1 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, \(({\mathscr {F}}_n)\) an increasing sequence of \(\sigma \)-algebras of sets in [0, 1] where each \({\mathscr {F}}_n\) is generated by a finite partition of [0, 1] into intervals of positive length. Moreover, define the spline space

$$\begin{aligned} {\mathscr {S}}_k({\mathscr {F}}_n) = \{f\in C^{k-2}[0,1] : f\text { is a polynomial of order }k \text { on each atom of }{\mathscr {F}}_n \} \end{aligned}$$

and let \(P_n^{(k)}\) be the orthogonal projection operator onto \({\mathscr {S}}_k({\mathscr {F}}_n)\) with respect to the \(L_2\) inner product on [0, 1] with the Lebesgue measure \(|\cdot |\). The space \({\mathscr {S}}_1({\mathscr {F}}_n)\) consists of piecewise constant functions and \(P_n^{(1)}\) is the conditional expectation operator with respect to the \(\sigma \)-algebra \({\mathscr {F}}_n\). Similarly to the definition of martingales, we introduce the following notion: let \((f_n)_{n\ge 0}\) be a sequence of integrable functions. We call this sequence a k-martingale spline sequence (adapted to \(({\mathscr {F}}_n)\)) if, for all n,

$$\begin{aligned} P_n^{(k)} f_{n+1} = f_n. \end{aligned}$$

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 (\({\mathscr {F}}_n\)) of the above type, just by replacing conditional expectation operators by the projection operators \(P_n^{(k)}\). Indeed, we have

  1. (i)

    (Shadrin’s theorem) there exists a constant \(C_k\) depending only on k such that

    $$\begin{aligned} \sup _n\Vert P_n^{(k)} : L_1 \rightarrow L_1 \Vert \le C_k, \end{aligned}$$
  2. (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 \(\lambda >0\),

    $$\begin{aligned} |\{ \sup _n |f_n| > \lambda \}| \le C_k \frac{\sup _n\Vert f_n\Vert _{1}}{ \lambda }, \end{aligned}$$
  3. (iii)

    (Doob’s \(L_p\) inequality for splines)

    for all \(p\in (1,\infty ]\) there exists a constant \(C_{p,k}\) depending only on p and k such that for all k-martingale spline sequences \((f_n)\),

    $$\begin{aligned} \big \Vert \sup _n |f_n| \big \Vert _{p} \le C_{p,k} \sup _n\Vert f_n\Vert _{p},\ \end{aligned}$$
  4. (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,

  5. (v)

    (Spline convergence theorem, \(L_p\)-version)

    for \(1<p<\infty \), 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 \(L_1(\ell _2)\)-inequality [12], which reads

$$\begin{aligned} \Big \Vert \big ( \sum _n {\mathbb {E}}[f_n | \mathscr {F}_{n-1}]^2 \big )^{1/2}\Big \Vert _1 \le 2\cdot \Big \Vert \big ( \sum _n f_n^2 \big )^{1/2} \Big \Vert _1, \end{aligned}$$

provided the sequence of (real-valued) random variables \(f_n\) is adapted to the filtration \(({\mathscr {F}}_n)\), i.e. each \(f_n\) is \({\mathscr {F}}_n\)-measurable. Different proofs of (1.1) were given by Bourgain [3, Proposition 5], Delbaen and Schachermayer [4, Lemma 1] and Müller [13, Proposition 4.1]. The spline version of inequality (1.1) is contained in Theorem 4.1.

This inequality is an \(L_1\) extension of the following result for \(1<p<\infty \), proved by Stein [24], that holds for arbitrary integrable functions \(f_n\):

$$\begin{aligned} \Big \Vert \big ( \sum _n {\mathbb {E}}[f_n |{\mathscr {F}}_{n-1}]^2 \big )^{1/2}\Big \Vert _p \le a_p \Big \Vert \big ( \sum _n f_n^2\big )^{1/2} \Big \Vert _p, \end{aligned}$$

for some constant \(a_p\) depending only on p. This can be seen as a dual version of Doob’s inequality \(\Vert \sup _{\ell } |{\mathbb {E}}[f | {\mathscr {F}}_\ell ]| \Vert _p \le c_p \Vert f\Vert _p\) for \(p>1\), see [1]. Once we know Doob’s inequality for spline projections, which is point (iii) above, the same proof as in [1] works for spline projections if we use suitable positive operators \(T_n\) instead of \(P_n^{(k)}\) that also satisfy Doob’s inequality and dominate the operators \(P_n^{(k)}\) pointwise (cf. Sects. 3.1, 3.2).

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 \({\mathbb {E}} [f_n \cdot h_n]\) (for \(f_n\) being \({\mathscr {F}}_n\)-measurable) by Cauchy–Schwarz after introducing the square function \(S_n^2 = \sum _{\ell \le n} f_\ell ^2\):

$$\begin{aligned} ({\mathbb {E}}[f_n\cdot h_n])^2 \le {\mathbb {E}}[ f_n^2/S_n ] \cdot {\mathbb {E}}[S_n h_n^2] \end{aligned}$$

and estimating both factors on the right hand side separately. A key point in estimating the second factor is that \(S_n\) is \(\mathscr {F}_n\)-measurable, and therefore, \({\mathbb {E}}[S_n|{\mathscr {F}}_n]=S_n\). If we want to allow \(f_n\in {\mathscr {S}}_k({\mathscr {F}}_n)\), \(S_n\) will not be contained in \({\mathscr {S}}_k({\mathscr {F}}_n)\) in general. Under certain conditions on the filtration \(({\mathscr {F}}_n)\), we will show in this article how to substitute \(S_n\) in estimate (1.3) by a function \(g_n\in {\mathscr {S}}_k(\mathscr {F}_n)\) 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 \(H^1\)-\({{\,\mathrm{BMO}\,}}\) duality. For its martingale version, we refer to A. M. Garsia’s book [8] on Martingale Inequalities.

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 \(V\subset {\mathbb {R}}\) be a compact interval in \({\mathbb {R}}\) and \(E\subset V\) a measurable subset. Then, for all polynomials p of order k (i.e. degree \(k-1\)) on V,

$$\begin{aligned} \Vert p \Vert _{L_\infty (V)} \le \bigg ( 4 \frac{|V|}{|E|}\bigg )^{k-1} \Vert p \Vert _{L_\infty (E)}. \end{aligned}$$

Applying this theorem with the set \(E = \{x\in V : |p(x)| \le 8^{-k+1}\Vert p\Vert _{L_\infty (V)} \}\) immediately yields the following corollary:

Corollary 2.2

Let p be a polynomial of order k on a compact interval \(V\subset {\mathbb {R}}\). Then

$$\begin{aligned} \big |\big \{ x \in V : |p(x)| \ge 8^{-k+1} \Vert p\Vert _{L_\infty (V)} \big \}\big | \ge |V|/2. \end{aligned}$$

2.2 Properties of spline functions

For an interval \(\sigma \)-algebra \({\mathscr {F}}\) (i.e. \({\mathscr {F}}\) is generated by a finite collection of intervals having positive length), the space \({\mathscr {S}}_k({\mathscr {F}})\) 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 \({\mathscr {F}}\). Moreover, \((N_i)\) is a partition of unity, i.e. for all \(x\in [0,1]\), there exist at most k functions \(N_i\) so that \(N_i(x)\ne 0\) and \(\sum _i N_i(x)=1\). 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, \(\inf E_i \le \inf E_{i+1}\) and \(\sup E_i \le \sup E_{i+1}\).

We write \(A(t)\lesssim B(t)\) to denote the existence of a constant C such that for all t, \(A(t)\le C B(t)\), 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 \(\gtrsim \) and \(\simeq \) 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.

Theorem 2.3

(B-spline stability, local and global) Let \(1\le p\le \infty \) and \(g=\sum _{j} a_j N_j\). Then, for all j,

$$\begin{aligned} |a_j|\lesssim |J_j|^{-1/p}\Vert g\Vert _{L_p(J_j)}, \end{aligned}$$

where \(J_j\) is an atom of \({\mathscr {F}}\) contained in \(E_j\) having maximal length. Additionally,

$$\begin{aligned} \Vert g\Vert _p\simeq \Vert (a_j|E_j|^{1/p})\Vert _{\ell _p}, \end{aligned}$$

where in both (2.1) and (2.2), the implied constants depend only on the spline order k.

Observe that (2.1) implies for \(g\in \mathscr {S}_k({\mathscr {F}})\) and any measurable set \(A\subset [0,1]\)

$$\begin{aligned} \Vert g\Vert _{L_\infty (A)} \lesssim \max _{j : |E_j\cap A|>0} \Vert g\Vert _{L_\infty (J_j)}. \end{aligned}$$

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 \({\mathscr {G}}\subset {\mathscr {F}}\) be two interval \(\sigma \)-algebras and denote by \((N_{{\mathscr {G}},i})_i\) the B-spline basis of the coarser space \({\mathscr {S}}_k({\mathscr {G}})\) and by \((N_{\mathscr {F},i})_i\) the B-spline basis of the finer space \(\mathscr {S}_k({\mathscr {F}})\). Then, given \(f=\sum _{j} a_j N_{{\mathscr {G}},j}\), we can expand f in the basis \((N_{{\mathscr {F}},i})_i\)

$$\begin{aligned} \sum _j a_j N_{{\mathscr {G}},j} = \sum _i b_i N_{{\mathscr {F}},i}, \end{aligned}$$

where for each i, \(b_i\) is a convex combination of the coefficients \(a_j\) with \({\text {supp}}N_{{\mathscr {G}},j} \supseteq {\text {supp}}N_{{\mathscr {F}},i}\).

For those results and more information on spline functions, in particular B-splines, we refer to [21] or [5].

2.3 Spline orthoprojectors

We now use the B-spline basis of \({\mathscr {S}}_k({\mathscr {F}})\) and expand the orthogonal projection operator P onto \(\mathscr {S}_k({\mathscr {F}})\) in the form

$$\begin{aligned} Pf = \sum _{i,j} a_{ij} \Big (\int _0^1 f(x) N_i(x)\,\mathrm {d}x\Big )\cdot N_j \end{aligned}$$

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 estimate for \(a_{ij}\) [19]: there exist constants C and \(0<q<1\) depending only on k so that for each interval \(\sigma \)-algebra \({\mathscr {F}}\) and each ij,

$$\begin{aligned} |a_{ij}| \le C \frac{q^{|i-j|}}{|E_{ij}|}. \end{aligned}$$

2.4 Spline square functions

Let \(({\mathscr {F}}_n)\) be a sequence of increasing interval \(\sigma \)-algebras in [0, 1] and we assume that each \(\mathscr {F}_{n+1}\) is generated from \({\mathscr {F}}_n\) by the subdivision of exactly one atom of \({\mathscr {F}}_n\) into two atoms of \(\mathscr {F}_{n+1}\). Let \(P_n\) be the orthogonal projection operator onto \({\mathscr {S}}_k({\mathscr {F}}_n)\). We denote \(\Delta _n f = P_n f - P_{n-1} f\) and define the spline square function

$$\begin{aligned} Sf = \Big (\sum _n |\Delta _n f|^2\Big )^{1/2}. \end{aligned}$$

We have Burkholder’s inequality for the spline square function, i.e. for all \(1<p<\infty \) [16], the \(L_p\)-norm of the square function Sf is comparable to the \(L_p\)-norm of f:

$$\begin{aligned} \Vert Sf\Vert _p \simeq \Vert f\Vert _p,\qquad f\in L_p \end{aligned}$$

with constants depending only on p and k. Moreover, for \(p=1\), it is shown in [9] that

$$\begin{aligned} \Vert Sf\Vert _1 \simeq \sup _{\varepsilon \in \{-1,1\}^{{\mathbb {Z}}}} \Vert \sum _n \varepsilon _n \Delta _n f \Vert _1, \qquad Sf\in L_1, \end{aligned}$$

with constants depending only on k and where the proof of the \(\lesssim \)-part only uses Khintchine’s inequality whereas the proof of the \(\gtrsim \)-part uses fine properties of the functions \(\Delta _n f\).

2.5 \(L_p(\ell _q)\)-spaces

For \(1\le p,q\le \infty \), we denote by \(L_p(\ell _q)\) the space of sequences of measurable functions \((f_n)\) on [0, 1] so that the norm

$$\begin{aligned} \Vert (f_n)\Vert _{L_p(\ell _q)} = \Big (\int _0^1 \Big ( \sum _{n} |f_n(t)|^q \Big )^{p/q}\,\mathrm {d}t\Big )^{1/p} \end{aligned}$$

is finite (with the obvious modifications if \(p=\infty \) or \(q=\infty \)). For \(1\le p,q <\infty \), the dual space (see [2]) of \(L_p(\ell _q)\) is \(L_{p'}(\ell _{q'})\) with \(p'=p/(p-1)\), \(q'=q/(q-1)\) and the duality pairing

$$\begin{aligned} \langle (f_n), (g_n)\rangle = \int _0^1 \sum _n f_n(t) g_n(t) \,\mathrm {d}t. \end{aligned}$$

Hölder’s inequality takes the form \(|\langle (f_n), (g_n)\rangle | \le \Vert (f_n)\Vert _{L_p(\ell _q)} \Vert (g_n)\Vert _{L_{p'}(\ell _{q'})}\).

3 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 \(P_n= P_n^{(k)}\) 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 \(\mathbb E_{{\mathscr {G}}} {\mathbb {E}}_{{\mathscr {F}}} f = {\mathbb {E}}_{{\mathscr {G}}} f\) for \({\mathscr {G}}\subset {\mathscr {F}}\). 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 \(H_1\)-\({{\,\mathrm{BMO}\,}}\) duality in Sect. 4.

3.1 The positive operators T

As above, let \({\mathscr {F}}\) be an interval \(\sigma \)-algebra on [0, 1], \((N_i)\) the B-spline basis of \({\mathscr {S}}_k({\mathscr {F}})\), \(E_i\) the support of \(N_i\) and \(E_{ij}\) 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 \(T = T_{{\mathscr {F}}, q, k}\) by

$$\begin{aligned} Tf(x) := \sum _{i,j} \frac{q^{|i-j|}}{|E_{ij}|} \langle f, \mathbb {1}_{E_i}\rangle \mathbb {1}_{E_j}(x) = \int _0^1 K(x,t) f(t)\,\mathrm {d}t, \end{aligned}$$

where the kernel \(K=K_{T}\) is given by

$$\begin{aligned} K(x,t) = \sum _{i,j} \frac{q^{|i-j|}}{|E_{ij}|}\mathbb {1}_{E_i}(t)\cdot \mathbb {1}_{E_j}(x). \end{aligned}$$

We observe that the operator T is selfadjoint (w.r.t the standard inner product on \(L_2\)) and

$$\begin{aligned} k\le K_x := \int _0^1 K(x,t) \,\mathrm {d}t\le \frac{2(k+1)}{1-q},\qquad x\in [0,1], \end{aligned}$$

which, in particular, implies the boundedness of the operator T on \(L_1\) and \(L_\infty \):

$$\begin{aligned} \Vert Tf\Vert _1 \le \frac{2(k+1)}{1-q} \Vert f\Vert _1,\qquad \Vert Tf\Vert _\infty \le \frac{2(k+1)}{1-q}\Vert f\Vert _\infty . \end{aligned}$$

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 in the form

$$\begin{aligned} \varphi (Tf(x)) \le K_x^{-1} T\big (\varphi ( K_x \cdot f)\big )(x),\qquad f\in L_1, x\in [0,1], \end{aligned}$$

for convex functions \(\varphi \). This is seen by applying the classical Jensen inequality to the probability measure \(K(t,x)\,\mathrm {d}t/K_x\).

Let \({\mathscr {Mf}}f \) denote the Hardy–Littlewood maximal function of \(f\in L_1\), i.e.

$$\begin{aligned} {\mathscr {M}} f(x) = \sup _{I\ni x} \frac{1}{|I|}\int _I |f(y)|\,\mathrm {d}y, \end{aligned}$$

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.

$$\begin{aligned} |\{ {\mathscr {M}} f> \lambda \}| \le C \lambda ^{-1} \Vert f\Vert _1, \qquad f\in L_1, \lambda >0 \end{aligned}$$

for some constant C. Since trivially we have the estimate \(\Vert {\mathscr {M}}f\Vert _\infty \le \Vert f\Vert _\infty \), by Marcinkiewicz interpolation, for any \(p>1\), there exists a constant \(C_p\) depending only on p so that

$$\begin{aligned} \Vert {\mathscr {M}f}\Vert _p \le C_p \Vert f\Vert _p. \end{aligned}$$

For those assertions about \({\mathscr {M}}\), we refer to (for instance) [23].

The significance of T and \({\mathscr {M}}\) at this point is that we can use formula (2.4) and estimate (2.5) to obtain the pointwise bound

$$\begin{aligned} |Pf(x)| \le C_1 (T|f|)(x) \le C_2 {\mathscr {Mf}}(x),\qquad f\in L_1, x\in [0,1], \end{aligned}$$

where \(T=T_{{\mathscr {F}},q,k}\) 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 \({\mathscr {M}}\) pointwise and independently of \({\mathscr {F}}\).

3.2 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 \((\mathscr {F}_n)\) be an interval filtration on [0, 1] and \(P_n\) be the orthogonal projection operator onto the space \({\mathscr {S}}_k(\mathscr {F}_n)\) of splines of order k corresponding to \({\mathscr {F}}_n\). Working with the positive operators \(T_{{\mathscr {F}}_n, q, k}\) instead of the non-positive 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 [0, 1]. Then, for \(1\le r\le p<\infty \) or \(1<p\le r\le \infty \),

$$\begin{aligned} \Vert (P_n f_n) \Vert _{L_p(\ell _r)} \lesssim \Vert (f_n) \Vert _{L_p(\ell _r)} \end{aligned}$$

where the implied constant depends only on pr and k.


By (3.3), it suffices to prove this inequality for the operators \(T_n=T_{{\mathscr {F}}_n,q,k}\) 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 \({\mathscr {M}}\) for \(1<p'\le \infty \) imply that

$$\begin{aligned} \big \Vert \sup _{1\le n\le N} |T_n f| \big \Vert _{p'} \le C_{p',k} \Vert f \Vert _{p'}, \qquad f\in L_{p'} \end{aligned}$$

with a constant \(C_{p',k}\) depending on \(p'\) and k. Let \(1\le p<\infty \) and \(U_N : L_{p}(\ell _1^N) \rightarrow L_{p}\) be given by \((g_1,\ldots ,g_N)\mapsto \sum _{j=1}^N T_j g_j\). Inequality (3.5) implies the boundedness of the adjoint \(U_N^* : L_{p'}\rightarrow L_{p'}(\ell _\infty ^N)\), \(f\mapsto (T_j f)_{j=1}^N\) for \(p'=p/(p-1)\) by a constant independent of N and therefore also the boundedness of \(U_N\). Since \(|T_j f| \le T_j|f|\) by the positivity of \(T_j\), letting \(N\rightarrow \infty \) implies (3.4) for \(T_n\) instead of \(P_n\) in the case \(r=1\) and outer parameter \(1\le p < \infty \).

If \(1<r\le p\), we use Jensen’s inequality (3.2) and estimate (3.1) to obtain

$$\begin{aligned} \sum _{j=1}^N |T_j g_j|^r \lesssim \sum _{j=1}^N T_j(|g_j|^r) \end{aligned}$$

and apply the result for \(r=1\) and the outer parameter p / r to get the result for \(1\le r\le p<\infty \). The cases \(1<p\le r\le \infty \) now just follow from this result using duality and the self-adjointness of \(T_j\). \(\square \)

3.3 Tower property of T

Next, we will prove a substitute of the tower property \(\mathbb E_{{\mathscr {G}}} {\mathbb {E}}_{{\mathscr {F}}} f={\mathbb {E}}_{{\mathscr {G}}}f\)\(({\mathscr {G}}\subset {\mathscr {F}})\) for conditional expectations that applies to the operators T.

To formulate this result, we need a suitable notion of regularity for \(\sigma \)-algebras which we now describe. Let \({\mathscr {F}}\) be an interval \(\sigma \)-algebra, let \((N_j)\) be the B-spline basis of \({\mathscr {S}}_k({\mathscr {F}})\) and denote by \(E_{j}\) the support of the function \(N_j\). The k-regularity parameter\(\gamma _k({\mathscr {F}})\) is defined as

$$\begin{aligned} \gamma _k({\mathscr {F}}) := \max _i \max ( |E_i| / |E_{i+1}|, |E_{i+1}| / |E_i| ), \end{aligned}$$

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 \(\sigma \)-algebra \({\mathscr {F}}\).

Proposition 3.2

(Tower property of T) Let \({\mathscr {G}}\subset {\mathscr {F}}\) be two interval \(\sigma \)-algebras on [0, 1]. Let \(S = T_{{\mathscr {G}},\sigma ,k}\) and \(T=T_{\mathscr {F},\tau ,k'}\) for some \(\sigma ,\tau \in (0,1)\) and some positive integers \(k,k'\). Then, for all \(q>\max (\tau ,\sigma )\), there exists a constant C depending on \(q,k,k'\) so that

$$\begin{aligned} |ST f(x)| \le C \cdot \gamma ^k \cdot (T_{\mathscr {G},q,k} |f|)(x),\qquad f\in L_1, x\in [0,1], \end{aligned}$$

where \(\gamma = \gamma _{k}({\mathscr {G}})\) denotes the k-regularity parameter of \({\mathscr {G}}\).


Let \((F_i)\) be the collection of B-spline supports in \(\mathscr {S}_{k'}({\mathscr {F}})\) and \((G_i)\) the collection of B-spline supports in \({\mathscr {S}}_{k}({\mathscr {G}})\). 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)

$$\begin{aligned} \int _0^1 K_{S}(x,t) K_{T}(t,s)\,\mathrm {d}t \le C \gamma ^k \sum _{i,j} \frac{q^{|i-j|}}{|G_{ij}|} \mathbb {1}_{G_i}(x) \mathbb {1}_{G_j}(s),\qquad x,s\in [0,1]\nonumber \\ \end{aligned}$$

for all \(q>\max (\tau ,\sigma )\) and some constant C depending on \(q,k,k'\). In order to prove this inequality, we first fix \(x,s\in [0,1]\) and choose i such that \(x\in G_i\) and \(\ell \) such that \(s\in F_\ell \). Moreover, based on \(\ell \), we choose j so that \(s\in G_j\) and \(G_j \supset F_\ell \). There are at most \(\max (k,k')\) choices for each of the indices \(i,\ell ,j\) and without restriction, we treat those choices separately, i.e. we only have to estimate the expression

$$\begin{aligned} \sum _{m,r} \frac{\sigma ^{|m-i|} \tau ^{|r-\ell |} |G_m \cap F_r|}{|G_{im}| |F_{\ell r}|}. \end{aligned}$$

Since, for each r, there are also at most \(k+k'-1\) indices m so that \(|G_m\cap F_r| >0\) (recall that \({\mathscr {G}}\subset \mathscr {F}\)), we choose one such index \(m=m(r)\) and estimate

$$\begin{aligned} \Sigma = \sum _r \frac{\sigma ^{|m(r)-i|} \tau ^{|r-\ell |} |G_{m(r)} \cap F_r|}{|G_{i,m(r)}| |F_{\ell r}|}. \end{aligned}$$

Now, observe that for any parameter choice of r in the above sum,

$$\begin{aligned} G_{i,m(r)} \cup F_{\ell r} \supseteq (G_{ij}{\setminus } G_j) \cup G_i \end{aligned}$$

and therefore, since also \(G_{m(r)}\cap F_r \subset G_{i,m(r)} \cap F_{\ell r}\),

$$\begin{aligned} \Sigma \le \frac{2}{|(G_{ij}{\setminus } G_j) \cup G_i|}\sum _r \sigma ^{|m(r) - i|}\tau ^{|r-\ell |}, \end{aligned}$$

which, using the k-regularity parameter \(\gamma = \gamma _{k}({\mathscr {G}})\) of the \(\sigma \)-algebra \({\mathscr {G}}\) and denoting \(\lambda = \max (\tau ,\sigma )\), we estimate by

$$\begin{aligned} \Sigma&\le \frac{2\gamma ^k}{|G_{ij}|} \sum _m \lambda ^{|m-i|}\sum _{r: m(r)=m} \lambda ^{|r-\ell |} \lesssim \frac{\gamma ^k}{|G_{ij}|}\sum _m \lambda ^{|i-m| + |m-j|} \\&\lesssim \frac{\gamma ^k}{|G_{ij}|} \big (|i-j|+1\big )\lambda ^{|i-j|}, \end{aligned}$$

where the implied constants depend on \(\lambda ,k,k'\) and the estimate \(\sum _{r: m(r)=m} \lambda ^{|r-\ell |} \lesssim \lambda ^{|m-j|}\) used the fact that, essentially, there are more atoms of \({\mathscr {F}}\) between \(F_r\) and \(F_\ell \) (for r as in the sum) than atoms of \({\mathscr {G}}\) between \(G_m\) and \(G_j\). Finally, we see that for any \(q>\lambda \),

$$\begin{aligned} \Sigma \lesssim C\gamma ^k\frac{q^{|i-j|}}{|G_{ij}|} \end{aligned}$$

for some constant C depending on \(q,k,k'\), and, as \(x\in G_i\) and \(s\in G_j\), this shows inequality (3.7). \(\square \)

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 \({\mathscr {S}}_{k}({\mathscr {F}}_n)\) and by \(P_n'\) the orthogonal projection onto \({\mathscr {S}}_{k'}({\mathscr {F}}_n)\) for some positive integers \(k,k'\). Moreover, let \(T_n\) be the operator \(T_{{\mathscr {F}}_n, q, k}\) from (3.3) dominating \(P_n\) pointwise.

Then, for any integer n and for any \(1\le p\le \infty \),

$$\begin{aligned} \Big \Vert \sum _{\ell \ge n} P_n \big ((P_{\ell -1}' f_\ell )^2\big ) \Big \Vert _p \lesssim \Big \Vert \sum _{\ell \ge n} T_n \big ((P_{\ell -1}' f_\ell )^2\big ) \Big \Vert _p \lesssim \gamma _{k}({\mathscr {F}}_n)^k\cdot \Big \Vert \sum _{\ell \ge n} f_\ell ^2\Big \Vert _p, \end{aligned}$$

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 \({\mathbb {E}}(\cdot | \mathscr {F}_n)\) instead of the operators \(P_n, T_n, P_{\ell -1}'\) even with an absolute constant on the right hand side.


We denote by \(T_n\) the operator \(T_{{\mathscr {F}}_n, q, k}\) and by \(T_n'\) the operator \(T_{{\mathscr {F}}_n, q', k'}\), where the parameters \(q,q'<1\) are given by inequality (3.3) depending on k and \(k'\) respectively. Setting \(U_n := T_{{\mathscr {F}}_n, \max (q,q')^{1/2}, k}\), we perform the following chain of inequalities, where we use the positivity of \(T_n\) and (3.3), Jensen’s inequality for \(T_{\ell -1}'\), the tower property for \(T_n T_{\ell -1}'\) and the \(L_p\)-boundedness of \(U_n\), respectively:

$$\begin{aligned} \Big \Vert \sum _{\ell \ge n} T_n\big ( (P_{\ell -1}' f_\ell )^2\big )\Big \Vert _p&\lesssim \Big \Vert \sum _{\ell \ge n} T_n \big ( (T_{\ell -1}' |f_\ell |)^2\big )\Big \Vert _p \\&\lesssim \Big \Vert \sum _{\ell \ge n} T_n \big (T_{\ell -1}' f_\ell ^2\big )\Big \Vert _p \\&\le \Vert T_n(T_{n-1}' f_n^2) \Vert _p + \Big \Vert \sum _{\ell> n} T_n \big (T_{\ell -1}' f_\ell ^2\big )\Big \Vert _p \\&\lesssim \Vert f_n^2 \Vert _p + \gamma _{k}({\mathscr {F}}_n)^k \cdot \Big \Vert \sum _{\ell > n} U_n(f_\ell ^2)\Big \Vert _p \\&\lesssim \gamma _{k}({\mathscr {F}}_n)^k \cdot \Big \Vert \sum _{\ell \ge n} f_\ell ^2 \Big \Vert _p, \end{aligned}$$

where the implied constants only depend on k and \(k'\). \(\square \)

3.4 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 \(g_n\in {\mathscr {S}}_k({\mathscr {F}}_n)\) based on a spline square function.

Proposition 3.4

Let \((f_n)\) be a sequence of functions with \(f_n\in \mathscr {S}_k({\mathscr {F}}_n)\) for all n and set

$$\begin{aligned} X_n:=\sum _{\ell \le n} f_\ell ^2. \end{aligned}$$

Then, there exists a sequence of non-negative functions \(g_n\in {\mathscr {S}}_{k}({\mathscr {F}}_n)\) so that for each n,

  1. (1)

    \(g_n \le g_{n+1}\),

  2. (2)

    \(X_n^{1/2} \le g_n\)

  3. (3)

    \({\mathbb {E}} g_n \lesssim {\mathbb {E}} X_n^{1/2}\), where the implied constant depends on k and on \(\sup _{m\le n}\gamma _{k}({\mathscr {F}}_m)\).

For the proof of this result, we need the following simple lemma.

Lemma 3.5

Let \(c_1\) be a positive constant and let \((A_j)_{j=1}^N\) be a sequence of atoms in \({\mathscr {F}}_n\). Moreover, let \(\ell : \{1,\ldots , N\} \rightarrow \{1,\ldots , n\}\) and, for each \(j\in \{1,\ldots , N\}\), let \(B_j\) be a subset of an atom \(D_j\) of \(\mathscr {F}_{\ell (j)}\) with

$$\begin{aligned} |B_j| \ge c_1 \sum _{\begin{array}{c} i:\ell (i)\ge \ell (j), \\ D_i\subseteq D_j \end{array}} |A_i |. \end{aligned}$$

Then, there exists a map \(\varphi \) on \(\{1,\ldots , N\}\) so that

  1. (1)

    \(|\varphi (j)| = c_1 |A_j|\) for all j,

  2. (2)

    \(\varphi (j) \subseteq B_j\) for all j,

  3. (3)

    \(\varphi (i) \cap \varphi (j) = \emptyset \) for all \(i\ne j\).


Without restriction, we assume that the sequence \((A_j)\) is enumerated such that \(\ell (j+1) \le \ell (j)\) for all \(1\le j\le N-1\). We first choose \(\varphi (1)\) 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 \(1\le j\le j_0\), we have constructed \(\varphi (j)\) with the properties

  1. (1)

    \(|\varphi (j)| = c_1|A_j|\),

  2. (2)

    \(\varphi (j)\subseteq B_j\),

  3. (3)

    \(\varphi (j) \cap \cup _{i<j} \varphi (i) = \emptyset \).

Based on that, we now construct \(\varphi (j_0 + 1)\). Define the index sets \(\Gamma = \{ i : \ell (i) \ge \ell (j_0 +1), D_i \subseteq D_{j_0+1} \}\) and \(\Lambda = \{ i : i\le j_0+1, D_i\subseteq D_{j_0+1} \}\). Since we assumed that \(\ell \) is decreasing, we have \(\Lambda \subseteq \Gamma \) and by the nestedness of the \(\sigma \)-algebras \({\mathscr {F}}_n\), we have for \(i\le j_0+1\) that either \(D_i\subset D_{j_0+1}\) or \(|D_i \cap D_{j_0+1}|=0\). This implies

$$\begin{aligned} \Big | B_{j_0+1}{\setminus } \bigcup _{i\le j_0} \varphi (i) \Big |&= |B_{j_0 + 1}| - \Big | B_{j_0+1}\cap \bigcup _{i\le j_0} \varphi (i) \Big | \\&\ge c_1 \sum _{i\in \Gamma } |A_i| - \Big | D_{j_0+1} \cap \bigcup _{i\le j_0} \varphi (i) \Big | \\&\ge c_1 \sum _{i\in \Lambda } |A_i| - \Big | \bigcup _{i\in \Lambda {\setminus }\{j_0+1\}} \varphi (i) \Big | \\&\ge c_1\sum _{i\in \Lambda } |A_i| - \sum _{i\in \Lambda {\setminus } \{j_0+1\}} c_1 |A_i| = c_1 |A_{j_0+1}|. \end{aligned}$$

Therefore, we can choose \(\varphi (j_0 +1) \subseteq B_{j_0+1}\) that is disjoint to \(\varphi (i)\) for any \(i\le j_0\) and \(|\varphi (j_0+1)| = c_1 |A_{j_0+1}|\) which completes the proof. \(\square \)

Proof of Proposition 3.4

Fix n and let \((N_{n,j})\) be the B-spline basis of \(\mathscr {S}_{k}({\mathscr {F}}_n)\). Moreover, for any j, set \(E_{n,j} = {\text {supp}}N_{n,j}\) and \(a_{n,j} := \max _{\ell \le n} \max _{r : E_{\ell ,r} \supset E_{n,j}}\Vert X_{\ell } \Vert _{L_\infty (E_{\ell ,r})}^{1/2}\) and we define \(\ell (j)\le n\) and r(j) so that \(E_{\ell (j),r(j)} \supseteq E_{n,j} \) and \(a_{n,j} = \Vert X_{\ell (j)} \Vert _{L_\infty (E_{\ell (j),r(j)})}^{1/2}\). Set

$$\begin{aligned} g_n := \sum _j a_{n,j} N_{n,j} \in {\mathscr {S}}_{k}({\mathscr {F}}_n) \end{aligned}$$

and it will be proved subsequently that this \(g_n\) has the desired properties.

Property (1): In order to show \(g_n\le g_{n+1}\), we use Theorem 2.4 to write

$$\begin{aligned} g_n = \sum _j a_{n,j} N_{n,j} = \sum _j \beta _{n,j} N_{n+1,j}, \end{aligned}$$

where \(\beta _{n,j}\) is a convex combination of those \(a_{n,r}\) with \(E_{n+1,j} \subseteq E_{n,r}\), and thus

$$\begin{aligned} g_n \le \sum _j \big (\max _{r: E_{n+1,j}\subseteq E_{n,r}} a_{n,r}\big ) N_{n+1,j}. \end{aligned}$$

By the very definition of \(a_{n+1,j}\), we have

$$\begin{aligned} \max _{r: E_{n+1,j} \subseteq E_{n,r}} a_{n,r} \le a_{n+1,j}, \end{aligned}$$

and therefore, \(g_n\le g_{n+1}\) pointwise, since the B-splines \((N_{n+1,j})_j\) are nonnegative functions.

Property (2): Now we show that \(X_n^{1/2}\le g_n\). Indeed, for any \(x\in [0,1]\),

$$\begin{aligned} g_n(x) = \sum _j a_{n,j} N_{n,j}(x) \ge \min _{j : E_{n,j}\ni x} a_{n,j} \ge \min _{j: E_{n,j}\ni x} \Vert X_n\Vert _{L_\infty (E_{n,j})}^{1/2}\ge X_n(x)^{1/2}, \end{aligned}$$

since the collection of B-splines \((N_{n,j})_j\) forms a partition of unity.

Property (3): Finally, we show \({\mathbb {E}} g_n \lesssim {\mathbb {E}} X_n^{1/2}\), where the implied constant depends only on k and on \(\sup _{m\le n} \gamma _k({\mathscr {F}}_m)\). By B-spline stability (Theorem 2.3), we estimate the integral of \(g_n\) by

$$\begin{aligned} {\mathbb {E}} g_n \lesssim \sum _j |E_{n,j}| \cdot \Vert X_{\ell (j)}\Vert _{L_\infty (E_{\ell (j),r(j)})}^{1/2}, \end{aligned}$$

where the implied constant only depends on k. In order to continue the estimate, we next show the inequality

$$\begin{aligned} \Vert X_{\ell } \Vert _{L_\infty (E_{\ell ,r})} \lesssim \max _{s : |E_{\ell ,r} \cap E_{\ell ,s}| >0} \Vert X_{\ell }\Vert _{L_\infty (J_{\ell ,s})}, \end{aligned}$$

where by \(J_{\ell ,s}\) we denote an atom of \({\mathscr {F}}_\ell \) with \(J_{\ell ,s} \subset E_{\ell ,s}\) 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 (\(f_m\in {\mathscr {S}}_k({\mathscr {F}}_\ell )\) for \(m\le \ell \))

$$\begin{aligned} \Vert X_\ell \Vert _{L_\infty (E_{\ell ,r})}\le & {} \sum _{m\le \ell } \Vert f_m\Vert _{L_\infty (E_{\ell ,r})}^2 \nonumber \\\lesssim & {} \sum _{m\le \ell } \sum _{s : |E_{\ell ,s} \cap E_{\ell ,r}|>0} \Vert f_m\Vert _{L_\infty (J_{\ell ,s})}^2 = \sum _{s : |E_{\ell ,s}\cap E_{\ell ,r}|>0} \sum _{m\le \ell } \Vert f_m \Vert _{L_\infty (J_{\ell ,s})}^2.\nonumber \\ \end{aligned}$$

Now observe that for atoms I of \({\mathscr {F}}_\ell \), due to the equivalence of p-norms of polynomials (cf. Corollary 2.2),

$$\begin{aligned} \sum _{m\le \ell } \Vert f_m\Vert _{L_\infty (I)}^2 \lesssim \sum _{m\le \ell }\frac{1}{|I|} \int _I f_m^2 = \frac{1}{|I|} \int _I X_\ell \le \Vert X_\ell \Vert _{L_\infty (I)}, \end{aligned}$$

which means that, inserting this in estimate (3.11),

$$\begin{aligned} \Vert X_\ell \Vert _{L_\infty (E_{\ell ,r})} \lesssim \sum _{s : |E_{\ell ,s}\cap E_{\ell ,r}|>0} \Vert X_\ell \Vert _{L_\infty (J_{\ell ,s})}, \end{aligned}$$

and, since there are at most k indices s so that \(|E_{\ell ,s} \cap E_{\ell ,r}|>0\), we have established (3.10).

We define \(K_{\ell ,r}\) to be an interval \(J_{\ell ,s}\) with \(|E_{\ell ,r}\cap E_{\ell ,s}|>0\) so that

$$\begin{aligned} \max _{s : |E_{\ell ,r} \cap E_{\ell ,s}| >0} \Vert X_{\ell }\Vert _{L_\infty (J_{\ell ,s})} = \Vert X_{\ell }\Vert _{L_\infty (K_{\ell ,r})}. \end{aligned}$$

This means, after combining (3.9) with (3.10), we have

$$\begin{aligned} {\mathbb {E}} g_n \lesssim \sum _j|J_{n,j}|\cdot \Vert X_{\ell (j)}\Vert _{L_\infty (K_{\ell (j),r(j)})}^{1/2}. \end{aligned}$$

We now apply Lemma 3.5 with the function \(\ell \) and the choices

$$\begin{aligned} A_j&= J_{n,j}, \qquad D_j = K_{\ell (j),r(j)}, \\ B_j&= \Big \{ t\in D_j : X_{\ell (j)}(t) \ge 8^{-2(k-1)} \Vert X_{\ell (j)} \Vert _{L_\infty (D_j)} \Big \}. \end{aligned}$$

In order to see Assumption (3.8) of Lemma 3.5, fix the index j and let i be such that \(\ell (i)\ge \ell (j)\). By definition of \(D_i = K_{\ell (i), r(i)}\), the smallest interval containing \(J_{n,i}\) and \(D_i\) contains at most \(2k-1\) atoms of \({\mathscr {F}}_{\ell (i)}\) and, if \(D_{i}\subset D_j\), the smallest interval containing \(J_{n,i}\) and \(D_j\) contains at most \(2k-1\) atoms of \({\mathscr {F}}_{\ell (j)}\). This means that, in particular, \(J_{n,i}\) is a subset of the union V of 4k atoms of \({\mathscr {F}}_{\ell (j)}\) with \(D_j\subset V\). Since each atom of \({\mathscr {F}}_n\) occurs at most k times in the sequence \((A_j)\), there exists a constant \(c_1\) depending on k and \(\sup _{u\le \ell (j)} \gamma _k({\mathscr {F}}_u)\le \sup _{u\le n} \gamma _k(\mathscr {F}_u)\) so that

$$\begin{aligned} |D_j| \ge c_1 \sum _{\begin{array}{c} i:\ell (i)\ge \ell (j) \\ D_i\subset D_j \end{array}} |A_i|, \end{aligned}$$

which, since \(|B_j|\ge |D_j|/2\) by Corollary 2.2, shows that the assumption of Lemma 3.5 holds true and we get a function \(\varphi \) so that \(|\varphi (j)| = c_1|J_{n,j}|/2\), \(\varphi (j) \subset B_j\), \(\varphi (i)\cap \varphi (j)=\emptyset \) for all ij. Using these properties of \(\varphi \), we continue the estimate in (3.12) and write

$$\begin{aligned} {\mathbb {E}} g_n&\lesssim \sum _j |J_{n,j}| \cdot \Vert X_{\ell (j)} \Vert _{L_\infty ( D_j)}^{1/2} \le 8^{k-1}\cdot \sum _{j} \frac{|J_{n,j}|}{|\varphi (j)|} \int _{\varphi (j)} X_{\ell (j)}^{1/2} \\&=\frac{2}{c_1} \cdot 8^{k-1} \cdot \sum _j \int _{\varphi (j)} X_{\ell (j)}^{1/2} \\&\lesssim \sum _j \int _{\varphi (j)} X_n^{1/2} \le {\mathbb {E}} X_n^{1/2}, \end{aligned}$$

with constants depending only on k and \(\sup _{u\le n}\gamma _k({\mathscr {F}}_u)\). \(\square \)

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 \(H^1\)-\({{\,\mathrm{BMO}\,}}\) duality.

Theorem 3.6

Let \(({\mathscr {F}}_n)\) be such that \(\gamma :=\sup _{n} \gamma _k({\mathscr {F}}_n) < \infty \) and \((f_n)_{n\ge 1}\) a sequence of functions with \(f_n\in {\mathscr {S}}_k({\mathscr {F}}_n)\) for each n. Additionally, let \(h_n\in L_1\) be arbitrary. Then, for any N,

$$\begin{aligned} \sum _{n\le N} {\mathbb {E}}[|f_n \cdot h_n|] \lesssim \sqrt{2}\cdot {\mathbb {E}}\Big [ \Big (\sum _{\ell \le N} f_\ell ^2\Big )^{1/2}\Big ] \cdot \sup _{n\le N} \Vert P_n\big ( \sum _{\ell =n}^N h_\ell ^2 \big )\Vert _\infty ^{1/2}, \end{aligned}$$

where the implied constant is the same constant that appears in (3) of Proposition 3.4 and therefore only depends on k and \(\gamma \).


Let \(X_n:= \sum _{\ell \le n} f_\ell ^2\) and \(f_\ell \equiv 0\) for \(\ell > N\) and \(\ell \le 0\). By Proposition 3.4, we choose an increasing sequence \((g_n)\) of functions with \(g_0=0\), \(g_n\in {\mathscr {S}}_{k}({\mathscr {F}}_n)\) and the properties \(X_n^{1/2}\le g_n\) and \({\mathbb {E}} g_n \lesssim {\mathbb {E}} X_n^{1/2}\) where the implied constant depends only on k and \(\gamma \). Then, apply Cauchy–Schwarz inequality by introducing the factor \(g_n^{1/2}\) to get

$$\begin{aligned} \sum _n{\mathbb {E}} [ |f_n \cdot h_n|]= & {} \sum _n {\mathbb {E}} \left[ \left| \frac{f_n}{g_n^{1/2}}\cdot g_n^{1/2} h_n\right| \right] \\\le & {} \left[ \sum _n {\mathbb {E}} [f_n^2/g_n] \right] ^{1/2} \cdot \left[ \sum _n {\mathbb {E}}[g_n h_n^2] \right] ^{1/2}. \end{aligned}$$

We estimate each of the factors on the right hand side separately and set

$$\begin{aligned} \Sigma _1 :=\sum _n {\mathbb {E}} [f_n^2/g_n], \qquad \Sigma _2:=\sum _n{\mathbb {E}}[g_n h_n^2]. \end{aligned}$$

The first factor is estimated by the pointwise inequality \(X_n^{1/2}\le g_n\):

$$\begin{aligned} \Sigma _1 = {\mathbb {E}} \left[ \sum _n \frac{f_n^2}{g_n}\right]&\le {\mathbb {E}} \left[ \sum _n \frac{f_n^2}{X_n^{1/2}}\right] \\&= {\mathbb {E}} \left[ \sum _n \frac{X_n - X_{n-1}}{X_n^{1/2}} \right] \le 2{\mathbb {E}} \sum _n (X_n^{1/2} - X_{n-1}^{1/2}) = 2 {\mathbb {E}} X_N^{1/2}. \end{aligned}$$

We continue with \(\Sigma _2\):

$$\begin{aligned} \Sigma _2&= {\mathbb {E}} \left[ \sum _{\ell =1}^N g_\ell h_\ell ^2 \right] = {\mathbb {E}} \left[ \sum _{\ell =1}^N \sum _{n=1}^\ell (g_n - g_{n-1}) h_\ell ^2 \right] \\&= {\mathbb {E}} \left[ \sum _{n=1}^N (g_n - g_{n-1}) \cdot \sum _{\ell =n}^N h_\ell ^2 \right] \\&= {\mathbb {E}} \left[ \sum _{n=1}^N P_n(g_n - g_{n-1}) \cdot \sum _{\ell =n}^N h_\ell ^2 \right] \\&={\mathbb {E}} \left[ \sum _{n=1}^N (g_n - g_{n-1}) \cdot P_n\left( \sum _{\ell =n}^N h_\ell ^2\right) \right] \\&\le {\mathbb {E}} \left[ \sum _{n=1}^N (g_n - g_{n-1})\right] \cdot \sup _{1\le n\le N} \left\| P_n \left( \sum _{\ell =n}^N h_\ell ^2 \right) \right\| _\infty , \end{aligned}$$

where the last inequality follows from \(g_n\ge g_{n-1}\). Noting that by the properties of \(g_n\), \({\mathbb {E}} \big [\sum _{n=1}^N (g_n - g_{n-1})\big ] = {\mathbb {E}} g_N \lesssim {\mathbb {E}} X_N^{1/2}\) and combining the two parts \(\Sigma _1\) and \(\Sigma _2\), we obtain the conclusion. \(\square \)

4 Applications

We give two applications of Theorem 3.6, (i) D. Lépingle’s inequality and (ii) an analogue of C. Fefferman’s \(H_1\)-\({{\,\mathrm{BMO}\,}}\) duality in the setting of splines. Once the results 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.

4.1 Lépingle’s inequality for splines

Theorem 4.1

Let \(k,k'\) be positive integers. Let \(({\mathscr {F}}_n)\) be an interval filtration with \(\sup _n \gamma _k({\mathscr {F}}_n)<\infty \) and, for any n, \(f_n \in {\mathscr {S}}_{k}({\mathscr {F}}_n)\) and \(P_n'\) be the orthogonal projection operator on \({\mathscr {S}}_{k'}({\mathscr {F}}_n)\). Then,

$$\begin{aligned} \Vert (P_{n-1}' f_n) \Vert _{L_1(\ell _2)} = \left\| \left( \sum _n (P_{n-1}' f_n)^2 \right) ^{1/2} \right\| _1 \lesssim \left\| \left( \sum _n f_n^2\right) ^{1/2} \right\| _1 = \Vert (f_n) \Vert _{L_1(\ell _2)}, \end{aligned}$$

where the implied constant depends only on k, \(k'\) and \(\sup _n \gamma _{k}({\mathscr {F}}_n)\).

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 \(k'\)-regularity parameter \(\sup _n \gamma _{k'}({\mathcal {F}}_n)\).


We first assume that \(f_n=0\) for \(n>N\) and begin by using duality

$$\begin{aligned} {\mathbb {E}} \left[ \left( \sum _n (P_{n-1}'f_n)^2\right) ^{1/2}\right] = \sup _{(H_n)} {\mathbb {E}} \left[ \sum _n (P_{n-1}' f_n) \cdot H_n\right] , \end{aligned}$$

where sup is taken over all \((H_n)\in L_\infty (\ell _2)\) with \(\Vert (H_n)\Vert _{L_\infty (\ell _2)} =1\). By the self-adjointness of \(P_{n-1}'\),

$$\begin{aligned} {\mathbb {E}}\big [(P_{n-1}'f_n) \cdot H_n\big ] = {\mathbb {E}}\big [ f_n \cdot (P_{n-1}' H_n) \big ]. \end{aligned}$$

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 \({\mathscr {S}}_k({\mathscr {F}}_n)\))

$$\begin{aligned} \sum _{n\le N} | {\mathbb {E}} [f_n\cdot h_n] | \lesssim {\mathbb {E}}\left[ \left( \sum _{\ell \le N} f_\ell ^2\right) ^{1/2}\right] \cdot \sup _{n\le N} \left\| P_n \left( \sum _{\ell =n}^N (P_{\ell -1}'H_\ell )^2\right) \right\| _\infty ^{1/2}. \end{aligned}$$

To estimate the right hand side, we note that for any n, by Corollary 3.3,

$$\begin{aligned} \left\| P_n \left( \sum _{\ell =n}^N (P_{\ell -1}' H_\ell )^2\right) \right\| _\infty \lesssim \left\| \sum _{\ell =n}^N H_\ell ^2 \right\| _\infty . \end{aligned}$$

Therefore, (4.1) implies

$$\begin{aligned} {\mathbb {E}} \left[ \left( \sum _n (P_{n-1}'f_n)^2\right) ^{1/2}\right] = \sup _{(H_n)} {\mathbb {E}} \left[ \sum _n f_n \cdot ( P_{n-1}' H_n)\right] \lesssim {\mathbb {E}}\left[ \left( \sum _{\ell \le N} f_\ell ^2\right) ^{1/2}\right] , \end{aligned}$$

with a constant depending only on k,\(k'\) and \(\sup _{n\le N} \gamma _k({\mathscr {F}}_n)\). Letting N tend to infinity, we obtain the conclusion. \(\square \)

4.2 \(H_1\)-\({{\,\mathrm{BMO}\,}}\) duality for splines

We fix an interval filtration \(({\mathscr {F}}_n)_{n=1}^\infty \), a spline order k and the orthogonal projection operators \(P_n\) onto \({\mathscr {S}}_k({\mathscr {F}}_n)\) and additionally, we set \(P_0=0\). For \(f\in L_1\), we introduce the notation

$$\begin{aligned} \Delta _n f := P_n f - P_{n-1} f,\qquad S_n(f) := \left( \sum _{\ell =1}^n (\Delta _\ell f)^2\right) ^{1/2}, \qquad S(f) = \sup _n S_n(f). \end{aligned}$$

Observe that for \(\ell < n\) and \(f,g\in L_1\),

$$\begin{aligned} {\mathbb {E}} [ \Delta _\ell f \cdot \Delta _n g ] = {\mathbb {E}} [ P_\ell (\Delta _\ell f) \cdot \Delta _n g] = {\mathbb {E}} [ \Delta _\ell f \cdot P_\ell (\Delta _n g)] = 0. \end{aligned}$$

Let V be the \(L_1\)-closure of \(\cup _n {\mathscr {S}}_k({\mathscr {F}}_n)\). Then, the uniform boundedness of \(P_n\) on \(L_1\) implies that \(P_n f\rightarrow f\) in \(L_1\) for \(f\in V\). Next, set

$$\begin{aligned} H_{1,k} = H_{1,k}( ({\mathscr {F}}_n) ) = \{ f\in V : {\mathbb {E}} ( S(f) ) < \infty \} \end{aligned}$$

and equip \(H_{1,k}\) with the norm \(\Vert f \Vert _{H_{1,k}} = {\mathbb {E}} S(f)\). Define

$$\begin{aligned} {{\,\mathrm{BMO}\,}}_k = {{\,\mathrm{BMO}\,}}_k ( ({\mathscr {F}}_n) ) = \left\{ f\in V : \sup _n \Vert \sum _{\ell \ge n}T_n\big ( (\Delta _\ell f)^2 \big )\Vert _\infty < \infty \right\} \end{aligned}$$

and the corresponding quasinorm

$$\begin{aligned} \Vert f \Vert _{{{\,\mathrm{BMO}\,}}_k} = \sup _n \big \Vert \sum _{\ell \ge n} T_n\big ((\Delta _\ell f)^2 \big )\big \Vert _\infty ^{1/2}, \end{aligned}$$

where \(T_n\) is the operator from (3.3) that dominates \(P_n\) pointwise.

Let us now assume \(\sup _n \gamma _k({\mathscr {F}}_n) < \infty \). In this case we identify, similarly to \(H_1\)-\({{\,\mathrm{BMO}\,}}\)-duality (cf. [7, 8, 10]), \({{\,\mathrm{BMO}\,}}_k\) as the dual space of \(H_{1,k}\).

First, we use the duality estimate Theorem 3.6 and (4.2) to prove, for \(f\in H_{1,k}\) and \(h\in {{\,\mathrm{BMO}\,}}_k\),

$$\begin{aligned} \big | {\mathbb {E}} \big [ (P_n f) \cdot (P_n h) \big ] \big | \le \sum _{\ell \le n} {\mathbb {E}} \big [ |\Delta _\ell f| \cdot |\Delta _\ell h|\big ] \lesssim \Vert S_n(f) \Vert _1 \cdot \Vert h\Vert _{{{\,\mathrm{BMO}\,}}_k}. \end{aligned}$$

This estimate also implies that the limit \(\lim _n {\mathbb {E}} \big [ (P_nf)\cdot (P_nh) \big ]\) exists and satisfies

$$\begin{aligned} \big |\lim _n {\mathbb {E}} \big [ (P_nf)\cdot (P_nh) \big ]\big | \lesssim \Vert f\Vert _{H_{1,k}} \cdot \Vert h\Vert _{{{\,\mathrm{BMO}\,}}_k}. \end{aligned}$$

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 \(\Lambda \) on \(L_1(\ell _2)\), which, by Sect. 2.5, has the form

$$\begin{aligned} \Lambda (\eta ) = {\mathbb {E}} \left[ \sum _\ell \sigma _\ell \eta _\ell \right] , \qquad \eta \in L_1(\ell _2) \end{aligned}$$

for some \(\sigma \in L_\infty (\ell _2)\). The k-martingale spline sequence \(h_n= \sum _{\ell \le n} \Delta _\ell \sigma _\ell \) is bounded in \(L_2\) and therefore, by the spline convergence theorem ((v) on page 2), has a limit \(h\in L_2\) with \(P_n h = h_n\) and which is also contained in \({{\,\mathrm{BMO}\,}}_k\). Indeed, by using Corollary 3.3, we obtain \(\Vert h\Vert _{{{\,\mathrm{BMO}\,}}_k} \lesssim \Vert \sigma \Vert _{L_\infty (\ell _2)} = \Vert \Lambda \Vert = \Vert L\Vert \) with a constant depending only on k and \(\sup _{n}\gamma _k({\mathscr {F}}_n)\). Moreover, for \(f\in H_{1,k}\), since L is continuous on \(H_{1,k}\),

$$\begin{aligned} L(f) =\lim _n L(P_n f)&= \lim _n \Lambda \big ((\Delta _1 f,\ldots , \Delta _n f,0,0,\ldots ) \big ) \\&=\lim _n \sum _{\ell =1}^n {\mathbb {E}}[ \sigma _\ell \cdot \Delta _\ell f ]= \lim _n {\mathbb {E}}\big [ (P_nf)\cdot (P_nh) \big ]. \end{aligned}$$

This yields the following

Theorem 4.2

If \(\sup _n \gamma _k({\mathscr {F}}_n)<\infty \), the linear mapping

$$\begin{aligned} u : {{\,\mathrm{BMO}\,}}_k \rightarrow H_{1,k}^*, \qquad h \mapsto \big ( f \mapsto \lim _n {\mathbb {E}}\big [(P_n f)\cdot (P_n h)\big ] \big ) \end{aligned}$$

is bijective and satisfies

$$\begin{aligned} \Vert u(h) \Vert _{H_{1,k}^*} \simeq \Vert h\Vert _{{{\,\mathrm{BMO}\,}}_k}, \end{aligned}$$

where the implied constants depend only on k and \(\sup _n \gamma _k({\mathscr {F}}_n)\).

Remark 4.3

We close with a few remarks concerning the above result and we assume that \(({\mathscr {F}}_n)\) is a sequence of increasing interval \(\sigma \)-algebras with \(\sup _n \gamma _k({\mathscr {F}}_n) < \infty \).

  1. (1)

    By Khintchine’s inequality, \(\Vert Sf\Vert _1 \lesssim \sup _{\varepsilon \in \{-1,1\}^{{\mathbb {Z}}}} \Vert \sum _n\varepsilon _n\Delta _n f\Vert _1\). Based on the interval filtration \(({\mathscr {F}}_n)\), we can generate an interval filtration \(({\mathscr {G}}_n)\) that contains \(({\mathscr {F}}_n)\) as a subsequence and each \({\mathscr {G}}_{n+1}\) is generated from \({\mathscr {G}}_n\) by dividing exactly one atom of \({\mathscr {G}}_n\) into two atoms of \(\mathscr {G}_{n+1}\). Denoting by \(P_n^{{\mathscr {G}}}\) the orthogonal projection operator onto \({\mathscr {S}}_k({\mathscr {G}}_n)\) and \(\Delta _j^{\mathscr {G}}= P_j^{{\mathscr {G}}}-P_{j-1}^{{\mathscr {G}}}\), we can write

    $$\begin{aligned} \sum _n\varepsilon _n\Delta _n f = \sum _n \varepsilon _n \sum _{j=a_n}^{a_{n+1}-1} \Delta _j^{{\mathscr {G}}} f \end{aligned}$$

    for some sequence \((a_n)\). By using inequalities (2.7) and (2.6) and writing \((S^{{\mathscr {G}}}f)^2= \sum _j |\Delta _j^{{\mathscr {G}}} f|^2\), we obtain for \(p>1\)

    $$\begin{aligned} \Vert Sf\Vert _1 \lesssim \Vert S^{{\mathscr {G}}} f\Vert _1 \le \Vert S^{{\mathscr {G}}} f\Vert _p \lesssim \Vert f\Vert _p. \end{aligned}$$

    This implies \(L_p\subset H_{1,k}\) for all \(p>1\) and, by duality, \({{\,\mathrm{BMO}\,}}_k \subset L_p\) for all \(p<\infty \).

  2. (2)

    If \(({\mathscr {F}}_n)\) is of the form that each \(\mathscr {F}_{n+1}\) is generated from \({\mathscr {F}}_n\) by splitting exactly one atom of \({\mathscr {F}}_n\) into two atoms of \({\mathscr {F}}_{n+1}\) and under the condition \(\sup _n \gamma _{k-1}({\mathscr {F}}_n) < \infty \) (which is stronger than \(\sup _n \gamma _k({\mathscr {F}}_n)<\infty \)), it is shown in [9] that

    $$\begin{aligned} \Vert Sf\Vert _{1} \simeq \Vert f\Vert _{H_1}, \end{aligned}$$

    where \(H_1\) denotes the atomic Hardy space on [0, 1], i.e. in this case, \(H_{1,k}\) coincides with \(H_1\).