## 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}
(1.1)

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}
(1.2)

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}
(1.3)

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}
(2.1)

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}
(2.2)

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

Observe that (2.1) implies for $$g\in \mathscr {S}_k({\mathscr {F}})$$ 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}
(2.3)

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}
(2.4)

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}
(2.5)

### 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}
(2.6)

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}
(2.7)

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}
(3.1)

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}
(3.2)

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}
(3.3)

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}
(3.4)

where the implied constant depends only on pr and k.

### Proof

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}
(3.5)

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}
(3.6)

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

### Proof

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}
(3.7)

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.

### Proof

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}
(3.8)

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 $$i\ne j$$.

### Proof

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}
(3.9)

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}
(3.10)

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}
(3.11)

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}
(3.12)

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 $$\gamma$$.

### Proof

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.

### 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 $$\sup _n \gamma _{k'}({\mathcal {F}}_n)$$.

### Proof

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}
(4.1)

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}
(4.2)

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 $$H_1$$.