Martingale inequalities for spline sequences

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Introduction
This article is part of a series of papers that extend martingale results to polynomial spline sequences of arbitrary order (see e.g. [11,14,[16][17][18][19]22]). In order to explain those martingale type results, we have to introduce a little bit of terminology: Let k be a 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 n be the orthogonal projection operator onto S k (F n ) with respect to the L 2 inner product on [0, 1] with the Lebesgue measure | · |. The space S 1 (F n ) consists of piecewise constant functions and P (1) n is the conditional expectation operator with respect to the σ -algebra F n . Similarly to the definition of martingales, we introduce the following notion: let ( f n ) n≥0 be a sequence of integrable functions. We call this sequence a k-martingale spline sequence (adapted to (F n )) if, for all n, For basic facts about martingales and conditional expectations, we refer to [15].
Classical martingale theorems such as Doob's inequality or the martingale convergence theorem 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 by the projection operators P (k) n . Indeed, we have (i) (Shadrin's theorem) there exists a constant C k depending only on k such that sup n P (k) n : (ii) (Doob's weak type inequality for splines) there exists a constant C k depending only on k such that for any k-martingale spline sequence ( f n ) and any λ > 0, (iv) (Spline convergence theorem) if ( f n ) is an L 1 -bounded k-martingale spline sequence, then ( f n ) converges almost surely to some L 1 -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) was proved by Shadrin in the groundbreaking paper [22]. We also refer to the paper [25] by von Golitschek, who gives a substantially shorter proof of (i). Properties (ii) and (iii) are proved in [19] and properties (iv) and (v) in [14], but see also [18], where it is shown that, in analogy to the martingale case, the validity of (iv) and (v) for all k-martingale spline sequences with values in a Banach space X characterize the Radon-Nikodým property of X (for background information on that material, we refer to the monographs [6,20]).
Here, we continue this line of transferring martingale results to k-martingale spline sequences and extend Lépingle's L 1 ( 2 )-inequality [12], which reads 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. Different proofs of (1. This inequality is an L 1 extension of the following result for 1 < p < ∞, proved by Stein [24], that holds for arbitrary integrable functions f n : The usage of those operators T n is also necessary in the extension of inequality (1.1) to splines. Lépingle's proof of (1.1) rests on an idea by Herz [10] of splitting E[ f n · h n ] (for f n being F n -measurable) by Cauchy-Schwarz after introducing the square function S 2 n = ≤n f 2 : and estimating both factors on the right hand side separately. A key point in estimating the second factor is that S n is F n -measurable, and therefore, E[S n |F n ] = S n . If we want to allow f n ∈ S k (F n ), S n will not be contained in S k (F n ) in general. Under certain conditions on the filtration (F n ), we will show in this article how to substitute S n in estimate (1.3) by a function g n ∈ S k (F n ) that enjoys similar properties to S n and allows us to proceed (cf. Sect. 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 [7] on H 1 -BMO duality. For its martingale version, we refer to A. M. Garsia's book [8] on Martingale Inequalities.

Preliminaries
In this section, we collect all tools that are needed subsequently.

Properties of polynomials
We will need Remez' inequality for polynomials: Then, for all polynomials p of order k (i.e. degree k − 1) on V , Applying this theorem with the set E = {x ∈ V : |p(x)| ≤ 8 −k+1 p L ∞ (V ) } immediately yields the following corollary: Corollary 2.2 Let p be a polynomial of order k on a compact interval V ⊂ R. Then

Properties of spline functions
For an interval σ -algebra F (i.e. F is generated by a finite collection of intervals having positive length), the space S k (F ) is spanned by a 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, 1], there exist at most k functions N i so that N i (x) = 0 and i N i (x) = 1. In the following, we denote by E i the support of the B-spline function N i . The usual ordering of the B-splines (N i )-which we also employ here-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, 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 indicate this in the text. Similarly, the symbols and are used.
Another important property of B-splines is the following relation between B-spline coefficients and the L p -norm of the corresponding B-spline expansions.
where J j is an atom of F contained in E j having maximal length. Additionally,

2)
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, 1] We will also need the following relation between the B-spline expansion of a function and 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 Bspline basis 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 = j a j N G , j , we can expand f in the basis For those results and more information on spline functions, in particular B-splines, we refer to [21] or [5].

Spline orthoprojectors
We now use the B-spline basis of S k (F ) and expand the orthogonal projection operator P onto S k (F ) in the form for some coefficients (a i j ). Denoting by E i j 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 estimate for a i j [19]: there exist constants C and 0 < q < 1 depending only on k so that for each interval σ -algebra F and each i, j,

Spline square functions
Let (F n ) be a sequence of increasing interval σ -algebras in [0, 1] and we assume that each F n+1 is generated from F n by the subdivision of exactly one atom of F n into two atoms of F n+1 . Let P n be the orthogonal projection operator onto S k (F n ). We denote n f = P n f − P n−1 f and define the spline square function We have Burkholder's inequality for the spline square function, i.e. for all 1 < p < ∞ [16], the L p -norm of the square function S f is comparable to the L p -norm of f : with constants depending only on p and k. Moreover, for p = 1, it is shown in [9] that with constants depending only on k and where the proof of the -part only uses Khintchine's inequality whereas the proof of the -part uses fine properties of the functions n f .

L p ( q )-spaces
and the duality pairing Hölder's inequality takes the form

Main results
In this section, we prove our main results. Section 3.1 defines and gives properties of suitable positive operators that dominate our (non-positive) operators P n = P (k) n pointwise. In Sect. 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 In this form, it extends to the operators (P n ), but not to the operators T from Sect. 3.1. In Sect. 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 Lépingle's inequality (1.1) and H 1 -BMO duality in Sect. 4.

The positive operators T
As above, let F be an interval σ -algebra on [0, 1], (N i ) the B-spline basis of S k (F ), E i the support of N i and E i j the smallest interval containing both E i and E j . Moreover, let q be a positive number smaller than 1. Then, we define the linear operator T = T F ,q,k by where the kernel K = K T is given by We observe that the operator T is selfadjoint (w.r.t the standard inner product on L 2 ) and which, in particular, implies the boundedness of the operator T on L 1 and L ∞ : Another very important property of T is that it is a positive operator, i.e. it maps nonnegative functions to non-negative functions and that T satisfies Jensen's inequality in the form for convex functions ϕ. This is seen by applying the classical Jensen inequality to the probability measure where the supremum is taken over all subintervals of [0, 1] that contain the point x. This operator is of weak type (1, 1), i.e.
for some constant C. Since trivially we have the estimate M f ∞ ≤ f ∞ , by Marcinkiewicz interpolation, for any p > 1, there exists a constant C p depending only on p so that For those assertions about M , we refer to (for instance) [23]. 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 where T = T F ,q,k with q given by (2.5), C 1 is a constant that depends only on k and C 2 is a constant that depends only on k and the geometric progression q. But as the parameter q < 1 in (2.5) depends only on k, the constant C 2 will also only depend on k.
In other words, (3.3) tells us that the positive operator T dominates the non-positive operator P pointwise, but at the same time, T is dominated by the Hardy-Littlewood maximal function M pointwise and independently of F .

Stein's inequality for splines
We now use this pointwise dominating, positive operator T to prove Stein's inequality for spline projections. For this, let (F n ) be an interval filtration 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 nonpositive operators P n , the proof of Stein's inequality (1.2) for spline projections can be carried over from the martingale case (cf. [1,24]). For completeness, we include it here.

Theorem 3.1 Suppose that ( f n ) is a sequence of arbitrary integrable functions on
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 assertion follows from Shadrin's theorem ((i) on page 1). Inequality (3.3) and the L p -boundedness of M for 1 < p ≤ ∞ imply that with a constant C p ,k depending on p and k. Let 1 ≤ p < ∞ and U N : L p ( N 1 ) → L p be given by (g 1 , . . . , g N ) → N j=1 T j g j . Inequality (3.5) implies the boundedness of the adjoint U * N : by a constant independent of N and therefore also 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 parameter 1 ≤ p < ∞.
If 1 < r ≤ p, we use Jensen's inequality (3.2) and estimate (3.1) 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 the self-adjointness of T j .

Tower property of T
Next, we will prove a substitute of the tower property conditional expectations that applies to the operators T .
To formulate this result, we need a suitable notion of regularity for σ -algebras which we 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 ) is defined as 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 of at 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, 1]. Let S = T G ,σ,k and T = T F ,τ,k for some σ, τ ∈ (0, 1) 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 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 of B-spline supports in S k (G ). Moreover, we denote by F i j the smallest interval containing F i and F j and by G i j 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) for all q > max(τ, σ ) and some constant C depending on q, k, k . In order to prove this inequality, we first fix x, s ∈ [0, 1] and choose i such that x ∈ G i and such that s ∈ F . Moreover, based on , we choose j so that s ∈ G j and G j ⊃ F . There are at most max(k, k ) choices for each of the indices i, , j and without restriction, we treat those choices separately, i.e. we only have to estimate the expression m,r Since, for each r , there are also at most k + k − 1 indices m so that |G m ∩ F r | > 0 (recall that G ⊂ F ), we choose one such index m = m(r ) and estimate Now, observe that for any parameter choice of r in the above sum, which, using the k-regularity parameter γ = γ k (G ) of the σ -algebra G and denoting λ = max(τ, σ ), we estimate by where the implied constants depend on λ, k, k and the estimate r :m(r )=m λ |r − | λ |m− j| used the fact that, essentially, there are more atoms of F between F r and F (for r as in the sum) than atoms of G between G m and G j . Finally, we see that for any q > λ, for some constant C depending on q, k, k , and, as x ∈ G i and s ∈ G j , this shows inequality (3.7).
As a corollary of Proposition 3.2, we have Corollary 3.3 Let ( f n ) be functions in L 1 . We denote by P n the orthogonal projection onto 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 1 ≤ 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 conditional expectations E(·|F n ) instead of the operators P n , T n , P −1 even with an absolute constant 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 the parameters q, q < 1 are given by inequality (3.3) depending on k and k respectively. Setting U n := T F n ,max(q,q ) 1/2 ,k , we perform the following chain of inequalities, where we use the positivity of T n and (3.3), Jensen's inequality for T −1 , the tower property for T n T −1 and the L p -boundedness of U n , respectively: where the implied constants only depend on k and k .

A duality estimate using a spline square function
In order to give the desired duality estimate contained in Theorem 3.6, we need the following construction of a function g n ∈ S k (F n ) based on a spline square function. Then, there exists a sequence of non-negative functions g n ∈ S k (F n ) so that for each n, For the proof of this result, we need the following simple lemma.  Moreover, let : {1, . . . , N } → {1, . . . , n} and, for each j ∈ {1, . . . , N }, Then, there exists a map ϕ on {1, . . . , N } so that Proof Without restriction, we assume that the sequence (A j ) is enumerated such that ( j +1) ≤ ( j) for all 1 ≤ j ≤ N −1. We first choose ϕ(1) as an arbitrary (measurable) subset of B 1 with measure c 1 |A 1 |, which is possible by assumption (3.8). Next, we assume that for 1 ≤ j ≤ j 0 , we have constructed ϕ( j) with the properties Based on that, we now construct ϕ( j 0 + 1). Define the index sets = {i : (i) ≥ ( j 0 + 1), D i ⊆ D j 0 +1 } and = {i : i ≤ j 0 + 1, D i ⊆ D j 0 +1 }. Since we assumed that is decreasing, we have ⊆ and by the nestedness of the σ -algebras F n , we have for i ≤ j 0 + 1 that either D i ⊂ D j 0 +1 or |D i ∩ D j 0 +1 | = 0. This implies Therefore, we can choose ϕ( j 0 + 1) ⊆ B j 0 +1 that is disjoint to ϕ(i) for any i ≤ j 0 and |ϕ( j 0 + 1)| = c 1 |A j 0 +1 | which completes the proof. 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 ≤n max r :E ,r ⊃E n, j X 1/2 L ∞ (E ,r ) and we define ( j) ≤ n and r ( j) so that E ( j),r ( j) ⊇ E n, j and a n, j = X ( j) r ( j) ) . Set g n := 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.

Proof of Proposition 3.4 Fix
Property (1): In order to show g n ≤ g n+1 , we use Theorem 2.4 to write 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 ≤ j max r :E n+1, j ⊆E n,r a n,r N n+1, j .
By the very definition of a n+1, j , we have max 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 1/2 n ≤ g n . Indeed, for any x ∈ [0, 1], g n (x) = j a n, j N n, j (x) ≥ min j:E n, j x a n, j ≥ min j:E n, j x X n  (3.9) where the implied constant only depends on k. In order to continue the estimate, we next show the inequality where by J ,s we denote an atom of F with J ,s ⊂ E ,s of maximal length and the implied constant depends only on k. Indeed, we use Theorem 2.3 in the form of (2.3) to get ( f m ∈ S k (F ) for m ≤ ) (3.11) Now observe that for atoms I of F , due to the equivalence of p-norms of polynomials (cf. Corollary 2.2), which means that, inserting this in estimate (3.11), and, since there are at most k indices s so that |E ,s ∩ E ,r | > 0, we have established (3.10). We define K ,r to be an interval J ,s with |E ,r ∩ E ,s | > 0 so that max s:|E ,r ∩E ,s |>0 This means, after combining (3.9) with (3.10), we have We now apply Lemma 3.5 with the function and the choices In order to see Assumption (3.8) of Lemma 3.5, fix the index j and let i be such that (i) ≥ ( j). By definition of D i = K (i),r (i) , the smallest interval containing J n,i and D i contains at most 2k − 1 atoms of F (i) and, if D i ⊂ D j , the smallest interval containing J n,i and D j contains at most 2k − 1 atoms of F ( j) . This means that, in particular, J n,i is a subset of the union V of 4k atoms of F ( 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 1 depending on k and sup u≤ ( which, since |B j | ≥ |D j |/2 by Corollary 2.2, shows that the assumption of Lemma 3.5 holds true and we get a function ϕ so that |ϕ( j)| = c 1 |J n, j |/2, 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 with constants depending only on k and sup u≤n γ k (F u ).
Employing this construction of g n , we now give the following duality estimate for spline projections (for the martingale case, see for instance [8]). The martingale version of this result is the essential estimate in the proof of both Lépingle's inequality (1.1) and the H 1 -BMO duality. Theorem 3.6 Let (F n ) be such that γ := sup n γ k (F n ) < ∞ and ( f n ) n≥1 a sequence of functions with f n ∈ S k (F n ) for each n. Additionally, let h n ∈ L 1 be arbitrary. Then, for any N , where the implied constant is the same constant that appears in (3) of Proposition 3.4 and therefore only depends on k and γ .
Proof Let X n := ≤n f 2 and f ≡ 0 for > N and ≤ 0. By Proposition 3.4, we choose an increasing sequence (g n ) of functions with g 0 = 0, g n ∈ S k (F n ) and the properties X 1/2 n ≤ g n and Eg n EX 1/2 n where the implied constant depends only on k and γ . Then, apply Cauchy-Schwarz inequality by introducing the factor g 1/2 We estimate each of the factors on the right hand side separately and set The first factor is estimated by the pointwise inequality X 1/2 n ≤ g n : We continue with 2 : where the last inequality follows from g n ≥ g n−1 . Noting that by the properties of g n , E N n=1 (g n − g n−1 ) = Eg N EX 1/2 N and combining the two parts 1 and 2 , we obtain the conclusion. from Sect. 3 are known, the proofs of the subsequent results proceed similarly to their martingale counterparts in [8,12] by using spline properties instead of martingale properties.

Lépingle'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 projection operator on S k (F n ). Then, where the implied constant depends only on k, k and sup n γ k (F n ).
We emphasize that the parameters k and k can be different here, k being the spline order of the sequence ( f n ) and k being the spline order of the projection operators P n−1 . In particular, the constant on the right hand side does not depend on the kregularity parameter sup n γ k (F n ).

Proof
We first assume that f n = 0 for n > N and begin by using duality where sup is taken over all (H n ) ∈ L ∞ ( 2 ) with (H n ) L ∞ ( 2 ) = 1. By the selfadjointness of P n−1 , E (P n−1 f n ) · H n = E f n · (P n−1 H n ) .
Then we apply Theorem 3.6 for f n and h n = P n−1 H n to obtain (denoting by P n the orthogonal projection operator onto To estimate the right hand side, we note that for any n, by Corollary 3.3, 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.

H 1 -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 = 0. For f ∈ L 1 , we introduce the notation Observe that for < n and f , g ∈ L 1 , Let V be the L 1 -closure of ∪ n S k (F n ). Then, the uniform boundedness of P n on L 1 implies that P n f → f in L 1 for f ∈ V . Next, set and the corresponding quasinorm 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 1 -BMO-duality (cf. [7,8,10]), BMO k as the dual space of H 1,k .
First, we use the duality estimate Theorem 3.6 and (4.2) to prove, for f ∈ H 1,k and h ∈ BMO k , This estimate also implies that the limit lim n E (P n f ) · (P n h) exists and satisfies lim n E (P n f ) · (P n h) On the other hand, similarly to the martingale case (see [8]), given a continuous linear functional L on H 1,k , we extend L norm-preservingly to a continuous linear functional on L 1 ( 2 ), which, by Sect. 2.5, has the form for some σ ∈ L ∞ ( 2 ). The k-martingale spline sequence h n = ≤n σ is bounded in L 2 and therefore, by the spline convergence theorem ((v) on page 2), has a limit h ∈ L 2 with P n h = h n and which is also contained in BMO k . Indeed, by using Corollary 3.3, we obtain h BMO k σ L ∞ ( 2 ) = = L with a constant depending only on k and sup n γ k (F n ). Moreover, for f ∈ H 1,k , since L is continuous on 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 assume that (F n ) is a sequence of increasing interval σ -algebras with sup n γ k (F n ) < ∞.
(1) By Khintchine's inequality, S f 1 sup ε∈{−1,1} Z n ε n n f 1 . Based on the interval 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 exactly one atom of G n into two atoms of G n+1 . Denoting by P G n the orthogonal projection operator onto S k (G n ) and G j = P G j − P G j−1 , we can write n ε n n f = n ε n a n+1 −1 j=a n G j f for some sequence (a n ). By using inequalities (2.7) and (2.6) and writing (S G f ) 2 = j | G j f | 2 , we obtain for p > 1 This implies L p ⊂ H 1,k for all p > 1 and, by duality, BMO 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 one atom of F n into two atoms of F n+1 and under the condition sup n γ k−1 (F n ) < ∞ (which is stronger than sup n γ k (F n ) < ∞), it is shown in [9] that where H 1 denotes the atomic Hardy space on [0, 1], i.e. in this case, H 1,k coincides with H 1 .