aa r X i v : . [ m a t h . F A ] D ec MARTINGALE INEQUALITIES FOR SPLINE SEQUENCES
MARKUS PASSENBRUNNER
Abstract.
We show that D. L´epingle’s L ( ℓ )-inequality (cid:13)(cid:13)(cid:13)(cid:0) X n E [ f n | F n − ] (cid:1) / (cid:13)(cid:13)(cid:13) ≤ · (cid:13)(cid:13)(cid:13)(cid:0) X n f n (cid:1) / (cid:13)(cid:13)(cid:13) , f n ∈ F n , extends to the case where we substitute the conditional expectation operators with or-thogonal projection operators onto spline spaces and where we can allow that f n iscontained in a suitable spline space S ( F n ). This is done provided the filtration ( F n )satisfies a certain regularity condition depending on the degree of smoothness of the func-tions contained in S ( F n ). As a by-product, we also obtain a spline version of H -BMOduality under this assumption. Introduction
This article is part of a series of papers that extend martingale results to polynomialspline sequences of arbitrary order (see e.g. [17, 15, 12, 10, 14, 13, 8]). In order to explainthose martingale type results, we have to introduce a little bit of terminology: Let k bea positive integer, ( F n ) an increasing sequence of σ -algebras of sets in [0 ,
1] where each F n is generated by a finite partition of [0 ,
1] into intervals of positive length. Moreover,define the spline space S k ( F n ) = { f ∈ C k − [0 ,
1] : f is a polynomial of order k on each atom of F n } and let P ( k ) n be the orthogonal projection operator onto S k ( F n ) with respect to the L inner product on [0 ,
1] with the Lebesgue measure | · | . The space S ( F n ) consistsof piecewise constant functions and P (1) n is the conditional expectation operator withrespect to the σ -algebra F n . Similarly to the definition of martingales, we introduce thefollowing notion: let ( f n ) n ≥ be a sequence of integrable functions. We call this sequencea k -martingale spline sequence (adapted to ( F n )) if, for all n , P ( k ) n f n +1 = f n . For basic facts about martingales and conditional expectations, we refer to [11].Classical martingale theorems such as Doob’s inequality or the martingale convergencetheorem in fact carry over to k -martingale spline sequences corresponding to arbitrary filtrations ( F n ) of the above type, just by replacing conditional expectation operators bythe projection operators P ( k ) n . Indeed, we have(i) (Shadrin’s theorem) there exists a constant C k depending only on k such thatsup n k P ( k ) n : L → L k ≤ C k , Mathematics Subject Classification. (ii) (Doob’s weak type inequality for splines)there exists a constant C k depending only on k such that for any k -martingale splinesequence ( f n ) and any λ > |{ sup n | f n | > λ }| ≤ C k sup n k f n k λ , (iii) (Doob’s L p inequality for splines)for all p ∈ (1 , ∞ ] there exists a constant C p,k depending only on p and k such thatfor all k -martingale spline sequences ( f n ), (cid:13)(cid:13) sup n | f n | (cid:13)(cid:13) p ≤ C p,k sup n k f n k p , (iv) (Spline convergence theorem)if ( f n ) is an L -bounded k -martingale spline sequence, then ( f n ) converges almostsurely to some L -function,(v) (Spline convergence theorem, L p -version)for 1 < p < ∞ , if ( f n ) is an L p -bounded k -martingale spline sequence, then ( f n )converges almost surely and in L p .Property (i) is proved in [17], properties (ii) and (iii) in [15] and properties (iv) and(v) in [10], but see also [14].Here, we continue this line of transferring martingale results to k -martingale splinesequences and extend D. L´epingle’s L ( ℓ )-inequality [9], which reads(1.1) (cid:13)(cid:13)(cid:13)(cid:0) X n E [ f n | F n − ] (cid:1) / (cid:13)(cid:13)(cid:13) ≤ · (cid:13)(cid:13)(cid:13)(cid:0) X n f n (cid:1) / (cid:13)(cid:13)(cid:13) , provided the sequence of (real-valued) random variables f n is adapted to the filtration( F n ), i.e., each f n is F n -measurable. The spline version of this inequality is contained inTheorem 4.1.This inequality is an L extension of the following result for 1 < p < ∞ , proved byE. M. Stein [19], that holds for arbitrary integrable functions f n :(1.2) (cid:13)(cid:13)(cid:13)(cid:0) X n E [ f n | F n − ] (cid:1) / (cid:13)(cid:13)(cid:13) p ≤ a p (cid:13)(cid:13)(cid:13)(cid:0) X n f n (cid:1) / (cid:13)(cid:13)(cid:13) p , for some constant a p depending only on p . This can be seen as a dual version of Doob’sinequality k sup ℓ | E [ f | F ℓ ] |k p ≤ c p k f k p for p >
1, see [1]. Once we know Doob’s inequalityfor spline projections, which is point (iii) above, the same proof as in [1] works for splineprojections if we use suitable positive operators T n instead of P ( k ) n that also satisfy Doob’sinequality and dominate the operators P ( k ) n pointwise (cf. Sections 3.1 and 3.2).The usage of those operators T n is also necessary in the extension of inequality (1.1)to splines. D. L´epingle’s proof of (1.1) rests on an idea by C. Herz [7] of splitting E [ f n · h n ](for f n being F n -measurable) by Cauchy-Schwarz after introducing the square function S n = P ℓ ≤ n f ℓ :(1.3) ( E [ f n · h n ]) ≤ E [ f n /S n ] · E [ S n h n ]and estimating both factors on the right hand side separately. A key point in estimatingthe second factor is that S n is F n -measurable, and therefore, E [ S n | F n ] = S n . If we wantto allow f n ∈ S k ( F n ), S n will not be contained in S k ( F n ) in general. Under certainconditions on the filtration ( F n ), we will show in this article how to substitute S n in ARTINGALE INEQUALITIES FOR SPLINE SEQUENCES 3 estimate (1.3) by a function g n ∈ S k ( F n ) that enjoys similar properties to S n and allowsus to proceed (cf. Section 3.4, in particular Proposition 3.4 and Theorem 3.6). As a by-product, we obtain a spline version (Theorem 4.2) of C. Fefferman’s theorem [4] on H -BMO duality. For its martingale version, we refer to A. M. Garsia’s book [5] on MartingaleInequalities. 2. Preliminaries
In this section, we collect all tools that are needed subsequently.2.1.
Properties of polynomials.
We will need Remez’ inequality for polynomials:
Theorem 2.1.
Let V ⊂ R be a compact interval in R and E ⊂ V a measurable subset.Then, for all polynomials p of order k (i.e. degree k − ) on V , k p k L ∞ ( V ) ≤ (cid:18) | V || E | (cid:19) k − k p k L ∞ ( E ) . Applying this theorem with the set E = { x ∈ V : | p ( x ) | ≤ − k +1 k p k L ∞ ( V ) } immediatelyyields the following corollary: Corollary 2.2.
Let p be a polynomial of order k on a compact interval V ⊂ R . Then (cid:12)(cid:12)(cid:8) x ∈ V : | p ( x ) | ≥ − k +1 k p k L ∞ ( V ) (cid:9)(cid:12)(cid:12) ≥ | V | / . Properties of spline functions.
For an interval σ -algebra F (i.e., F is generatedby a finite collection of intervals having positive length), the space S k ( F ) is spanned bya very special local basis ( N i ), the so called B-spline basis. It has the properties that each N i is non-negative and each support of N i consists of at most k neighboring atoms of F .Moreover, ( N i ) is a partition of unity, i.e., for all x ∈ [0 , k functions N i so that N i ( x ) = 0 and P i N i ( x ) = 1. In the following, we denote by E i the support ofthe B-spline function N i . The usual ordering of the B-splines ( N i )–which we also employhere–is such that for all i , inf E i ≤ inf E i +1 and sup E i ≤ sup E i +1 .We write A ( t ) . B ( t ) to denote the existence of a constant C such that for all t , A ( t ) ≤ CB ( t ), where t denote all implicit and explicit dependencies the expression A and B might have. If the constant C additionally depends on some parameter, we will indicatethis in the text. Similarly, the symbols & and ≃ are used.Another important property of B-splines is the following relation between B-splinecoefficients and the L p -norm of the corresponding B-spline expansions. Theorem 2.3 (B-spline stability, local and global) . Let ≤ p ≤ ∞ and g = P j a j N j .Then, for all j , (2.1) | a j | . | J j | − /p k g k L p ( J j ) , where J j is an atom of F contained in E j having maximal length. Additionally, (2.2) k g k p ≃ k ( a j | E j | /p ) k ℓ p , where in both (2.1) and (2.2) , the implied constants depend only on the spline order k . Observe that (2.1) implies for g ∈ S k ( F ) and any measurable set A ⊂ [0 , k g k L ∞ ( A ) . max j : | E j ∩ A | > k g k L ∞ ( J j ) . M. PASSENBRUNNER
We will also need the following relation between the B-spline expansion of a functionand its expansion using B-splines of a finer grid.
Theorem 2.4.
Let G ⊂ F be two interval σ -algebras and denote by ( N G ,i ) i the B-splinebasis of the coarser space S k ( G ) and by ( N F ,i ) i the B-spline basis of the finer space S k ( F ) . Then, given f = P j a j N G ,j , we can expand f in the basis ( N F ,i ) i X j a j N G ,j = X i b i N F ,i , where for each i , b i is a convex combination of the coefficients a j with supp N G ,j ⊇ supp N F ,i . For those results and more information on spline functions, in particular B-splines, werefer to [16] or [3]. We now use the B-spline basis of S k ( F ) and expand the orthogonalprojection operator P onto S k ( F ) in the form(2.4) P f = X i,j a ij (cid:16) Z f ( x ) N i ( x ) d x (cid:17) · N j for some coefficients ( a ij ). Denoting by E ij the smallest interval containing both supports E i and E j of the B-spline functions N i and N j respectively, we have the following estimatefor a ij [15]: there exist constants C and 0 < q < k so that for eachinterval σ -algebra F and each i, j ,(2.5) | a ij | ≤ C q | i − j | | E ij | . Spline square functions.
Let ( F n ) be a sequence of increasing interval σ -algebrasin [0 ,
1] and we assume that each F n +1 is generated from F n by the subdivision of exactlyone atom of F n into two atoms of F n +1 . Let P n be the orthogonal projection operatoronto S k ( F n ). We denote ∆ n f = P n f − P n − f and define the spline square function Sf = (cid:16) X n | ∆ n f | (cid:17) / . We have Burkholder’s inequality for the spline square function, i.e., for all 1 < p < ∞ ([12]), the L p -norm of the square function Sf is comparable to the L p -norm of f :(2.6) k Sf k p ≃ k f k p , f ∈ L p with constants depending only on p and k . Moreover, for p = 1, it is shown in [6] that(2.7) k Sf k ≃ sup ε ∈{− , } Z k X n ε n ∆ n f k , Sf ∈ L , with constants depending only on k and where the proof of the . -part only uses Khint-chine’s inequality whereas the proof of the & -part uses fine properties of the functions∆ n f . ARTINGALE INEQUALITIES FOR SPLINE SEQUENCES 5 L p ( ℓ q ) -spaces. For 1 ≤ p, q ≤ ∞ , we denote by L p ( ℓ q ) the space of sequences ofmeasurable functions ( f n ) on [0 ,
1] so that the norm k ( f n ) k L p ( ℓ q ) = (cid:16) Z (cid:16) X n | f n ( t ) | q (cid:17) p/q d t (cid:17) /p is finite (with the obvious modifications if p = ∞ or q = ∞ ). For 1 ≤ p, q < ∞ , the dualspace (see [2]) of L p ( ℓ q ) is L p ′ ( ℓ q ′ ) with p ′ = p/ ( p − q ′ = q/ ( q −
1) and the dualitypairing h ( f n ) , ( g n ) i = Z X n f n ( t ) g n ( t ) d t. H¨older’s inequality takes the form |h ( f n ) , ( g n ) i| ≤ k ( f n ) k L p ( ℓ q ) k ( g n ) k L p ′ ( ℓ q ′ ) .3. Main Results
In this section, we prove our main results. Section 3.1 defines and gives properties ofsuitable positive operators that dominate our (non-positive) operators P n pointwise. InSection 3.2, we use those operators to give a spline version of Stein’s inequality (1.2).A useful property of conditional expectations is the tower property E G E F f = E G f for G ⊂ F . In this form, it extends to the operators ( P n ), but not to the operators T fromSection 3.1. In Section 3.3 we prove a version of the tower property for those operators.Section 3.4 is devoted to establishing a duality estimate using a spline square function,which is the crucial ingredient in the proofs of the spline versions of both L´epingle’sinequality (1.1) and H -BMO duality in Section 4.3.1. The positive operators T . As above, let F be an interval σ -algebra on [0 , N i )the B-spline basis of S k ( F ), E i the support of N i and E ij the smallest interval containingboth E i and E j . Moreover, let q be a positive number smaller than 1. Then, we define thelinear operator T = T F ,q,k by T f ( x ) := X i,j q | i − j | | E ij | h f, E i i E j ( x ) = Z K ( x, t ) f ( t ) d t, where the kernel K = K T is given by K ( x, t ) = X i,j q | i − j | | E ij | E i ( t ) · E j ( x ) . We observe that the operator T is selfadjoint (w.r.t the standard inner product on L )and(3.1) k ≤ K x := Z K ( t, x ) d t ≤ k + 1)1 − q , x ∈ [0 , , which, in particular, implies the boundedness of the operator T on L and L ∞ : k T f k ≤ k + 1)1 − q k f k , k T f k ∞ ≤ k + 1)1 − q k f k ∞ . M. PASSENBRUNNER
Another very important property of T is that it is a positive operator, i.e. it maps non-negative functions to non-negative functions and that T satisfies Jensen’s inequality inthe form(3.2) ϕ ( T f ( x )) ≤ K − x T (cid:0) ϕ ( K x · f ) (cid:1) ( x ) , f ∈ L , x ∈ [0 , , for convex functions ϕ . This is seen by applying the classical Jensen inequality to theprobability measure K ( t, x ) d t/K x .Let M f denote the Hardy-Littlewood maximal function of f ∈ L , i.e., M f ( x ) = sup I ∋ x | I | Z I | f ( y ) | d y, where the supremum is taken over all subintervals of [0 ,
1] that contain the point x . Thisoperator is of weak type (1 , |{ M f > λ }| ≤ Cλ − k f k , f ∈ L , λ > C . Since trivially we have the estimate k M f k ∞ ≤ k f k ∞ , by Marcin-kiewicz interpolation, for any p >
1, there exists a constant C p depending only on p sothat k M f k p ≤ C p k f k p . For those assertions about M , we refer to (for instance) [18].The significance of T and M at this point is that we can use formula (2.4) and estimate(2.5) to obtain the pointwise bound(3.3) | P f ( x ) | ≤ C ( T | f | )( x ) ≤ C M f ( x ) , f ∈ L , x ∈ [0 , , where T = T F ,q,k with q given by (2.5) and C , C are two constants solely depending on k .In other words, the positive operator T dominates the non-positive operator P pointwise,but at the same time, T is dominated by M pointwise independently of F .3.2. Stein’s inequality for splines.
We now use this pointwise dominating, positive op-erator T to prove Stein’s inequality for spline projections. For this, let ( F n ) be an intervalfiltration on [0 ,
1] and P n be the orthogonal projection operator onto the space S k ( F n )of splines of order k corresponding to F n . Working with the positive operators T F n ,q,k instead of the non-positive operators P n , the proof of Stein’s inequality (1.2) for splineprojections can be carried over from the martingale case (cf. [19, 1]). For completeness,we include it here. Theorem 3.1.
Suppose that ( f n ) is a sequence of arbitrary integrable functions on [0 , .Then, for ≤ r ≤ p < ∞ or < p ≤ r ≤ ∞ , (3.4) k ( P n f n ) k L p ( ℓ r ) . k ( f n ) k L p ( ℓ r ) where the implied constant depends only on p, r and k .Proof. By (3.3), it suffices to prove this inequality for the operators T n = T F n ,q,k with q given by (2.5) instead of the operators P n . First observe that for r = p = 1, the assertionfollows from Shadrin’s theorem ((i) on page 1). Inequality (3.3) and the L p ′ -boundednessof M for 1 < p ′ ≤ ∞ imply that(3.5) (cid:13)(cid:13) sup ≤ n ≤ N | T n f | (cid:13)(cid:13) p ′ ≤ C p ′ ,k k f k p ′ , f ∈ L p ′ ARTINGALE INEQUALITIES FOR SPLINE SEQUENCES 7 with a constant C p ′ ,k depending on p ′ and k . Let 1 ≤ p < ∞ and U N : L p ( ℓ N ) → L p be given by ( g , . . . , g N ) P Nj =1 T j g j . Inequality (3.5) implies the boundedness of theadjoint U ∗ N : L p ′ → L p ′ ( ℓ N ∞ ), f ( T j f ) Nj =1 for p ′ = p/ ( p −
1) by a constant independentof N and a fortiori the boundedness of U N . Since | T j f | ≤ T j | f | by the positivity of T j ,letting N → ∞ implies (3.4) for T n instead of P n in the case r = 1 and outer parameter1 ≤ p < ∞ .If 1 < r ≤ p , we use Jensen’s inequality (3.2) and estimate (3.1) to obtain N X j =1 | T j g j | r . N X j =1 T j ( | g j | r )and apply the result for r = 1 and the outer parameter p/r to get the result for 1 ≤ r ≤ p < ∞ . The cases 1 < p ≤ r ≤ ∞ now just follow from this result using duality and theself-adjointness of T j . (cid:3) Tower property of T . Next, we will prove a substitute of the tower property E G E F f = E G f ( G ⊂ F ) for conditional expectations that applies to the operators T .To formulate this result, we need a suitable notion of regularity for σ -algebras whichwe now describe. Let F be an interval σ -algebra, let ( N j ) be the B-spline basis of S k ( F )and denote by E j the support of the function N j . The k -regularity parameter γ k ( F ) isdefined as γ k ( F ) := max i max( | E i | / | E i +1 | , | E i +1 | / | E i | ) , where the first maximum is taken over all i so that E i and E i +1 are defined. The name k -regularity is motivated by the fact that each B-spline support E i of order k consists ofat most k (neighboring) atoms of the σ -algebra F . Proposition 3.2 (Tower property of T ) . Let G ⊂ F be two interval σ -algebras on [0 , .Let S = T G ,σ,k and T = T F ,τ,k ′ for some σ, τ ∈ (0 , and some positive integers k, k ′ .Then, for all q > max( τ, σ ) , there exists a constant C depending on q, k, k ′ so that (3.6) | ST f ( x ) | ≤ C · γ k · ( T G ,q,k | f | )( x ) , f ∈ L , x ∈ [0 , , where γ = γ k ( G ) denotes the k -regularity parameter of G .Proof. Let ( F i ) be the collection of B-spline supports in S k ′ ( F ) and ( G i ) the collection ofB-spline supports in S k ( G ). Moreover, we denote by F ij the smallest interval containing F i and F j and by G ij the smallest interval containing G i and G j .We show (3.6) by showing the following inequality for the kernels K S of S and K T of T (cf. (3.1))(3.7) Z K S ( x, t ) K T ( t, s ) d t ≤ Cγ k X i,j q | i − j | | G ij | G i ( x ) G j ( s ) , x, s ∈ [0 , q > max( τ, σ ) and some constant C depending on q, k, k ′ . In order to prove thisinequality, we first fix x, s ∈ [0 ,
1] and choose i such that x ∈ G i and ℓ such that s ∈ F ℓ .Moreover, based on ℓ , we choose j so that s ∈ G j and G j ⊃ F ℓ . There are at most k choicesfor each of the indices i, ℓ, j and without restriction, we treat those choices separately, i.e.,we only have to estimate the expression X m,r σ | m − i | τ | r − ℓ | | G m ∩ F r || G im || F ℓr | . M. PASSENBRUNNER
Since, for each r , there are also at most k + k ′ − m so that | G m ∩ F r | > G ⊂ F ), we choose one such index m = m ( r ) and estimateΣ = X r σ | m ( r ) − i | τ | r − ℓ | | G m ( r ) ∩ F r || G i,m ( r ) || F ℓr | . Now, observe that for any parameter choice of r in the above sum, G i,m ( r ) ∪ F ℓr ⊇ ( G ij \ G j ) ∪ G i and therefore, since also G m ( r ) ∩ F r ⊂ G i,m ( r ) ∩ F ℓr ,Σ ≤ | ( G ij \ G j ) ∪ G i | X r σ | m ( r ) − i | τ | r − ℓ | , which, using the k -regularity parameter γ = γ k ( G ) of the σ -algebra G and denoting λ = max( τ, σ ), we estimate byΣ ≤ γ k | G ij | X m λ | m − i | X r : m ( r )= m λ | r − ℓ | . γ k | G ij | X m λ | i − m | + | m − j | . γ k | G ij | (cid:0) | i − j | + 1 (cid:1) λ | i − j | , where the implied constants depend on λ, k, k ′ and the estimate P r : m ( r )= m λ | r − ℓ | . λ | m − j | used the fact that, essentially, there are more atoms of F between F r and F ℓ (for r as inthe sum) than atoms of G between G m and G j . Finally, we see that for any q > λ ,Σ . Cγ k q | i − j | | G ij | for some constant C depending on q, k, k ′ , and, as x ∈ G i and s ∈ G j , this shows inequality(3.7). (cid:3) As a corollary of Proposition 3.2, we have
Corollary 3.3.
Let ( f n ) be functions in L . We denote by P n the orthogonal projectiononto S k ( F n ) and by P ′ n the orthogonal projection onto S k ′ ( F n ) for some positive integers k, k ′ . Moreover, let T n be the operator T F n ,q,k from (3.3) dominating P n pointwise.Then, for any integer n and for any ≤ p ≤ ∞ , (cid:13)(cid:13)(cid:13) X ℓ ≥ n P n (cid:0) ( P ′ ℓ − f ℓ ) (cid:1)(cid:13)(cid:13)(cid:13) p . (cid:13)(cid:13)(cid:13) X ℓ ≥ n T n (cid:0) ( P ′ ℓ − f ℓ ) (cid:1)(cid:13)(cid:13)(cid:13) p . γ k ( F n ) k · (cid:13)(cid:13)(cid:13) X ℓ ≥ n f ℓ (cid:13)(cid:13)(cid:13) p , where the implied constants only depend on k and k ′ . We remark that by Jensen’s inequality and the tower property, this is trivial for con-ditional expectations E ( ·| F n ) instead of the operators P n , T n , P ′ ℓ − even with an absoluteconstant on the right hand side. Proof.
We denote by T n the operator T F n ,q,k and by T ′ n the operator T F n ,q ′ ,k ′ , where theparameters q, q ′ < k and k ′ respectively.Setting U n := T F n , max( q,q ′ ) / ,k , we perform the following chain of inequalities, where we ARTINGALE INEQUALITIES FOR SPLINE SEQUENCES 9 use the positivity of T n and (3.3), Jensen’s inequality for T ′ ℓ − , the tower property for T n T ′ ℓ − and the L p -boundedness of U n , respectively: (cid:13)(cid:13)(cid:13) X ℓ ≥ n T n (cid:0) ( P ′ ℓ − f ℓ ) (cid:1)(cid:13)(cid:13)(cid:13) p . (cid:13)(cid:13)(cid:13) X ℓ ≥ n T n (cid:0) ( T ′ ℓ − | f ℓ | ) (cid:1)(cid:13)(cid:13)(cid:13) p . (cid:13)(cid:13)(cid:13) X ℓ ≥ n T n (cid:0) T ′ ℓ − f ℓ (cid:1)(cid:13)(cid:13)(cid:13) p ≤ k T n ( T ′ n − f n ) k p + (cid:13)(cid:13)(cid:13) X ℓ>n T n (cid:0) T ′ ℓ − f ℓ (cid:1)(cid:13)(cid:13)(cid:13) p . k f n k p + γ k ( F n ) k · (cid:13)(cid:13)(cid:13) X ℓ>n U n ( f ℓ ) (cid:13)(cid:13)(cid:13) p . γ k ( F n ) k · (cid:13)(cid:13)(cid:13) X ℓ ≥ n f ℓ (cid:13)(cid:13)(cid:13) p , where the implied constants only depend on k and k ′ . (cid:3) A duality estimate using a spline square function.
In order to give the de-sired duality estimate contained in Theorem 3.6, we need the following construction of afunction g n ∈ S k ( F n ) based on a spline square function. Proposition 3.4.
Let ( f n ) be a sequence of functions with f n ∈ S k ( F n ) for all n and set X n := X ℓ ≤ n f ℓ . Then, there exists a sequence of non-negative functions g n ∈ S k ( F n ) so that for each n ,(1) g n ≤ g n +1 ,(2) X / n ≤ g n (3) E g n . E X / n , where the implied constant depends on k and on sup m ≤ n γ k ( F m ) . For the proof of this result, we need the following simple lemma.
Lemma 3.5.
Let c be a positive constant and let ( A j ) Nj =1 be a sequence of atoms in F n .Moreover, let ℓ : { , . . . , N } → { , . . . , n } and, for each j ∈ { , . . . , N } , let B j be a subsetof an atom D j of F ℓ ( j ) with (3.8) | B j | ≥ c X i : ℓ ( i ) ≥ ℓ ( j ) ,D i ⊆ D j | A i | . Then, there exists a set-valued mapping ϕ on { , . . . , N } so that(1) | ϕ ( j ) | = c | A j | for all j ,(2) ϕ ( j ) ⊆ B j for all j ,(3) ϕ ( i ) ∩ ϕ ( j ) = ∅ for all i = j .Proof. Without restriction, we assume that the sequence ( A j ) is enumerated such that ℓ ( j + 1) ≤ ℓ ( j ) for all 1 ≤ j ≤ N −
1. We first choose ϕ (1) as an arbitrary (measurable)subset of B with measure c | A | , which is possible by assumption (3.8). Next, we assumethat for 1 ≤ j ≤ j , we have constructed ϕ ( j ) with the properties(1) | ϕ ( j ) | = c | A j | , (2) ϕ ( j ) ⊆ B j ,(3) ϕ ( j ) ∩ ∪ i Fix n and let ( N n,j ) be the B-spline basis of S k ( F n ). Moreover,for any j , set E n,j = supp N n,j and a n,j := max ℓ ≤ n max r : E ℓ,r ⊃ E n,j k X ℓ k / L ∞ ( E ℓ,r ) and wedefine ℓ ( j ) ≤ n and r ( j ) so that E ℓ ( j ) ,r ( j ) ⊇ E n,j and a n,j = k X ℓ ( j ) k / L ∞ ( E ℓ ( j ) ,r ( j ) ) . Set g n := X j a n,j N n,j ∈ S k ( F n )and it will be proved subsequently that this g n has the desired properties. Property (1): In order to show g n ≤ g n +1 , we use Theorem 2.4 to write g n = X j a n,j N n,j = X j β n,j N n +1 ,j , where β n,j is a convex combination of those a n,r with E n +1 ,j ⊆ E n,r , and thus g n ≤ X j (cid:0) max r : E n +1 ,j ⊆ E n,r a n,r (cid:1) N n +1 ,j . By the very definition of a n +1 ,j , we havemax r : E n +1 ,j ⊆ E n,r a n,r ≤ a n +1 ,j , and therefore, g n ≤ g n +1 pointwise, since the B-splines ( N n +1 ,j ) j are nonnegative functions. Property (2): Now we show that X / n ≤ g n . Indeed, for any x ∈ [0 , g n ( x ) = X j a n,j N n,j ( x ) ≥ min j : E n,j ∋ x a n,j ≥ min j : E n,j ∋ x k X n k / L ∞ ( E n,j ) ≥ X n ( x ) / , since the collection of B-splines ( N n,j ) j forms a partition of unity. ARTINGALE INEQUALITIES FOR SPLINE SEQUENCES 11 Property (3): Finally, we show E g n . E X / n , where the implied constant dependsonly on k and on sup m ≤ n γ k ( F m ). By B-spline stability (Theorem 2.3), we estimate theintegral of g n by(3.9) E g n . X j | E n,j | · k X ℓ ( j ) k / L ∞ ( E ℓ ( j ) ,r ( j ) ) , where the implied constant only depends on k . In order to continue the estimate, we nextshow the inequality(3.10) k X ℓ k L ∞ ( E ℓ,r ) . max s : | E ℓ,r ∩ E ℓ,s | > k X ℓ k L ∞ ( J ℓ,s ) , where by J ℓ,s we denote an atom of F ℓ with J ℓ,s ⊂ E ℓ,s of maximal length and the impliedconstant depends only on k . Indeed, we use Theorem 2.3 in the form of (2.3) to get( f m ∈ S k ( F ℓ ) for m ≤ ℓ )(3.11) k X ℓ k L ∞ ( E ℓ,r ) ≤ X m ≤ ℓ k f m k L ∞ ( E ℓ,r ) . X m ≤ ℓ X s : | E ℓ,s ∩ E ℓ,r | > k f m k L ∞ ( J ℓ,s ) = X s : | E ℓ,s ∩ E ℓ,r | > X m ≤ ℓ k f m k L ∞ ( J ℓ,s ) . Now observe that for atoms I of F ℓ , due to the equivalence of p -norms of polynomials(cf. Corollary 2.2), X m ≤ ℓ k f m k L ∞ ( I ) . X m ≤ ℓ | I | Z I f m = 1 | I | Z I X ℓ ≤ k X ℓ k L ∞ ( I ) , which means that, inserting this in estimate (3.11), k X ℓ k L ∞ ( E ℓ,r ) . X s : | E ℓ,s ∩ E ℓ,r | > k X ℓ k L ∞ ( J ℓ,s ) , and, since there are at most k indices s so that | E ℓ,s ∩ E ℓ,r | > 0, we have established(3.10).We define K ℓ,r to be an interval J ℓ,s with | E ℓ,r ∩ E ℓ,s | > s : | E ℓ,r ∩ E ℓ,s | > k X ℓ k L ∞ ( J ℓ,s ) = k X ℓ k L ∞ ( K ℓ,r ) . This means, after combining (3.9) with (3.10), we have(3.12) E g n . X j | J n,j | · k X ℓ ( j ) k / L ∞ ( K ℓ ( j ) ,r ( j ) ) . We now apply Lemma 3.5 with the function ℓ and the choices A j = J n,j , D j = K ℓ ( j ) ,r ( j ) ,B j = n t ∈ D j : X ℓ ( j ) ( t ) ≥ − k − k X ℓ ( j ) k L ∞ ( D j ) o . In order to see assumption (3.8) of Lemma 3.5, fix the index j and let i be such that ℓ ( i ) ≥ ℓ ( j ). By definition of D i = K ℓ ( i ) ,r ( i ) , the smallest interval containing J n,i and D i contains at most 2 k − F ℓ ( i ) and, if D i ⊂ D j , the smallest interval containing J n,i and D j contains at most 2 k − F ℓ ( j ) . This means that, in particular, J n,i is a subset of the union V of 4 k atoms of F ℓ ( j ) with D j ⊂ V . Since each atom of F n occurs at most k times in the sequence ( A j ), there exists a constant c depending on k and sup u ≤ ℓ ( j ) γ k ( F u ) ≤ sup u ≤ n γ k ( F u ) so that | D j | ≥ c X i : ℓ ( i ) ≥ ℓ ( j ) D i ⊂ D j | A i | , which, since | B j | ≥ | D j | / ϕ so that | ϕ ( j ) | = c | J n,j | / ϕ ( j ) ⊂ B j , ϕ ( i ) ∩ ϕ ( j ) = ∅ for all i, j . Using these properties of ϕ , we continue the estimate in (3.12) and write E g n . X j | J n,j | · k X ℓ ( j ) k / L ∞ ( D j ) ≤ k − · X j | J n,j || ϕ ( j ) | Z ϕ ( j ) X / ℓ ( j ) = 2 c · k − · X j Z ϕ ( j ) X / ℓ ( j ) . X j Z ϕ ( j ) X / n ≤ E X / n , with constants depending only on k and sup u ≤ n γ k ( F u ). (cid:3) Employing this construction of g n , we now give the following duality estimate for splineprojections (for the martingale case, see for instance [5]). The martingale version of thisresult is the essential estimate in the proof of both L´epingle’s inequality (1.1) and the H -BMO duality. Theorem 3.6. Let ( F n ) be such that γ := sup n γ k ( F n ) < ∞ and ( f n ) n ≥ a sequence offunctions with f n ∈ S k ( F n ) for each n . Additionally, let h n ∈ L be arbitrary. Then, forany N , X n ≤ N E [ | f n · h n | ] . E h(cid:16) X ℓ ≤ N f ℓ (cid:17) / i · sup n ≤ N k P n (cid:0) N X ℓ = n h ℓ (cid:1) k / ∞ , where the implied constant depends only on k and γ .Proof. Let X n := P ℓ ≤ n f ℓ and f ℓ ≡ ℓ > N and ℓ ≤ 0. By Proposition 3.4, we choosean increasing sequence ( g n ) of functions with g = 0, g n ∈ S k ( F n ) and the properties X / n ≤ g n and E g n . E X / n where the implied constant depends only on k and γ . Then,apply Cauchy-Schwarz inequality by introducing the factor g / n to get X n E [ | f n · h n | ] = X n E h(cid:12)(cid:12)(cid:12) f n g / n · g / n h n (cid:12)(cid:12)(cid:12)i ≤ h X n E [ f n /g n ] i / · h X n E [ g n h n ] i / . We estimate each of the factors on the right hand side separately and setΣ := X n E [ f n /g n ] , Σ := X n E [ g n h n ] . The first factor is estimated by the pointwise inequality X / n ≤ g n :Σ = E (cid:2) X n f n g n (cid:3) ≤ E (cid:2) X n f n X / n (cid:3) ARTINGALE INEQUALITIES FOR SPLINE SEQUENCES 13 = E (cid:2) X n X n − X n − X / n (cid:3) ≤ E X n ( X / n − X / n − ) = 2 E X / N . We continue with Σ :Σ = E (cid:2) N X ℓ =1 g ℓ h ℓ (cid:3) = E h N X ℓ =1 ℓ X n =1 ( g n − g n − ) h ℓ i = E h N X n =1 ( g n − g n − ) · N X ℓ = n h ℓ i = E h N X n =1 P n ( g n − g n − ) · N X ℓ = n h ℓ i = E h N X n =1 ( g n − g n − ) · P n (cid:0) N X ℓ = n h ℓ (cid:1)i ≤ E h N X n =1 ( g n − g n − ) i · sup ≤ n ≤ N (cid:13)(cid:13) P n (cid:0) N X ℓ = n h ℓ (cid:1)(cid:13)(cid:13) ∞ , where the last inequality follows from g n ≥ g n − . Noting that by the properties of g n , E (cid:2) P Nn =1 ( g n − g n − ) (cid:3) = E g N . E X / N and combining the two parts Σ and Σ , we obtainthe conclusion. (cid:3) Applications We give two applications of Theorem 3.6, (i) D. L´epingle’s inequality and (ii) ananalogue of C. Fefferman’s H -BMO duality in the setting of splines. Once the resultsfrom Section 3 are known, the proofs of the subsequent results proceed similarly to theirmartingale counterparts in [9] and [5] by using spline properties instead of martingaleproperties.4.1. L´epingle’s inequality for splines.Theorem 4.1. Let k, k ′ be positive integers. Let ( F n ) be an interval filtration with sup n γ k ( F n ) < ∞ and, for any n , f n ∈ S k ( F n ) and P ′ n be the orthogonal projectionoperator on S k ′ ( F n ) . Then, k ( P ′ n − f n ) k L ( ℓ ) = (cid:13)(cid:13)(cid:13)(cid:0) X n ( P ′ n − f n ) (cid:1) / (cid:13)(cid:13)(cid:13) . (cid:13)(cid:13)(cid:13)(cid:0) X n f n (cid:1) / (cid:13)(cid:13)(cid:13) = k ( f n ) k L ( ℓ ) , where the implied constant depends only on k , k ′ and sup n γ k ( F n ) .Proof. We first assume that f n = 0 for n > N and begin by using duality E (cid:2)(cid:0) X n ( P ′ n − f n ) (cid:1) / (cid:3) = sup ( H n ) E (cid:2) X n ( P ′ n − f n ) · H n (cid:3) , where sup is taken over all ( H n ) ∈ L ∞ ( ℓ ) with k ( H n ) k L ∞ ( ℓ ) = 1. By the self-adjointnessof P ′ n − , E (cid:2) ( P ′ n − f n ) · H n (cid:3) = E (cid:2) f n · ( P ′ n − H n ) (cid:3) . Then we apply Theorem 3.6 for f n and h n = P ′ n − H n to obtain (denoting by P n theorthogonal projection operator onto S k ( F n ))(4.1) X n ≤ N | E [ f n · h n ] | . E h(cid:0) X ℓ ≤ N f ℓ (cid:1) / i · sup n ≤ N (cid:13)(cid:13)(cid:13) P n (cid:0) N X ℓ = n ( P ′ ℓ − H ℓ ) (cid:1)(cid:13)(cid:13)(cid:13) / ∞ . To estimate the right hand side, we note that for any n , by Corollary 3.3, (cid:13)(cid:13) P n (cid:0) N X ℓ = n ( P ′ ℓ − H ℓ ) (cid:1)(cid:13)(cid:13) ∞ . (cid:13)(cid:13) N X ℓ = n H ℓ (cid:13)(cid:13) ∞ . Therefore, (4.1) implies E (cid:2)(cid:0) X n ( P ′ n − f n ) (cid:1) / (cid:3) = sup ( H n ) E (cid:2) X n f n · ( P ′ n − H n ) (cid:3) . E h(cid:0) X ℓ ≤ N f ℓ (cid:1) / i , with a constant depending only on k , k ′ and sup n ≤ N γ k ( F n ). Letting N tend to infinity,we obtain the conclusion. (cid:3) H - BMO duality for splines. We fix an interval filtration ( F n ) ∞ n =1 , a spline order k and the orthogonal projection operators P n onto S k ( F n ) and additionally, we set P = 0.For f ∈ L , we introduce the notation∆ n f := P n f − P n − f, S n ( f ) := (cid:16) n X ℓ =1 (∆ ℓ f ) (cid:17) / , S ( f ) = sup n S n ( f ) . Observe that for ℓ < n and f, g ∈ L ,(4.2) E [∆ ℓ f · ∆ n g ] = E [ P ℓ (∆ ℓ f ) · ∆ n g ] = E [∆ ℓ f · P ℓ (∆ n g )] = 0 . Let V be the L -closure of ∪ n S k ( F n ). Then, the uniform boundedness of P n on L implies that P n f → f in L for f ∈ V . Next, set H ,k = H ,k (( F n )) = { f ∈ V : E ( S ( f )) < ∞} and equip H ,k with the norm k f k H ,k = E S ( f ). DefineBMO k = BMO k (( F n )) = { f ∈ V : sup n k X ℓ ≥ n T n (cid:0) (∆ ℓ f ) (cid:1) k ∞ < ∞} and the corresponding quasinorm k f k BMO k = sup n (cid:13)(cid:13) X ℓ ≥ n T n (cid:0) (∆ ℓ f ) (cid:1)(cid:13)(cid:13) / ∞ , where T n is the operator from (3.3) that dominates P n pointwise.Let us now assume sup n γ k ( F n ) < ∞ . In this case we identify, similarly to H -BMO-duality (cf. [4, 7, 5]), BMO k as the dual space of H ,k .First, we use the duality estimate Theorem 3.6 and (4.2) to prove, for f ∈ H ,k and h ∈ BMO k , (cid:12)(cid:12) E (cid:2) ( P n f ) · ( P n h ) (cid:3)(cid:12)(cid:12) ≤ X ℓ ≤ n E (cid:2) | ∆ ℓ f | · | ∆ ℓ h | (cid:3) . k S n ( f ) k · k h k BMO k . This estimate also implies that the limit lim n E (cid:2) ( P n f ) · ( P n h ) (cid:3) exists and satisfies (cid:12)(cid:12) lim n E (cid:2) ( P n f ) · ( P n h ) (cid:3)(cid:12)(cid:12) . k f k H ,k · k h k BMO k . ARTINGALE INEQUALITIES FOR SPLINE SEQUENCES 15 On the other hand, similarly to the martingale case (see [5]), given a continuous linearfunctional L on H ,k , we extend L norm-preservingly to a continuous linear functional Λon L ( ℓ ), which, by Section 2.4, has the formΛ( η ) = E (cid:2) X ℓ σ ℓ η ℓ (cid:3) , η ∈ L ( ℓ )for some σ ∈ L ∞ ( ℓ ). The k -martingale spline sequence h n = P ℓ ≤ n ∆ ℓ σ ℓ is bounded in L and therefore, by the spline convergence theorem ((v) on page 2), has a limit h ∈ L with P n h = h n and which is also contained in BMO k . Indeed, by using Corollary 3.3,we obtain k h k BMO k . k σ k L ∞ ( ℓ ) = k Λ k = k L k with a constant depending only on k andsup n γ k ( F n ). Moreover, for f ∈ H ,k , since L is continuous on H ,k , L ( f ) = lim n L ( P n f ) = lim n Λ (cid:0) (∆ f, . . . , ∆ n f, , , . . . ) (cid:1) = lim n n X ℓ =1 E [ σ ℓ · ∆ ℓ f ] = lim n E (cid:2) ( P n f ) · ( P n h ) (cid:3) . This yields the following Theorem 4.2. If sup n γ k ( F n ) < ∞ , the linear mapping u : BMO k → H ∗ ,k , h (cid:0) f lim n E (cid:2) ( P n f ) · ( P n h ) (cid:3)(cid:1) is bijective and satisfies k u ( h ) k H ∗ ,k ≃ k h k BMO k , where the implied constants depend only on k and sup n γ k ( F n ) . Remark 4.3. We close with a few remarks concerning the above result and we assumethat ( F n ) is a sequence of increasing interval σ -algebras with sup n γ k ( F n ) < ∞ .(1) By Khintchine’s inequality, k Sf k . sup ε ∈{− , } Z k P n ε n ∆ n f k . Based on the in-terval filtration ( F n ), we can generate an interval filtration ( G n ) that contains( F n ) as a subsequence and each G n +1 is generated from G n by dividing exactlyone atom of G n into two atoms of G n +1 . Denoting by P G n the orthogonal projectionoperator onto S k ( G n ) and ∆ G j = P G j − P G j − , we can write X n ε n ∆ n f = X n ε n a n +1 − X j = a n ∆ G j f for some sequence ( a n ). By using inequalities (2.7) and (2.6) and writing ( S G f ) = P j | ∆ G j f | , we obtain for p > k Sf k . k S G f k ≤ k S G f k p . k f k p . This implies L p ⊂ H ,k for all p > k ⊂ L p for all p < ∞ .(2) If ( F n ) is of the form that each F n +1 is generated from F n by splitting exactly oneatom of F n into two atoms of F n +1 and under the condition sup n γ k − ( F n ) < ∞ (which is stronger than sup n γ k ( F n ) < ∞ ), it is shown in [6] that k Sf k ≃ k f k H , where H denotes the atomic Hardy space on [0 , H ,k coincideswith H . Acknowledgments. The author is supported by the Austrian Science Fund (FWF),Project F5509-N26, which is a part of the Special Research Program “Quasi-Monte CarloMethods: Theory and Applications”. References [1] N. Asmar and S. Montgomery-Smith. Littlewood-Paley theory on solenoids. Colloq. Math. , 65(1):69–82, 1993.[2] A. Benedek and R. Panzone. The space L p , with mixed norm. Duke Math. J. , 28:301–324, 1961.[3] R. A. DeVore and G. G. Lorentz. Constructive approximation , volume 303 of Grundlehren der Math-ematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] . Springer-Verlag,Berlin, 1993.[4] C. Fefferman. Characterizations of bounded mean oscillation. Bull. Amer. Math. Soc. , 77:587–588,1971.[5] A. M. Garsia. Martingale inequalities: Seminar notes on recent progress . W. A. Benjamin, Inc.,Reading, Mass.-London-Amsterdam, 1973. Mathematics Lecture Notes Series.[6] G. Gevorkyan, A. Kamont, K. Keryan, and M. Passenbrunner. Unconditionality of orthogonal splinesystems in H . Studia Math. , 226(2):123–154, 2015.[7] C. Herz. Bounded mean oscillation and regulated martingales. Trans. Amer. Math. Soc. , 193:199–215, 1974.[8] K. Keryan and M. Passenbrunner. Unconditionality of periodic orthonormal spline systems in L p . preprint arXiv:1708.09294, to appear in Studia Math. , 2017.[9] D. L´epingle. Une in´egalit´e de martingales. In S´eminaire de Probabilit´es, XII (Univ. Strasbourg, Stras-bourg, 1976/1977) , volume 649 of Lecture Notes in Math. , pages 134–137. Springer, Berlin, 1978.[10] P. F. X. M¨uller and M. Passenbrunner. Almost everywhere convergence of spline sequences. preprintarXiv:1711.01859 , 2017.[11] J. Neveu. Discrete-parameter martingales . North-Holland Publishing Co., Amsterdam-Oxford; Amer-ican Elsevier Publishing Co., Inc., New York, revised edition, 1975. Translated from the French byT. P. Speed, North-Holland Mathematical Library, Vol. 10.[12] M. Passenbrunner. Unconditionality of orthogonal spline systems in L p . Studia Math. , 222(1):51–86,2014.[13] M. Passenbrunner. Orthogonal projectors onto spaces of periodic splines. Journal of Complexity ,42:85–93, 2017.[14] M. Passenbrunner. Spline Characterizations of the Radon-Nikod´ym property. preprintarXiv:1807.01861 , 2018.[15] M. Passenbrunner and A. Shadrin. On almost everywhere convergence of orthogonal spline projec-tions with arbitrary knots. J. Approx. Theory , 180:77–89, 2014.[16] L. L. Schumaker. Spline functions: basic theory . John Wiley & Sons Inc., New York, 1981. Pure andApplied Mathematics, A Wiley-Interscience Publication.[17] A. Shadrin. The L ∞ -norm of the L -spline projector is bounded independently of the knot sequence:a proof of de Boor’s conjecture. Acta Math. , 187(1):59–137, 2001.[18] E. M. Stein. Singular integrals and differentiability properties of functions . Princeton MathematicalSeries, No. 30. Princeton University Press, Princeton, N.J., 1970.[19] E. M. Stein. Topics in harmonic analysis related to the Littlewood-Paley theory . Annals of Mathemat-ics Studies, No. 63. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo,1970. Institute of Analysis, Johannes Kepler University Linz, Austria, 4040 Linz, AltenbergerStrasse 69 E-mail address ::