLEV TYPE. 1. Introduction

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1 THE HAAR SYSTEM AS A SCHAUDER BASIS IN SPACES OF HARDY-SOBOLEV TYPE GUSTAVO GARRIGÓS ANDREAS SEEGER TINO ULLRICH Abstract. We show that, for suitable enumerations, the Haar system is a Schauder basis in the classical Sobolev saces in R d with integrability 1 < < and smoothness 1/ 1 < s < 1/. This comlements earlier work by the last two authors on the unconditionality of the Haar system and imlies that it is a conditional Schauder basis for a nonemty oen subset of the (1/, s)-diagram. The results extend to (quasi-)banach saces of Hardy-Sobolev and Triebel-Lizorkin tye in the range of arameters < < and max{d(1/ 1), 1/ 1} < s < min{1, 1/}, d d+1 which is otimal excet erhas at the end-oints. 1. Introduction We recall the definition of the (inhomogeneous) Haar system in R d. Consider the 1-variable functions h (0) = 1 [0,1) and h (1) = 1 [0,1/2) 1 [1/2,1). For every ε = (ε 1,..., ε d ) {0, 1} d one defines Finally, one sets h (ε) (x 1,..., x d ) = h (ε 1) (x 1 ) h (ε d) (x d ). h (ε) k,l (x) = h(ε) (2 k x l), k N 0, l Z d, Denoting Υ = {0, 1} d \ { 0}, the Haar system is then given by { H d = h ( 0) }l Zd { } 0,l h (ε) k,l : k N 0, l Z d, ε Υ. Observe that su h (ε) k,l is the dyadic cube I k,l := 2 k (l + [0, 1] d ). In this aer we consider basis roerties of H d in Besov saces B s,q, and Triebel-Lizorkin saces F s,q in R d. We refer to [13], [14] for definitions and 2010 Mathematics Subject Classification. 46E35, 46B15, 42C40. Key words and hrases. Schauder basis, Unconditional basis, Haar system, Sobolev sace, Triebel-Lizorkin sace. G.G. was suorted in art by grants MTM P, MTM C2-1- P, MTM P from MINECO (Sain), and grant 19368/PI/14 from Fundación Séneca (Región de Murcia, Sain). A.S. was suorted in art by NSF grant DMS T.U. was suorted the DFG Emmy-Noether rogram UL403/1-1. 1

2 2 G. GARRIGÓS A. SEEGER T. ULLRICH roerties of these saces and to [1] for terminology and general facts about bases in Banach saces. In the 1970 s, Triebel [11, 12] roved that the Haar system H d is a Schauder basis on B,q(R s d ) if (1) d d+1 < <, 0 < q <, max { d( 1 1), 1 1} < s < min { 1, 1 }, and that this range is maximal, excet erhas at the endoints. Moreover, the basis is unconditional when (1) holds; see [15, Theorem 2.21]. Concerning F s,q saces, however, in [15] it is only shown that H d is an unconditional basis for F s,q(r d ) when, besides (1), the additional assumtion (2) max { d( 1 q 1), 1 q 1} < s < 1 q is satisfied. Recently, two of the authors showed in [9, 10] that the additional restriction (2) is in fact necessary, at least when d = 1. It was left oen whether suitable enumerations of the Haar system can form a Schauder basis in F,q s in the larger range (1). We shall answer this question affirmatively. Given an enumeration {u 1, u 2,...} of the system H d, we let P N be the orthogonal rojection onto the subsace sanned by u 1,..., u N, i.e. (3) P N f = N n=1 u n 2 2 f, u n u n. The sequence {u n } n=1 is a Schauder basis on F s,q if (4) lim P Nf f F s N,q = 0, for all f F,q. s In view of the uniform boundedness rincile, density theorems and the result for Besov saces, (4) follows if we can show that the oerators P N have uniform F,q s F,q s oerator norms. Note, that the condition s < 1/ is necessary since the Haar functions need to belong to F,q. s By duality, if 1 < <, the condition s > 1/ 1 becomes also necessary, so the range in (1) is otimal in this case. If 1, then an interolation argument shows that (1) is also a maximal range, excet erhas at the end-oints; see 4 below. Definition. An enumeration U = {u 1, u 2,...} of the Haar system H d is admissible if the following condition holds for each cube I ν = ν + [0, 1] d, ν Z d. If u n and u n are both suorted in I ν and su(u n ) > su(u n ), then necessarily n < n. The table in the figure shows how to obtain an admissible (natural) enumeration of H d via a diagonalization of the intervals I ν versus the levels k. We first label the set Z d = {ν 1, ν 2,...}. Then, we follow the order indicated by the table, where being at osition (ν i, k) means to ick all the Haar functions with suort contained in I νi and size 2 kd, arbitrarily enumerated, before going to the subsequent table entry. Our main result reads as follows.

3 THE HAAR SYSTEM AS A SCHAUDER BASIS 3 k\i ν I ν0 I ν1 I ν2 I ν3 I ν Figure 1. An admissible enumeration of H d. Theorem 1.1. Let U = {u n } n=1 be an admissible enumeration of the Haar system H d. Assume that d (i) d+1 < <, (ii) 0 < q <, (iii) max{d( 1 1), 1 1} < s < min{1, 1 }. Then U is a Schauder basis in F s,q(r d ). s s d+1 d+1 1 2d d Figure 2. Unconditionality of the Haar system in Hardy- Sobolev saces in R and R d In the left art of Figure 2, the traezoid is the arameter domain for which the Haar system is a Schauder basis in the Hardy-Sobolev sace H(R) s (= F,2 s (R)) while the shaded art reresents the arameter domain for which the Haar system is an unconditional basis in H(R). s The right figure shows the resective arameter domain for H(R s d ). The heart of the matter is a boundedness result for the dyadic averaging oerators E N given by (5) E N f(x) = µ Z d 1 IN,µ (x) 2 N I N,µ f(t)dt with I N,µ = 2 N (µ + [0, 1) d ), µ Z d, N = 0, 1, 2,...

4 4 G. GARRIGÓS A. SEEGER T. ULLRICH Note that E N f is just the conditional exectation of f with resect to the σ-algebra generated by the set D N of all dyadic cubes of side length 2 N. There is a well known relation between the Haar system and the dyadic averaging oerators, namely for N = 0, 1, 2,..., (6) E N+1 f E N f = 2 Nd f, h (ε) N,µ h(ε) N,µ, ε Υ µ Z d i.e. E N+1 E N is the orthogonal rojection onto the sace generated by the Haar functions with Haar frequency 2 N. Now let η 0 be a Schwartz function on R d, suorted in { ξ < 3/8} and so that η 0 (ξ) = 1 for ξ 1/4. Let Π N be defined by (7) Π N f(ξ) = η 0 (2 N ξ) f(ξ). There is a basic standard inequality (almost immediate from the definition of Triebel-Lizorkin saces) (8) su Π N f F s,q C(, q, s) f F s,q N which is valid for all s R and for 0 < <, 0 < q. Moreover, (8) and the fact that Π N g g F s,q 0 for Schwartz functions g gives (9) lim N Π Nf f F s,q = 0 if f F,q s and 0 <, q <. The main tool in roving Theorem 1.1 is a similar bound for the oerators E N which of course follows from the corresonding bound for E N Π N. It turns out that the oerators E N Π N enjoy better maing roerties in Besov saces. Similar bounds are also satisfied by rojection oerators into sets of Haar functions with fixed Haar frequency. Namely, for N N and functions a l (Z d Υ), we define (10) T N [f, a] = a µ,ε 2 Nd f, h (ε) N,µ h(ε) N,µ. ε Υ µ Z d Observe that the choice a µ,ε 1 recovers the oerator E N+1 E N. Then, we shall rove the following. Theorem 1.2. Let d/(d + 1) <, 0 < r, and (11) max{d(1/ 1), 1/ 1} < s < min{1, 1/}. Then there is a constant C := C(, r, s) > 0 such that for all f B s, (12) su E N f Π N f B s,r C f B s,. N Moreover, (13) su T N [f, a] B s,r a f B s,. N

5 THE HAAR SYSTEM AS A SCHAUDER BASIS 5 We have the embedding F s,q F s, B s, which we use on the function side. For r we have the embedding B s,r F s,r (by Minkowski s inequality in L /r ) and if also r < q we have F s,r F s,q; these two are used for E N f Π N f, or T N [f, a]. In articular we conclude from Theorem 1.2 that E N Π N is bounded on F s,q, uniformly in N. Hence Corollary 1.3. Let, s be as in (11) and 0 < q. Then (14) su N E N f F s,q + su N su a l 1 T N [f, a] F s,q f F s,q. The roofs in this aer use basic rinciles in the theory of function saces, such as L inequalities for the Peetre maximal functions. A different aroach to Corollary 1.3 via wavelet theory is resented in the subsequent aer [3]. The main arguments and the roof of Theorem 1.2 are contained in 2. In 3 we show how estimates in the roof of Theorem 1.2 are used to deduce Theorem 1.1. Finally, in 4 we discuss the otimality of the results. 2. Proof of Theorem 1.2 We start with some reliminaries on convolution kernels which are used in Littlewood-Paley tye decomositions. Let β 0, β be Schwartz functions on R d, comactly suorted in ( 1/2, 1/2) d such that β 0 (ξ) > 0 when ξ 1 and β(ξ) > 0 when 1/8 ξ 1. Moreover assume β has vanishing moments u to a large order (15) M > d + s, that is, (16) R d β(x) x m 1 1 x m d d dx = 0 when m m d < M. For k = 1, 2,... let β k := 2 kd β(2 k ) and L k f = β k f. We shall use the inequality (17) g B s,r 2 ksr L k g r k=0 ) 1/r and aly it to g = E N f Π N f. Inequality (17) is of course just one art of a characterization of B s,r saces by sequences of comactly suorted kernels (or local means ), with sufficient cancellation assumtions, see for examle [14, 2.5.3].

6 6 G. GARRIGÓS A. SEEGER T. ULLRICH Let η 0 Cc (R d ) be as in (7), that is, suorted on { ξ < 3/8} and such that η 0 (ξ) = 1 when ξ 1/4. Define Λ 0, and Λ k for k 1 by Λ 0 f(ξ) = η 0(ξ) f(ξ) β 0 (ξ) Λ k f(ξ) = η 0(2 k ξ) η 0 (2 k+1 ξ) β(2 k ξ) Then j=0 L jλ j = Id with convergence in S, and su 2 js Λ j f f B s,. j 0 Moreover Π N = N j=0 L jλ j, and therefore (18) E N f Π N f = N (E N L j Λ j f L j Λ j f) + j=0 If we use the convenient notation E N := I E N, then the asserted estimate (12) will follow from (19) and (20) 2 ksr k=0 j=n+1 f(ξ), k 1. j=n+1 E N L j Λ j f. L k E N L j Λ j f r 1/r su 2 ) js Λ j f. j 2 ksr N L k E NL j Λ j f r 1/r su 2 ) js Λ j f. j k=0 j=0 Below we shall use variants of the Peetre maximal functions, which are a standard tool in the study of Besov and Triebel-Lizorkin saces. We define (21a) (21b) (21c) M j g(x) = M jg(x) = su g(x + h), h 2 j+1 su g(x + h), h 2 j+5 M g(x + h) A,jg(x) = su h R d (1 + 2 j h ) A, where h = max{ h 1,..., h d }, h = (h 1,..., h d ) R d. These different versions are introduced for technical uroses in the roofs. They satisfy obvious ointwise inequalities, M j g(x) M jg(x) C A M A,jg(x),

7 and (22) THE HAAR SYSTEM AS A SCHAUDER BASIS 7 M j g(x) inf M jg(x + h) h 2 j+4 ) 1/r, (2 (j 4)d [M jg(x + h)] r dh 0 < r. h 2 j+4 Below we shall use Peetre s inequality ([6], see also [13, 1.3.1]) (23) M A,jf C,A f, 0 <, A > d/, for f S (R d ) satisfying (24) su( f) {ξ : ξ 2 j+1 }. Throughout we shall assume that M A; we require secifically d/ < A < M s. The main estimates needed in the roof of (19) and (20) are summarized in Proosition 2.1. Let 0 < and (25) 2 N j 2 j k 2 (j N)(d 1)( 1 1) + if j, k N + 1, 2 N k 2 j N if j N, k N + 1, B(j, k, N) = 2 k N 2 j N 2 (N k)d( 1 1) + if 0 j, k N, j N k j+ 2 +[N k+(j k)(d 1)]( 1 1) + if j N + 1, k N. Then the following inequalities hold for all f S (R d ) whose Fourier transform is suorted in { ξ 2 j+1 }. (i) For j N + 1, { B(j, k, N) f if k N + 1, (26) L k E N [L j f] [B(j, k, N) + 2 j k (M A) ] f if 0 k N. (ii) For 0 j N, (27) L k E N[L j f] {[ B(j, k, N) + 2 j k (M A) ] f if k N + 1, B(j, k, N) f if 0 k N. (iii) The same bounds hold if the oerators E N in (i) and E N relaced by T N [, a], uniformly in a 1. in (ii) are We begin with two reliminary lemmata, the first a straightforward estimate for L k L j. Lemma 2.2. Let k, j 0 and suose that f is locally integrable. Let M be as in (16) with M > A > d/. Then (28) L k L j f(x) 2 k j (M A) M A,max{j,k} f(x).

8 8 G. GARRIGÓS A. SEEGER T. ULLRICH If f S (R d ) with f(ξ) = 0 for ξ 2 j+1 then L k L j f 2 k j (M A) f. Proof. The second assertion is an immediate consequence of (28), by (23). We have L k L j f = γ j,k f where γ j,k = β k β j. By symmetry we may assume k j. Using the cancellation assumtion (16) on the β j we get γ j,k (x) = = 2 kd kd[ M 1 β(2 k 1 ] (x y) m! 2k y, m β(2 k x) 2 jd β(2 j y)dy (1 s) M 1 (M 1)! 2 (k j)m 2 kd 1 [ 2 k,2 k ] d(x), and thus m=0 2 k y, M β(2 k x s2 k y) ds 2 jd β(2 j y)dy 2 (j k)m γ j,k f(x) 2 kd h 2 k f(x h) dh 2 kd Hence (28) holds. h 2 k 2 (j k)a f(x h) (1 + 2 j h ) A dh 2 (j k)a M A,jf(x). Some notation. (i) Below, when j > N we use the notation { U N,j = (y 1,..., y d ) R d : min dist(y i, 2 N Z) 2 j 1}. 1 i d That is, U N,j is a 2 j 1 -neighborhood of the set I DN I. (ii) For a dyadic cube I of side length 2 N and l > N we shall denote by D l [ I] the set of dyadic cubes J D l such that J I. (iii) For a dyadic cube I of side length 2 N denote by D N (I) the neighboring cubes of I, that is, the cubes I D N with Ī Ī. Lemma 2.3. (i) Let k > N 1 and g be locally integrable. Then (29) L k (E N g)(x) = 0, for all x U N,k = Rd \ U N,k. (ii) Let j > N 1, and f locally integrable. Then (30) E N [L j f] = E N [L j (1 UN,j f)]. Moreover, (31) E N (L j f) 2 (N j)d I D N J D j+1 [ I] f L (J) 1 I.

9 THE HAAR SYSTEM AS A SCHAUDER BASIS 9 Proof. (i) We use the suort and cancellation roerties of β k. Note that L k (E N g)(x) = β k (x y) E N g(y) dy, and su β k (x ) x + 2 k [ 1/2, 1/2] d. So, if I D N and x I UN,k, then su β k (x ) I, and hence L k (E N g)(x) = (E N g) I (x) β k (x y) dy = 0. (ii) One argues similarly. First note that, changing the order of integration, (32) E N (L j f) = [ ] f(y) β j (x y) dx dy 1 I R d I I. I D N Now if J D N and y J UN,k then su β j( y) J, and hence I β j(x y) dx = 0. Thus E N [L j (1 U f)] = 0. Finally, to rove (31) note N,j that, if I D N and x I, then from (32) it follows E N (L j f)(x) = I 1 [ ] f(y) β j (x y) dx dy 2 Nd J D j+1 [ I] J D j+1 [ I] I J f L (J)2 (j+1)d β j 1, I which gives the asserted (31). Proof of Proosition 2.1. Proof of (26) in the case j, k N + 1. By Lemma 2.3.i, L k E N [L j f](x) = 0 if x UN,K, so we assume that x U N,k I, for some I D N. Recall that D N (I) consists of the neighboring cubes of I. Then (31) and the suort roerty of β k give L k E N [L j f](x) β k (x y) E N (L j f)(y) dy 2 (N j)d f L (J) β k 1. I D N (I) J D j+1 [ I ] Hence L k E N [L j f] = (33) [ I D N 2 (N j)d[ I U N,k L k (E N L j f) dx I D N J D j+1 [ I] ] 1 f L (J)) I UN,k ] 1.

10 10 G. GARRIGÓS A. SEEGER T. ULLRICH Now, I U N,k 2 k 2 N(d 1), and card D j+1 [ I] 2 (j N)(d 1). Also, if we write J = 2 j 1 (l J + [0, 1] d ), then f L (J) inf M jf(l J + h) [2 jd M jf(l J + h) dh h 2 j 1 h 2 j 1 Therefore, using either Hölder s inequality (if > 1), or the embedding l l 1 (if 1), we have [ ) ] 1 (34) I D N J D j+1 [ I] f L (J) [ 2 (j N)(d 1)(1 1 ) + [ 2 (j N)(d 1)(1 1 ) + 2 (j N)(d 1)(1 1 ) + 2 jd I D N J D j+1 [ I] J D j+1 2 jd M jf L (R d ). ] f 1 L (J) Finally, inserting (34) into (33), and using (23), yields h 2 j 1 M jf(l J + h) dh L k E N [L j f] 2 (N j)d 2 (j N)(d 1)(1 1 ) + 2 jd f 2 k 2 N(d 1) = 2 N j 2 j k 2 (j N)(d 1)( 1 1) + f, using in the last ste the trivial identity (1 1 ) + = ( 1 1) + ( 1 1). This establishes (26) for j, k N + 1. Proof of (27) in the case j N, k N + 1. For w I with I D N we have E N(L j f)(w) = E N [L j f](w) L j f(w) = 2 Nd 2 jd[ β(2 j (v y)) β(2 j (w y)) ] f(y)dy dv I R d 1 = 2 (N+j)d β(2 j [(1 t)w + tv y]) 2 j (v w) dt f(y) dy dv I R d (N+j)d 2 j N f(y) β(2 j [(1 t)w + tv y]) dy dt dv I 0 R d 2 j N M j f(w), since for fixed w, v, t the exression involving β is suorted in the set {y : y w 2 j N }. Moreover, since k > N, when w I N,µ and z 2 k 1 we have (35) E N[L j f](w z) 2 j N inf M jf(2 N µ + h), h 2 j ] 1 ] 1.

11 and therefore, (36) L k ( E N [L j f] ) (w) Now Lemma 2.3.i gives ( (37) Lk E N [L j f] ) THE HAAR SYSTEM AS A SCHAUDER BASIS 11 L k L j f L (U N,k ) + [ E N(w z) β k (z) dz 2 j N[ M jf(2 N µ + h) dh h 2 j µ Z d L k ] 1 ( E N [L j f] ) L (U N,k I N,µ)] 1. Using (36), the last term is controlled by 2 j N[ I N,µ U N,k M jf(2 N µ + h) dh µ Z d h 2 j 2 j N [2 k 2 N(d 1) ] 1 2 Nd M jf 2 j N 2 N k f. Finally, the first term in (37) is controlled by Lemma 2.2, so overall one obtains ( L k E N [L j f] ) [2 (M A) k j + 2 j N 2 N k ] f, ] 1. establishing (27) in the case j N, k N + 1. Proof of (27) in the case 0 j, k N. We use E N[L j f](y) dy = 0, I D N, to write L k ( E N [L j f] ) (x) = µ For fixed x, we say that I I N,µ ( βk (x y) β k (x 2 N µ) ) E N[L j f](y) dy. (38) µ Λ(x) if x 2 N µ 2 N + 2 k 1. Observe that only these µ s contribute to the above sum. Notice also that β k (x y) β k (x 2 N µ) 2 kd 2 k N, if y I N,µ, and since j N, the estimate in (35) gives E N[L j f](y) 2 j N inf M jf(2 N µ + h), y I N,µ. h 2 j

12 12 G. GARRIGÓS A. SEEGER T. ULLRICH Combining all these bounds we obtain L k ( E N [L j f] ) (x) 2 (k N)(d+1) 2 j N µ Λ(x) ( 2 (k N)(d+1) 2 j N 2 (N k)d(1 1 ) + µ Λ(x) ( µ 2 N +[ 1 2 j, 1 ) 1 [M jf] 2 j ]d µ 2 N +[ 1 2 j, 1 2 j ]d [M jf] using in the last ste Hölder s inequality (or l l 1 if 1) and the fact that card Λ(x) 2 (N k)d. Observe also that the L -quasinorm of the last bracketed exression satisfies ( Rd µ Λ(x) µ 2 N +[ 1 2 j, 1 2 j ]d [M jf] ) 1 µ Z d 2 kd µ 2 N +[ 1 2 j, 1 2 (N k)d/ M j f. ) 1 ) 1 [M jf] 2 j ]d Thus, overall we obtain L k E N[L j f] 2 (k N)(d+1) 2 j N 2 (N k)d(1 1 ) + 2 (N k)d/ f = 2 k N 2 j N 2 (N k)d( 1 1) + f, after simlifying the indices in the last ste. This establishes (27) in the case 0 j, k N. Proof of (26) in the case j N + 1, k N. This condition and (30) in Lemma 2.3 imly that E N [L j f] = E N [L j (f1 UN,j )]. For simlicity, we denote f = f1 UN,j, and write (39) L k E N [L j f] = L k (E N [L j f] Lj f) + Lk L j f. Observe that, by Lemma 2.2, L k L j f 2 (M A) j k M A,jf(x) 2 (M A) j k f. So, we only need to estimate L k E N [L j f]. Proceeding as in the roof of the case j, k N, we write (with Λ(x) as in (38)), (40) ( ) L k E N [L j f] (x) β k (x y) β k (x 2 N µ) E N[L j f](y) dy µ Λ(x) I N,µ 2 kd 2 ( k N E N [L j f] + Lj ( f) ) µ Λ(x) = A 1 (x) + A 2 (x). I N,µ

13 THE HAAR SYSTEM AS A SCHAUDER BASIS 13 Now, using again (31), we have A 1 (x) 2 (k N)(d+1) 2 (N j)d (41) µ Λ(x) J D j+1 [ I N,µ ] [ 2 k N 2 (k j)d 2 (N k)d(1 1 ) + µ Λ(x) f L (J) J D j+1 [ I N,µ ] f L (J) ) ] 1, since card Λ(x) 2 (N k)d. Taking the L -quasinorm of the last bracketed exression gives [ x Rd ) ] 1 f L (J) dx (42) µ Λ(x) J D j+1 [ I N,µ ] [ I D N 2 kd( J D j+1 [ I] ) ] 1 f L (J) 2 (j k)d 2 (j N)(d 1)(1 1 ) + M j f L (R d ) Therefore, combining exonents in (41) and (42) one obtains (43) A 1 2 k N 2 (k j)d 2 (N k)d(1 1 ) + 2 (j k)d by (34). 2 (j N)(d 1)(1 1 ) + f = 2 k j 2 j N 2 (N k)( 1 1) + 2 (j k)(d 1)( 1 1) + f. Finally, we estimate the term A 2 (x) in (40). First notice that L j ( f)(y) β j (y z) f(z) dz = 0, if y UN,j 1, U N,j since su β j (y ) y + 2 j [ 1 2, 1 2 ]d UN,j. Moreover, if I D N, then for every cube J D j such that J I U N,j 1 we have L j ( f)(y) β j (z) f(y z) dz f L (J ), if y J where J = J + 2 j [ 1 2, 1 2 ]d. Therefore, L j ( f)(y) I J D j [ I] and overall we obtain A 2 (x) 2 (k j)d 2 k N f L (J ) J, µ Λ(x) J D j 1 [ I N,µ ] f L (J). But this is essentially the same exression we obtained in (41) for the term A 1 (x), so the same argument will give an estimate of A 2 in terms of the quantity in (43). This concludes the roof of (26) in the case j N + 1, k N. Finally, concerning (iii) in Proosition 2.1, we remark that the revious roofs can easily be adated relacing the oerators E N and E N by T N[, a],

14 14 G. GARRIGÓS A. SEEGER T. ULLRICH keeing in mind that T N [g, a] is now constant in cubes I D N+1, and enjoys an additional cancellation, I N,µ T N [g, a](x)dx = 0, which simlifies some of the revious stes. Proof of Theorem 1.2, conclusion. It remains to rove inequalities (19) and (20). By the embedding roerties for the sequence saces l r it suffices to verify both inequalities for very small r, say (44) r min{, 1}. In view of the embedding l r l 1 and Minkowski s inequality (in L /r ) it suffices then to rove (45) su 2 ksr L k E N L j Λ j f r 1/r ) su 2 js Λ j f N j and (46) su N k=0 j=n+1 N 2 ksr k=0 j=0 L k (E NL j Λ j f) r ) 1/r su j 2 js Λ j f. If we aly Proosition 2.1 to each of the functions Λ j f, we reduce matters to observe that (47) su 2 ksr [ 2 js B(j, k, N) ] r <, N k=0 j=0 with B(j, k, N) as in (25), and that N + j=n+1 k=0 N k=n+1 j=0 ) 2 j k (M A) < which is trivial. The verification of (47) under the assumtions in (11) is also elementary, but we carry out some details to clarify how the conditions on and s are used. When j, k > N, we have B(j, k, N) = 2 N j 2 j k 2 (j N)(d 1)( 1 1) + and thus [ 2 js B(j, k, N) ] r 2 ksr k>n j>n (48) = 2 kr( 1 s)) ) 2 rj[s+1 1 (d 1)( 1 1) +] 2 Nr[1 (d 1)( 1 1) +], k>n j>n and the series converge rovided s < 1/ and (49) s > (d 1)( 1 1) + = max {d( 1 1), 1 1 }. Further, being geometric sums, the final outcome in (48) is bounded uniformly in N.

15 THE HAAR SYSTEM AS A SCHAUDER BASIS 15 Next assume j N < k, then B(j, k, N) = 2 N k 2 j N and hence [ 2 js B(j, k, N) ] r = 2 kr( 1 s)) 2 rj(1 s)) 2 Nr( 1 1), k>n 2 ksr j N k>n which are finite exressions rovided s < min{1, 1/}. j N Consider j, k N, with B(j, k, N) = 2 k N 2 j N 2 (N k)d( 1 1) +. Then [ 2 js B(j, k, N) ] r = 2 ksr k N j N )( = 2 kr[s+1 d( 1 1) +] 2 rj(1 s)) 2 Nr[2 d( 1 1) +], k N j N which leads to uniform exressions in N under the assumtions s < 1 and (50) s > d( 1 1) + 1, the latter being weaker than (49). j N k j+ When k N < j we have B(j, k, N) = 2 +[N k+(j k)(d 1)]( 1 1) + and [ 2 js B(j, k, N) ] r = 2 ksr k N j>n )( = 2 kr[s+1 d( 1 1) ) +] 2 rj[s+1 1 (d 1)( 1 1) +] 2 Nr[ 1 ( 1 1) +], k N j>n where in the first series we would use (50) and in the second series (49). We have verified (47) in all cases. This finishes the roof of Theorem Schauder bases Let P N be defined as in (3). For the roof of Theorem 1.1 we need to rove that P N f f F s,q 0 for every f F,q, s with (, s) as in (11) and 0 < q <. We first discuss some reliminaries about localization and ointwise multilication by characteristic functions of cubes, then rove uniform bounds for the F,q s F,q s oerator norms of the P N and then establish the asserted limiting roerty. Preliminaries. For ν Z d let χ ν be the characteristic function of ν + [0, 1) d. Lemma 3.1. Assume that (51) d 1 d < <, 0 < q, and max{d( 1 1), 1 1} < s < 1. Then, the following holds for all g ν and f F,q: s χ ν g ν g F s ν ν Z d,q ν Z d F s,q ) 1/

16 16 G. GARRIGÓS A. SEEGER T. ULLRICH and ν Z d fχ ν F s,q ) 1/ f F s,q. Proof. Let ς Cc (R d ) so that su(ς) ( 1, 1) d and ν Zd ς(x ν) = 1 for all x R d. Let ς ν = ς( ν). We have, for all s R, (52) g F s,q ς ν g 1/ F ;,q) s see [14, 2.4.7]. Hence χ ν g ν = ( F ς F s ν χ ν g ν ς ν Z d,q ν s ν ν,q ν ) 1/ 1/. g ν F,q g s ν F,q) s ν ν ν 1 ν ν ν ν 1 χ ν g ν F s,q ) 1/ Here we have used that ς ν χ ν are ointwise multiliers of F,q, s with uniform bounds in (ν, ν ), in the range given by (51); see [7, Thm /1]. This roves the first inequality. For the second inequality we first observe that, by (52), ) 1/ fχ ν F s,q fχ ν ς ν F,q, ν Z d, s which yields fχ ν F,q s )1/ fχ ν ς ν F,q s ν ν ν ν ν ν ν ν 1 ) 1/ fχ ν ς ν F s,q ) 1/ ) 1/ fς ν F,q s f F s,,q where we have used the ointwise multilier assertion [7, Thm /1] and then again (52) in the last ste. Uniform boundedness of the P N. Observe that by the localization roerty of the Haar functions we have P N f = ν Z d χ νp N f = ν χ νp N [fχ ν ]. Thus by Lemma 3.1 P N f F s,q P N [fχ ν ] 1/ F.,q) s ν Since the enumeration of the Haar system is assumed to be admissible we have (53) P N [fχ ν ] = E Nν [fχ ν ] + T Nν [fχ ν, a N,ν ]

17 THE HAAR SYSTEM AS A SCHAUDER BASIS 17 for some N ν N, with N ν N and aroriate sequences a N,ν assuming only the values 1 and 0. We remark that for each ν, N ν = N ν (N) with (54) lim N N ν(n) =. By Theorem 1.2 P N [fχ ν ] F,q s ν ) 1/ E Nν [fχ ν ] ) 1/ F + T,q s Nν [fχ ν, a N,ν ] F,q s ν ν fχ ν F s,q ν ) 1/ f F s,q, where for the last inequality we have used Lemma 3.1 again. ) 1/ Proof of Theorem 1.1, conclusion. Let f F,q, s with (, s) as in (11) and 0 < q <. Let C = max{1, su N P N F s,q F,q s }. Since Schwartz functions are dense in F,q s when 0 <, q < there is f S(R) such that f f F s,q < (3C) 1 ɛ and hence P N f P N f F s,q < ɛ/3. Choose s 1 so that s < s 1 < max{1/, 1} then f B s 1,q F,q. s Since the Haar system is an unconditional basis on B s 1,q ([15]) we have lim N P N f f B s 1,q and therefore lim N P N f f F s,q = 0. Combining these facts we get P N f f F s,q < ɛ for sufficiently large N which shows that P N f f in F,q s. 4. Otimality away from the end-oints Proosition 4.1. Let 0 < q <. Then, the Haar system H d is not a Schauder basis of F s,q(r d ) in each of the following cases: (i) if 1 < < and s 1/ or s 1/ 1, (ii) if d/(d + 1) 1 and s > 1 or s < d(1/ 1), (iii) if 0 < < d/(d + 1) and s R. The same result for the saces B,q(R s d ) was roved by Triebel in [12]; see also [15, Proosition 2.24]. Proosition 4.1 can be obtained from this and Theorem 1.1 by suitable interolation. Indeed, assertion (i) was already discussed in the aragrah following (4), so we restrict to 1. Assume next that H d is a basis for F,q s for some d/(d + 1) < < 1 and s > 1 or s < d(1/ 1). By Theorem 1.1, H d is also a basis for F s 0,q, for any d(1/ 1) < s 0 < 1. By real interolation, see e.g. [13, Thm (ii)], for all 0 < θ < 1, the system H d will then be a basis of ( F s 0,q, F,q s ),q, with s θ = (1 θ)s 0 + θs. θ,q = Bs θ

18 18 G. GARRIGÓS A. SEEGER T. ULLRICH But when θ is close to 1 this would contradict Triebel s result. The remaining cases, = 1 and d/(d + 1) can be roved similarly using comlex interolation of F -saces; see [14, 1.6.7]. We remark that, in the aer [12], the failure of the Schauder basis roerty in the B-saces is sometimes due to the fact that san H d fails to be dense in B s,q. This is the case, for instance, in the region (55) (d 1)/d < < 1 and max { 1, d(1/ 1) } < s < 1/; see [12, Corollary 2]. Here we show that also a quantitative bound holds, therefore ruling out the ossibility that H d could be a basic sequence. Proosition 4.2. Let 0 < q, and (, s) be as in (55). Then, E N B s,q B s,q 2(s 1)N. Proof. Let η Cc (R d ) such that η 1 on [ 2, 2] d, and consider the Schwartz function f(x) = x 1 η(x). It suffices to show that (56) E N f B s 2 (s 1)N.,q Under (55) we have s > σ := d(1/ 1) +. Assume first that s < 2 (which is always the case if d > 1). Then we can use the equivalence of quasi-norms d ( 1 2 he g B s,q (R d ) g + j g q dh ) 1/q, h sq h j=1 with the usual modification in the case q =, see [14, 2.6.1]. In articular (57) E N f ( 2 N 1 2 ( he EN 1 f ) q ) 1/q L B s ([0,1] d ) dh,q h sq. h 0 Now, it is easily checked that, when x [0, 1) d, one has E N f = k+1/2 1 2 N [ k 2 N, k+1, 2 N ) [0,1)d 1 0 k<2 N and likewise, if we additionally assume 0 < h < 2 N 1, then ( he1 EN f ) 2N N 1 = 2 1 [ k 2 N h, k. 2 N ) [0,1)d 1 k=1 and 2 ( he 1 EN f ) 2N [ ] N 1 = 2 1 [ k 2 N 2h, k 1 2 N h) [0,1)d 1 [ k 2 N h, k. 2 N ) [0,1)d 1 Therefore, k=1 2 he 1 E N f L ([0,1] d ) = 2 (N+1)(1/ 1) h 1/, which, inserted into (57), gives (56). If d = 1 and s 2, one alies a similar argument to the functions L he 1 (E N f) with L = s +1 and h < 2 N /L. 0

19 THE HAAR SYSTEM AS A SCHAUDER BASIS 19 By interolation one obtains as well a quantitative bound for the relevant cases in Proosition 4.1(ii). Corollary 4.3. Let 0 < q, d/(d + 1) < < 1 and 1 < s < 1/. Then, for all ε > 0, (58) E N F s,q F s,q c ε 2 (s 1 ε)n. Proof. If d(1/ 1) < s 0 < 1 and θ (0, 1), then the real interolation inequalities give E N 1 θ F s 0,q F s 0,q E N θ F s,q F s,q c θ E N B s θ,q B s θ,q, with s θ = (1 θ)s 0 +θs. By Proosition 4.2 the right hand side is larger than a constant times 2 N(sθ 1), while by Corollary 1.3 we have E F N s 0,q F s 1. 0,q Choosing θ sufficiently close to 1 one derives (58). Acknowledgments. The authors worked on this aer while articiating in the 2016 summer rogram in Constructive Aroximation and Harmonic Analysis at the Centre de Recerca Matemàtica at the Universitat Autònoma de Barcelona, Sain. They would like to thank the organizers of the rogram for roviding a leasant and fruitful research atmoshere. We also thank the referee for various useful comments that have led to an imroved version of this aer. Finally, T.U. thanks Peter Oswald for discussions concerning [12] and the results in 4. References [1] F. Albiac, N. Kalton. Toics in Banach sace theory. Graduate Texts in Mathematics, 233. Sringer, New York, [2] C. Fefferman, E.M. Stein. Some maximal inequalities. Amer. J. Math. 93 (1971) [3] G. Garrigós, A. Seeger, T. Ullrich. On uniform boundedness of dyadic averaging oerators in saces of Hardy-Sobolev tye. Analysis Math. 43 (2) (2017), [4] A. Haar. Zur Theorie der orthogonalen Funktionensysteme. Math. Ann. 69 (1910), [5] J. Marcinkiewicz. Quelques théoremes sur les séries orthogonales. Ann. Sci. Polon. Math. 16 (1937), [6] J. Peetre. On saces of Triebel-Lizorkin tye. Ark. Mat. 13 (1975), [7] T. Runst, W. Sickel. Sobolev saces of fractional order, Nemytskij oerators, and nonlinear artial differential equations, volume 3 of de Gruyter Series in Nonlinear Analysis and Alications. Walter de Gruyter & Co., Berlin, [8] J. Schauder. Eine Eigenschaft der Haarschen Orthogonalsysteme. Math. Z. 28 (1928), [9] A. Seeger, T. Ullrich. Haar rojection numbers and failure of unconditional convergence in Sobolev saces. Math. Z. 285 (2017), [10]. Lower bounds for Haar rojections: Deterministic Examles. Constr. Ar. 46 (2017), [11] H. Triebel. Über die Existenz von Schauderbasen in Sobolev-Besov-Räumen. Isomorhiebeziehungen. Studia Math. 46 (1973), [12]. On Haar bases in Besov saces. Serdica 4 (1978), no. 4,

20 20 G. GARRIGÓS A. SEEGER T. ULLRICH [13]. Theory of function saces. Monograhs in Mathematics, 78. Birkhäuser Verlag, Basel, [14]. Theory of function saces II. Monograhs in Mathematics, 84. Birkhäuser Verlag, Basel, [15]. Bases in function saces, samling, discreancy, numerical integration. EMS Tracts in Mathematics, 11. Euroean Mathematical Society (EMS), Zürich, Gustavo Garrigós, Deartment of Mathematics, University of Murcia, Esinardo, Murcia, Sain address: Andreas Seeger, Deartment of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, WI, 53706, USA address: Tino Ullrich, Hausdorff Center for Mathematics, Endenicher Allee 62, Bonn, Germany address: