ON UNIFORM BOUNDEDNESS OF DYADIC AVERAGING OPERATORS IN SPACES OF HARDY-SOBOLEV TYPE. 1. Introduction

ON UNIFORM BOUNDEDNESS OF DYADIC AVERAGING OPERATORS IN SPACES OF HARDY-SOBOLEV TYPE GUSTAVO GARRIGÓS ANDREAS SEEGER TINO ULLRICH Abstract We give an alternative roof and a wavelet analog of recent results by the authors on uniform boundedness of dyadic averaging oerators in (quasi-)banach saces of Hardy-Sobolev and Triebel-Lizorkin tye This result served as the main tool to establish Schauder basis roerties of suitable enumerations of the univariate Haar system in the mentioned saces The roof here is based on characterizations of the resective saces in terms of orthogonal comactly suorted Daubechies wavelets 1 Introduction Consider the dyadic averaging oerators E N on the real line given by (1) E N f(x) = 1 IN,µ (x) 2 N f(t)dt µ Z I N,µ with I N,µ = [2 N µ, 2 N (µ + 1)) E N f is the conditional exectation of f with resect to the σ-algebra generated by the dyadic intervals of length 2 N The following theorem on uniform boundedness in Triebel-Lizorkin saces F s,q was roved by the authors in [2] and serves as the main tool to establish that suitably regular enumerations of the Haar system form a Schauder basis for the saces F s,q in the arameter ranges of the theorem Since the uniform boundedness result is interesting on its own we give an alternative roof based on wavelet theory to make it accessible for a broader readershi Theorem 11 [2] Let 1/2 < <, 0 < q, and 1/ 1 < s < min{1/, 1} Then there is a constant C := C(, q, s) > 0 such that for all f F s,q (2) su E N f F s,q C f F s,q N N 2010 Mathematics Subject Classification 46E35, 46B15, 42C40 Key words and hrases Schauder basis, Unconditional bases, Haar system, Hardy- Sobolev sace, Triebel-Lizorkin sace GG was suorted in art by grants MTM2013-40945-P and MTM2014-57838-C2-1-P from MINECO (Sain), and grant 19368/PI/14 from Fundación Séneca (Región de Murcia, Sain) AS was suorted in art by NSF grant DMS 1500162 TU was suorted the DFG Emmy-Noether rogram UL403/1-1 1

2 G GARRIGÓS A SEEGER T ULLRICH In [2], this result served as the main tool to establish that suitably regular enumerations of the Haar system form a Schauder basis for the saces F s,q in the arameter ranges of the theorem, see 3 The connection with the Haar system is given via the martingale difference oerators D N = E N+1 E N which are the orthogonal rojections to the saces generated by Haar functions with fixed Haar frequency 2 N In revious works stronger notions of convergence have been examined, such as unconditional convergence for the martingale difference series This is equivalent with the inequality (3) b n D n f b F s l (N) f F s,q n,q It follows from the results in Triebel [6] that (3) holds if we add the condition 1/q 1 < s < 1/q to the hyotheses in the theorem For the case q = 2 this corresonds to the shaded region in Figure 1 s 1 1 2 1 3 2 2 1 1 2 1 Figure 1 Unconditional convergence in Hardy-Sobolev saces It was shown in [3], [4] that the additional restriction on the q-arameter is necessary for (3) to hold If we dro it then Theorem 11 and a summation by arts argument imly that (3) holds with the larger norm b + b BV It should be interesting to establish shar results involving sequence saces that are intermediate between l (N) and BV (N) We remark that these roblems are interesting only for the F s,q saces since inequality (3) with B s,q in lace of F s,q holds in the full arameter range of Theorem 11, see [6] for further discussion and historical comments In 2 we give a roof of Theorem 11 using characterizations of Triebel- Lizorkin saces based on Daubechies wavelets Relying on this, the roof is rather elementary due to the orthogonality and locality roerties of the wavelet system In addition, a wavelet analog of [2, Thm 12] is rovided in Proosition 21 below In 3 we aly the methods to get an additional result needed to obtain the Schauder basis roerty of the Haar system

DYADIC AVERAGING OPERATORS 3 2 Proof of Theorem 11 We will exclusively use a characterization of Triebel-Lizorkin saces F s,q(r) and Besov saces B s,q via comactly suorted Daubechies wavelets [1], [7, Sect 4] Let ψ 0 and ψ be the orthogonal scaling function and corresonding wavelet of Daubechies tye such that ψ 0, ψ being sufficiently smooth (C K ) and ψ having sufficiently many vanishing moments (L) We denote ψ j,ν (x) := 1 2 ψ(2 j 1 x ν), j N, ν Z, and ψ 0,ν (x) := ψ 0 (x ν) for ν Z Let 0 < <, 0 < q and s R If K and L are large enough (deending on, q and s) then we have the equivalent characterization (usual modification in case q = ), (4) (5) f F s,q f B s,q ( 2 js q) 1/q λ j,ν (f)1 j,ν, j=0 ( 2 js q) 1/q λ j,ν (f)1 j,ν, j=0 where λ j,ν (f) := 2 j f, ψ j,ν and 1 j,ν denotes the characteristic function of the interval I j,ν := [2 j ν, 2 j (ν + 1)] See Triebel [5, Thm 164] and the references therein A corresonding characterization also holds true for Besov saces B,q s Since we also deal with distributions which are not locally integrable, the inner roduct f, ψ j,ν has to be interreted in the usual way Clearly, f can be decomosed into wavelet building blocks, ie (6) f = λ j,ν (f)ψ j,ν, if j 0, f j with f j = j Z 0 if j < 0 Let us denote the Nth artial sum of this reresentation by (7) P N f = j N f j, N N Note, that the functions f j and P N f reresent K times continuously differentiable functions due to the regularity assumtion on the wavelet In the sequel we will rove the following roosition Proosition 21 Let 1/2 <, 0 < r, and 1/ 1 < s < min{1/, 1} Let {ψ j,ν } j,ν reresent a Daubechies wavelet system such that (5) holds for all 0 < q and let P N be given by (7) Then there is a constant C := C(, r, s) > 0 such that for all f B s, (8) su E N f P N f B s,r C f B s, N N

4 G GARRIGÓS A SEEGER T ULLRICH Note, that for a fixed wavelet system satisfying (4) we clearly have (9) su P N f F s,q C f F s,q N N If this wavelet system in addition satisfies (5) for all 0 < q then Proosition 21 together with (9) imlies Theorem 11 A Proof of Proosition 21 in the case 1/2 < 1 Let 1/ 1 < s < 1 Using the decomosition (6) we can write with θ := min{1, } = E N f P N f B s,r (10) (11) ( 2 js 2 j r) 1/r E N f P N f, ψ j, 1 j, j=0 ( 2 js 2 j r) 1/r E N (P N f) P N f, ψ j, 1 j, j=0 ( + 2 js 2 j r) 1/r E N (f P N f), ψ j, 1 j, j=0 We slit the roof into several stes according to the cases we have to distinguish in the estimation of the quantities in (10) and (11) Ste A1 We deal with (11) and use that f P N f = f j+l Clearly, ( ( (12) (11) j=0 j+l N 2 js j+l>n 2 j θ) r/θ ] 1/r E N f j+l, ψ j, 1 j, We continue estimating 2 js 2j E N f j+l, ψ j, 1 j, Note first that due to 1 2 js 2 j E N f j+l, ψ j, 1 j, (13) ( λ j+l,ν (f) 2 js 2 j ) 1/ E N ψ j+l,ν, ψ j, 1 j, So it remains to deal with 2 js 2j E N ψ j+l,ν, ψ j, 1 j, Note, that due to j + l > N the function E N ψ j+l,ν is a ste function consisting of O(1) non-vanishing stes These stes have length 2 N and magnitude bounded by O(2 N (j+l) ) Case A11 Assume j N Due to the cancellation of ψ j, and j N we have that the function 2j E N ψ j+l,ν, ψ j, 1 j, is suorted on a union of intervals of total measure O(2 j ) and bounded from above by O(2 N (j+l) ) This gives (14) 2 js 2 j E N ψ j+l,ν, ψ j, 1 j, 2 js 2 j/ 2 N j l

DYADIC AVERAGING OPERATORS 5 Case A12 Assume j N Clearly, we have l > 0 since j + l > N Now 2j E N ψ j+l,ν, ψ j, 1 j, is suorted on an interval of size O(2 j ) As E N ψ j+l,ν consists of O(1) stes of length 2 N each and N j we get by straightforward size estimates 2 j E N ψ j+l,ν, ψ j, = O(2 l ) Hence (15) 2 js 2 j E N ψ j+l,ν, ψ j, 1 j, 2 js 2 j/ 2 l Ste A2 We consider (10) and observe first ( ( (16) (10) j=0 j+l N 2 js 2 j θ) r/θ ) 1/r E N f j+l f j+l, ψ j, 1 j, Analogously to (13) the matter reduces to estimate the L (quasi-)norm of the functions (17) 2 js 2 j E N ψ j+l,ν ψ j+l,ν, ψ j, 1 j, for the different cases resulting from j + l N Case A21 We first deal with the case j N Using the mean value theorem together with (1) we see for all x R that E N ψ j+l,ν (x) ψ j+l,ν (x) 2 j+l N Due to j + l N, its suort has length O(2 (j+l) ) around ν2 (j+l) We continue distinguishing the cases l 0 and l < 0 Case A211 Let l 0 Since j + l j the inner roduct with 2 j ψ j, gives an additional factor 2 l In addition, the suort of (17) is contained in an interval of size O(2 j ) Hence, we get (18) 2 js 2 j E N ψ j+l,ν ψ j+l,ν, ψ j, 1 j, 2 js 2 j+l N 2 l 2 j/ Case A212 Assume l 0 This time the inner roduct with 2 j ψ j, does not give an extra factor and the suort has length 2 (j+l) Thus, we have in this case (19) 2 js 2 j E N ψ j+l,ν ψ j+l,ν, ψ j, 1 j, 2 js 2 j+l N 2 (j+l)/

6 G GARRIGÓS A SEEGER T ULLRICH Case A22 Assume j > N j + l which imlies l < 0 Due to the orthogonality of the wavelets (l < 0) we can estimate (20) 2 js 2 js( 2 js( 2 j E N ψ j+l,ν ψ j+l,ν, ψ j, 1 j, µ Z x 2 N µ 2 j µ Z x 2 N µ 2 j 2 js 2 j+l N 2 [N (j+l) j]/, 2 j E N ψ j+l,ν, ψ j, 1 j, (x) dx ) 1/ 2 j E N ψ j+l,ν ψ j+l,ν, ψ j, 1 j, (x) dx ) 1/ where we took into account that the µ-sum consists of O(2 N (j+l) ) summands Ste A3 Estimation of (11) Plugging (13) and (14) into the right hand side of (12) yields (21) with [ ( j=n j+l N A N su j,l 2 js 2 j θ) r/θ ] 1/r E N f j+l, ψ j, 1 j, ( 2 (j+l)s λ j+l,ν (f) 2 (j+l)) 1/ A N f B s, A r N = j N 2 (N j)r( l N j 2 θl(1/ 1 s)) r/θ 1 by the assumtion 1/ > s > 1/ 1 Plugging (13) and (15) into into the right hand side of (12) leads to a similar estimate as above, only the sums over j and l change to à r N = ( 2 θl(1/ 1 s)) r/θ j N l N j which is uniformly bounded in N if s > 1/ 1 Ste A4 Estimation of (10) Combining (16), (18) and (19) we find [ N ( j=0 j+l N [( j N 2 js N j 2 (j N)r( 2 j θ) r/θ ] 1/r E N f j+l f j+l, ψ j, 1 j, l= 2 θl(1/ s)) r/θ] 1/r f B s,

DYADIC AVERAGING OPERATORS 7 The sums are finite and uniformly bounded if 1/ 1 < s < 1/ Finally, we combine (12), (13) and (20) to obtain ( ( 2 js 2 j θ) r/θ ] 1/r E N f j+l f j+l, ψ j, 1 j, (22) j=n [( j+l N j N N j 2 (j N)θ 2 (N j)θ/( l= 2 rl(1 s)) r/θ] 1/r f B s,, which is uniformly bounded if s < 1 This concludes the roof in the case 1 B Proof in the case 1 We follow the roof in the case 1 until (12) and (16), resectively Note, that we may use θ = 1 now Then we have to roceed differently Case B11 Assume N < j, j + l Taking (12) into account we relace (13) by 2 js 2 j E N f j+l, ψ j, 1 j, [ 2 js λ j+l,ν (f) 2 j (23) E N ψ j+l,ν, ψ j, 1 j, (x) ] dx 2 js λ j+l,ν (f) 2 j 2 (N j l) Indeed, since E N ψ j+l,ν = 0 if su ψ j+l,ν I N,µ the sum on the right-hand side of (23) is lacunary and the functions 2j E N ψ j+l,ν, ψ j, 1 j, have essentially disjoint suort (for different ν) Hence, we get 2 js 2 j E N f j+l, ψ j, 1 j, 2 ls 2 N j l 2 l/( 2 (j+l)s λ j+l,ν (f) 2 (j+l)) 1/ (24) 2 ls 2 N j l 2 l/ f B s, For 1/ 1 < s < 1/ the sum over the resective range of j and l is uniformly bounded Case B12 We now deal with j + l > N j Due to the orthogonality of the wavelet system and l > 0 we obtain (25) 2 js 2 j E N f j+l, ψ j, 1 j, = 2 js We continue exloiting the cancellation roerty (26) E N (f E N f) = 0 2 j E N f j+l f j+l, ψ j, 1 j,

8 G GARRIGÓS A SEEGER T ULLRICH to estimate the right-hand side of (25) We obtain the following identities (27) 2 js 1 j, (x)2 j ψ j, (y)(e N f j+l (y) f j+l (y)) dy = 2 js 1 j, (x) 2 j ψ j, (y)(e N f j+l (y) f j+l (y)) dy µ: 2 N µ x 2 j I N,µ = 2 js 1 j, (x) µ: 2 N µ x 2 j 2 I j (ψ j, (y) ψ j, (2 N µ))(e N f j+l (y) f j+l (y)) dy N,µ Let Z such that 1 j, (x) = 1 We continue estimating (27) by 2 js 2 j (ψ j, (y) ψ j, (2 N µ)) E N f j+l (y) dy µ: 2 N µ x 2 j I N,µ + 2 js 2 j (ψ j, (y) ψ j, (2 N µ)) f µ,1 j+l (y) dy µ: 2 N µ x 2 j I N,µ + 2 js (ψ j, (y) ψ j, (2 N µ)) f µ,2 j+l (y) dy I N,µ where µ: 2 N µ x 2 j 2 j =: F 0 (x) + F 1 (x) + F 2 (x), f µ j+l := f µ,1 j+l := ν:su ψ j+l,ν I N,µ λ j+l,ν (f)ψ j+l,ν, ν:su ψ j+l,ν I N,µ λ j+l,ν (f)ψ j+l,ν, f µ,2 j+l := f µ j+l f µ,1 j+l Note, that the function F 1 vanishes since l > 0 (use orthogonality) and j + l > 0 (use vanishing moments) F 0 (x) can be estimated by 2 js 2 2j 2N su λ j+l,ν (f)e N (ψ j+l,ν )(y) µ: 2 N µ x 2 j y I N,µ ν:su ψ j+l,ν I N,µ Here E N ψ j+l,ν is mostly vanishing, namely when su ψ j+l,ν I N,µ If it does not vanish then the boundary of I N,µ intersects su ψ j+l,ν and E N ψ j+l,ν 2 N (j+l) This haens only for a bounded number of ν s (indeendently of j, l) Thus for a fixed y only a bounded number of coefficients

contribute Hence, we have (28) F 0 (x) 2 js 2 2j 2N 2 N (j+l) DYADIC AVERAGING OPERATORS 9 µ: 2 N µ x 2 j su ν:su ψ j+l,ν I N,µ λ j+l,ν (f) Taking the L -norm and using Hölder s inequality with 1/+1/ = 1 yields (29) F 0 2 ls 2 2j 2N 2 N (j+l) 2 (N j)/ 2 l/ ( 2 (j+l)s λ j+l,ν (f) 2 (j+l)) 1/, where again f B s, dominates the sum on the right-hand side, see (5) Finally, we deal with F 2 (x) Since to f µ,2 j+l only a uniformly bounded number of coefficients λ j+l,ν contribute to the sum and the integrals are taken over an interval of length O(2 (j+l) ) we obtain, similar as above, by Hölder s inequality (30) F 2 2 ls 2 l+j N 2 (N j)/ 2 l/( 2 (j+l)s λ j+l,ν (f) 2 (j+l)) 1/ Putting the estimates from (25) to (30) together we observe that the sum over the resective range of j and l (see (11)) is uniformly bounded with resect to N if s > 1/ 1 Case B21 Here we deal with j + l, j N Starting from (16) (with θ = 1) we continue similarly as after (26) and obtain the ointwise identity (27) Note, that we already start with E N f j+l f j+l, so we do have to use the orthogonality argument (25), which does indeed not aly here since l = 0 is admitted Since j + l N there is only a bounded number of coefficients λ j+l,ν (f) contributing to f j+l on I N,µ Using the mean value theorem in both factors of the integral in (27) we obtain 2 js 2 j E N (f j+l ) f j+l, ψ j, 1 j, 2 js 2 2j 2N 2 j+l N ν ν µ: 2 N µ x 2 j su ν2 (j+l) 2 N µ 1 λ j+l,ν (f), which yields 2 js 2 j E N (f j+l ) f j+l, ψ j, 1 j, 2 ls 2 j+l N 2 2j 2N 2 (N j)/ 2 l/( 2 (j+l)s λ j+l,ν (f) 2 (j+l)) 1/ The sum over the resective j and l is uniformly bounded in N whenever 1 < s < 1 + 1/

10 G GARRIGÓS A SEEGER T ULLRICH Case B22 Finally j + l N < j Using again the orthogonality relation of the wavelets we may estimate as follows (similar to (20)) 2 js 2 j E N f j+l f j+l, ψ j, 1 j, (31) 2 js( 2 j E N f j+l f j+l, ψ j, 1 j, (x) 1/ dx), µ Z x 2 N µ 2 j which is bounded by (see (20)) (32) 2 ls 2 j+l N 2 (N j)/( 2 (j+l)s λ j+l,ν (f) 2 (j+l)) 1/ Altogether we encounter the condition 1/ 1 < s < 1/ for any 0 < r for the uniform boundedness of E N : B, s B,r s in case 1 3 On the Schauder basis roerty for the Haar system Let {h N,µ : µ Z} be the set of Haar functions with Haar frequency 2 N and define for N N 0 and sequences a l (Z), (33) T N [f, a] = µ Z a µ 2 N f, h N,µ h N,µ In articular for the choice of a = (1, 1, 1, ) one recovers the oerator E N+1 E N It was shown in [2] that Theorem 11 together with (34) su su N N a 1 T N [f, a] B s,r C f B s,, 1/2 <, 0 < r, and 1/ 1 < s < min{1/, 1}, imlies Schauder basis roerties for suitable enumerations of the Haar system For the sake of comleteness we give a sketch of this inequality which relies on the arguments in the revious section Proof of (34) We may assume a = 1 The modification of the roof of Proosition 21 is the fact that, due to the cancellation roerties of the Haar functions articiating in (33) we can work directly with T N [f, a] B s,r (instead of E N f P N f B s,r Case 11 Suose j + l, j > N The estimates in (23), (24) aly almost literally to T N [f, a] and yield estimates which are uniform for a = 1 Note, that we did not yet need any cancellation of the Haar functions Case 12 Suose j + l > N j We do not have to use (25) and work directly with 2 js T N [f j+l, a] An analogous identity to (27) holds true with E N (f j+l ) f j+l relaced by T N [f j+l, a] due to the cancellation of the Haar functions h N,µ In what follows we only have to care for a counterart of F 0 since F 1 and F 2 do not show u We end u with a counterart of (29) for 2 js T N [f j+l, a] Case 21 Suose N j+l, j Again, due to the cancellation of the Haar

DYADIC AVERAGING OPERATORS 11 function, we obtain a version of (27) as in Case 12 The mean value theorem alied to the first factor in the integral gives the factor 2 2j 2N, whereas the cancellation of h N,µ gives T N (ψ j+l,ν )(x) 2 j+l N We continue as in the roof of Proosition 21 Case 22 The remaining case j + l N < j goes analogously to Case B22 in the roof of Proosition 21 Note, that also here the slitting in (31) and the subsequent consideration for the second summand on the right-hand side is not necessary Acknowledgment The authors worked on this roject while articiating in the 2016 summer rogram in Constructive Aroximation and Harmonic Analysis at the Centre de Recerca Matemàtica They would like to thank the organizers of the rogram for roviding a leasant and fruitful research atmoshere The authors would also like to thank an anonymous referee for a careful roofreading and several valuable comments how to imrove the resentation of the roofs References [1] I Daubechies Ten lectures on wavelets, volume 61 of CBMS-NSF Regional Conference Series in Alied Mathematics Society for Industrial and Alied Mathematics (SIAM), Philadelhia, PA, 1992 [2] G Garrigós, A Seeger, T Ullrich The Haar system as a Schauder basis in saces of Hardy-Sobolev tye Prerint, arxiv:160908225 [3] A Seeger, T Ullrich Haar rojection numbers and failure of unconditional convergence in Sobolev saces Math Zeitschrift, 285, 91 119, 2017 See also arxiv:15070121 [4] Lower bounds for Haar rojections: Deterministic Examles Constructive Aroximation, ublished online (2016), DOI: 101007/s00365-016-9353-3 See also arxiv:151101470 [5] H Triebel Theory of function saces III, volume 100 of Monograhs in Mathematics Birkhäuser Verlag, Basel, 2006 [6] Bases in function saces, samling, discreancy, numerical integration EMS Tracts in Mathematics, 11 Euroean Mathematical Society (EMS), Zürich, 2010 [7] P Wojtaszczyk A mathematical introduction to wavelets, volume 37 of London Mathematical Society Student Texts Cambridge University Press, Cambridge, 1997 Gustavo Garrigós, Deartment of Mathematics, University of Murcia, 30100 Esinardo, Murcia, Sain E-mail address: gustavogarrigos@umes Andreas Seeger, Deartment of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, WI, 53706, USA E-mail address: seeger@mathwiscedu Tino Ullrich, Hausdorff Center for Mathematics, Endenicher Allee 62, 53115 Bonn, Germany E-mail address: tinoullrich@hcmuni-bonnde