Martingale inequalities for spline sequences

TL;DR

This paper extends a fundamental martingale inequality to spline sequences, establishing new bounds and duality results in the context of spline spaces under certain regularity conditions.

Contribution

It generalizes D. Lépingle's $L_1( ext{ell}_2)$-inequality to spline projections and introduces a spline version of $H_1$-$BMO$ duality.

Findings

Extended martingale inequality to spline projections

Established spline $H_1$-$BMO$ duality

Provided conditions based on spline space smoothness

Abstract

We show that D. L\'{e}pingle's $L_1(\ell_2)$-inequality \begin{equation*} \Big\| \big( \sum_n \mathbb E[f_n | \mathscr F_{n-1}]^2 \big)^{1/2}\Big\|_1 \leq 2\cdot \Big\| \big( \sum_n f_n^2 \big)^{1/2} \Big\|_1, \qquad f_n\in\mathscr F_n, \end{equation*} extends to the case where we substitute the conditional expectation operators with orthogonal projection operators onto spline spaces and where we can allow that $f_n$ is contained in a suitable spline space $\mathscr S(\mathscr F_n)$. This is done provided the filtration $(\mathscr F_n)$ satisfies a certain regularity condition depending on the degree of smoothness of the functions contained in $\mathscr S(\mathscr F_n)$. As a by-product, we also obtain a spline version of $H_1$-$BMO$ duality under this assumption.

Equations

$$\Big{\|}\big{(}\sum_{n}\mathbb{E}[f_{n}|\mathscr{F}_{n-1}]^{2}\big{)}^{1/2}\Big{\|}_{1}\leq 2\cdot\Big{\|}\big{(}\sum_{n}f_{n}^{2}\big{)}^{1/2}\Big{\|}_{1},\qquad f_{n}\in\mathscr{F}_{n},$$

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

$$P_n^{(k)}{f}_{n+1}=f_n.$$

$$\underset{n}{\sup }\Vert P_n^{(k)}:L_1\to L_1\Vert \le C_k,$$

$$|\{\underset{n}{\sup }|f_n|>\lambda \}|\le C_k\frac{{\sup }_n\Vert f_n{\Vert }_1}{\lambda },$$

$$\big{\|}\sup_{n}|f_{n}|\big{\|}_{p}\leq C_{p,k}\sup_{n}\|f_{n}\|_{p},\$$

$$\Big{\|}\big{(}\sum_{n}\mathbb{E}[f_{n}|\mathscr{F}_{n-1}]^{2}\big{)}^{1/2}\Big{\|}_{1}\leq 2\cdot\Big{\|}\big{(}\sum_{n}f_{n}^{2}\big{)}^{1/2}\Big{\|}_{1},$$

$$\Big{\|}\big{(}\sum_{n}\mathbb{E}[f_{n}|\mathscr{F}_{n-1}]^{2}\big{)}^{1/2}\Big{\|}_{p}\leq a_{p}\Big{\|}\big{(}\sum_{n}f_{n}^{2}\big{)}^{1/2}\Big{\|}_{p},$$

$$(\mathbb{E}[f_n\cdot h_n]{)}^2\le \mathbb{E}[f_n^2/S_n]\cdot \mathbb{E}[S_n{h}_n^2]$$

$$\|p\|_{L_{\infty}(V)}\leq\bigg{(}4\frac{|V|}{|E|}\bigg{)}^{k-1}\|p\|_{L_{\infty}(E)}.$$

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

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

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

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

$$\sum _j{a}_j{N}_{\mathcal{G},j}=\sum _i{b}_i{N}_{\mathcal{F},i},$$

$$Pf=\sum_{i,j}a_{ij}\Big{(}\int_{0}^{1}f(x)N_{i}(x)\,\mathrm{d}x\Big{)}\cdot N_{j}$$

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

$$Sf=\Big{(}\sum_{n}|\Delta_{n}f|^{2}\Big{)}^{1/2}.$$

$$\Vert Sf{\Vert }_p\simeq \Vert f{\Vert }_p,f\in L_p$$

$$\Vert Sf{\Vert }_1\simeq \underset{\epsilon \in \{-1,1{\}}^{\mathbb{Z}}}{\sup }\Vert \sum _n{\epsilon }_n{\Delta }_{n}f{\Vert }_1,Sf\in L_1,$$

$$\|(f_{n})\|_{L_{p}(\ell_{q})}=\Big{(}\int_{0}^{1}\Big{(}\sum_{n}|f_{n}(t)|^{q}\Big{)}^{p/q}\,\mathrm{d}t\Big{)}^{1/p}$$

$$⟨(f_n),(g_n)⟩={\int }_0^1\sum _n{f}_n(t)g_n(t)\mathrm{d}t.$$

$$Tf(x):=\sum _{i,j}\frac{q^{|i-j|}}{|E_{ij}|}⟨f,\text{mathbbm}{1}_{E_i}⟩\text{mathbbm}{1}_{E_j}(x)={\int }_0^{1}K(x,t)f(t)\mathrm{d}t,$$

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

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

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

$$\varphi(Tf(x))\leq K_{x}^{-1}T\big{(}\varphi(K_{x}\cdot f)\big{)}(x),\qquad f\in L_{1},x\in[0,1],$$

$$\mathcal{M}f(x)=\underset{I\ni x}{\sup }\frac{1}{|I|}{\int }_I|f(y)|\mathrm{d}y,$$

$$|\{\mathcal{M}f>\lambda \}|\le C{\lambda }^{-1}\Vert f{\Vert }_1,f\in L_1,\lambda >0$$

$$\Vert \mathcal{M}f{\Vert }_p\le C_p\Vert f{\Vert }_p.$$

Full text

[TABLE]

extends to the case where we substitute the conditional expectation operators with orthogonal projection operators onto spline spaces and where we can allow that $f_{n}$ is contained in a suitable spline space $\mathscr{S}(\mathscr{F}_{n})$. This is done provided the filtration $(\mathscr{F}_{n})$ satisfies a certain regularity condition depending on the degree of smoothness of the functions contained in $\mathscr{S}(\mathscr{F}_{n})$. As a by-product, we also obtain a spline version of $H_{1}$-$\operatorname{BMO}$ duality under this assumption.

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. [17, 15, 12, 10, 14, 13, 8]). 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

[TABLE]

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 $(f_{n})_{n\geq 0}$ be a sequence of integrable functions. We call this sequence a $k$-martingale spline sequence (adapted to $(\mathscr{F}_{n})$) if, for all $n$,

[TABLE]

For basic facts about martingales and conditional expectations, we refer to [11].

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

[TABLE] 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 $\lambda>0$,

[TABLE] 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 $(f_{n})$,

[TABLE] 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) is proved in [17], properties (ii) and (iii) in [15] and properties (iv) and (v) in [10], but see also [14].

Here, we continue this line of transferring martingale results to $k$-martingale spline sequences and extend D. Lépingle’s $L_{1}(\ell_{2})$-inequality [9], which reads

[TABLE]

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. The spline version of this inequality is contained in Theorem 4.1.

This inequality is an $L_{1}$ extension of the following result for $1<p<\infty$, proved by E. M. Stein [19], that holds for arbitrary integrable functions $f_{n}$:

[TABLE]

for some constant $a_{p}$ depending only on $p$. This can be seen as a dual version of Doob’s inequality $\|\sup_{\ell}|\mathbb{E}[f|\mathscr{F}_{\ell}]|\|_{p}\leq c_{p}\|f\|_{p}$ 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. Sections 3.1 and 3.2).

The usage of those operators $T_{n}$ is also necessary in the extension of inequality (1.1) to splines. D. Lépingle’s proof of (1.1) rests on an idea by C. Herz [7] 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 $S_{n}^{2}=\sum_{\ell\leq n}f_{\ell}^{2}$:

[TABLE]

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. Section 3.4, in particular Proposition 3.4 and Theorem 3.6). As a by-product, we obtain a spline version (Theorem 4.2) of C. Fefferman’s theorem [4] on $H^{1}$-$\operatorname{BMO}$ duality. For its martingale version, we refer to A. M. Garsia’s book [5] 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 $V$,

[TABLE]

Applying this theorem with the set $E=\{x\in V:|p(x)|\leq 8^{-k+1}\|p\|_{L_{\infty}(V)}\}$ 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

[TABLE]

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 $N_{i}(x)\neq 0$ 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 $i$, $\inf E_{i}\leq\inf E_{i+1}$ and $\sup E_{i}\leq\sup E_{i+1}$.

We write $A(t)\lesssim B(t)$ to denote the existence of a constant $C$ such that for all $t$, $A(t)\leq CB(t)$, where $t$ denote all implicit and explicit dependencies the expression $A$ and $B$ might have. If the constant $C$ additionally depends on some parameter, we will 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 $1\leq p\leq\infty$ and $g=\sum_{j}a_{j}N_{j}$. Then, for all $j$,

[TABLE]

where $J_{j}$ is an atom of $\mathscr{F}$ contained in $E_{j}$ having maximal length. Additionally,

[TABLE]

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 $A\subset[0,1]$

[TABLE]

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 $(N_{\mathscr{F},i})_{i}$

[TABLE]

where for each $i$, $b_{i}$ is a convex combination of the coefficients $a_{j}$ with $\operatorname{supp}N_{\mathscr{G},j}\supseteq\operatorname{supp}N_{\mathscr{F},i}$.

For those results and more information on spline functions, in particular B-splines, we refer to [16] or [3]. 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

[TABLE]

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 $a_{ij}$ [15]: there exist constants $C$ and $0<q<1$ depending only on $k$ so that for each interval $\sigma$-algebra $\mathscr{F}$ and each $i,j$,

[TABLE]

2.3. 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

[TABLE]

We have Burkholder’s inequality for the spline square function, i.e., for all $1<p<\infty$ ([12]), the $L_{p}$-norm of the square function $Sf$ is comparable to the $L_{p}$-norm of $f$:

[TABLE]

with constants depending only on $p$ and $k$. Moreover, for $p=1$, it is shown in [6] that

[TABLE]

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 $\Delta_{n}f$.

2.4. $L_{p}(\ell_{q})$-spaces

For $1\leq p,q\leq\infty$, we denote by $L_{p}(\ell_{q})$ the space of sequences of measurable functions $(f_{n})$ on $[0,1]$ so that the norm

[TABLE]

is finite (with the obvious modifications if $p=\infty$ or $q=\infty$). For $1\leq p,q<\infty$, the dual space (see [2]) of $L_{p}(\ell_{q})$ is $L_{p^{\prime}}(\ell_{q^{\prime}})$ with $p^{\prime}=p/(p-1)$, $q^{\prime}=q/(q-1)$ and the duality pairing

[TABLE]

Hölder’s inequality takes the form $|\langle(f_{n}),(g_{n})\rangle|\leq\|(f_{n})\|_{L_{p}(\ell_{q})}\|(g_{n})\|_{L_{p^{\prime}}(\ell_{q^{\prime}})}$.

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 $P_{n}$ pointwise. In Section 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 Section 3.1. In Section 3.3 we prove a version of the tower property for those operators. Section 3.4 is devoted to establishing a duality estimate using a spline square function, which is the crucial ingredient in the proofs of the spline versions of both Lépingle’s inequality (1.1) and $H_{1}$-$\operatorname{BMO}$ duality in Section 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

[TABLE]

where the kernel $K=K_{T}$ is given by

[TABLE]

We observe that the operator $T$ is selfadjoint (w.r.t the standard inner product on $L_{2}$) and

[TABLE]

which, in particular, implies the boundedness of the operator $T$ on $L_{1}$ and $L_{\infty}$:

[TABLE]

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

[TABLE]

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 $\mathscr{M}f$ denote the Hardy-Littlewood maximal function of $f\in L_{1}$, i.e.,

[TABLE]

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.,

[TABLE]

for some constant $C$. Since trivially we have the estimate $\|\mathscr{M}f\|_{\infty}\leq\|f\|_{\infty}$, by Marcinkiewicz interpolation, for any $p>1$, there exists a constant $C_{p}$ depending only on $p$ so that

[TABLE]

For those assertions about $\mathscr{M}$, we refer to (for instance) [18].

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

[TABLE]

where $T=T_{\mathscr{F},q,k}$ with $q$ given by (2.5) and $C_{1},C_{2}$ are two constants solely depending on $k$. In other words, the positive operator $T$ dominates the non-positive operator $P$ pointwise, but at the same time, $T$ is dominated by $\mathscr{M}$ pointwise 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. [19, 1]). 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 $1\leq r\leq p<\infty$ or $1<p\leq r\leq\infty$,

[TABLE]

where the implied constant depends only on $p,r$ and $k$.

Proof.

By (3.3), it suffices to prove this inequality for the operators $T_{n}=T_{\mathscr{F}_{n},q,k}$ 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 i). Inequality (3.3) and the $L_{p^{\prime}}$-boundedness of $\mathscr{M}$ for $1<p^{\prime}\leq\infty$ imply that

[TABLE]

with a constant $C_{p^{\prime},k}$ depending on $p^{\prime}$ and $k$. Let $1\leq p<\infty$ and $U_{N}:L_{p}(\ell_{1}^{N})\to L_{p}$ 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 $U_{N}^{*}:L_{p^{\prime}}\to L_{p^{\prime}}(\ell_{\infty}^{N})$, $f\mapsto(T_{j}f)_{j=1}^{N}$ for $p^{\prime}=p/(p-1)$ by a constant independent of $N$ and a fortiori the boundedness of $U_{N}$. Since $|T_{j}f|\leq T_{j}|f|$ by the positivity of $T_{j}$, letting $N\to\infty$ implies (3.4) for $T_{n}$ instead of $P_{n}$ in the case $r=1$ and outer parameter $1\leq p<\infty$.

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

[TABLE]

and apply the result for $r=1$ and the outer parameter $p/r$ to get the result for $1\leq r\leq p<\infty$. The cases $1<p\leq r\leq\infty$ now just follow from this result using duality and the self-adjointness of $T_{j}$. ∎

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

[TABLE]

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 $T=T_{\mathscr{F},\tau,k^{\prime}}$ for some $\sigma,\tau\in(0,1)$ and some positive integers $k,k^{\prime}$. Then, for all $q>\max(\tau,\sigma)$, there exists a constant $C$ depending on $q,k,k^{\prime}$ so that

[TABLE]

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 $\mathscr{S}_{k^{\prime}}(\mathscr{F})$ 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))

[TABLE]

for all $q>\max(\tau,\sigma)$ and some constant $C$ depending on $q,k,k^{\prime}$. 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 $k$ 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

[TABLE]

Since, for each $r$, there are also at most $k+k^{\prime}-1$ 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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

where the implied constants depend on $\lambda,k,k^{\prime}$ 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 $q>\lambda$,

[TABLE]

for some constant $C$ depending on $q,k,k^{\prime}$, and, as $x\in G_{i}$ and $s\in G_{j}$, this shows inequality (3.7). ∎

As a corollary of Proposition 3.2, we have

Corollary 3.3**.**

Let $(f_{n})$ be functions in $L_{1}$. We denote by $P_{n}$ the orthogonal projection onto $\mathscr{S}_{k}(\mathscr{F}_{n})$ and by $P_{n}^{\prime}$ the orthogonal projection onto $\mathscr{S}_{k^{\prime}}(\mathscr{F}_{n})$ for some positive integers $k,k^{\prime}$. 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 $1\leq p\leq\infty$,

[TABLE]

where the implied constants only depend on $k$ and $k^{\prime}$.

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 $P_{n},T_{n},P_{\ell-1}^{\prime}$ 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 $T_{n}^{\prime}$ the operator $T_{\mathscr{F}_{n},q^{\prime},k^{\prime}}$, where the parameters $q,q^{\prime}<1$ are given by inequality (3.3) depending on $k$ and $k^{\prime}$ respectively. Setting $U_{n}:=T_{\mathscr{F}_{n},\max(q,q^{\prime})^{1/2},k}$, we perform the following chain of inequalities, where we use the positivity of $T_{n}$ and (3.3), Jensen’s inequality for $T_{\ell-1}^{\prime}$, the tower property for $T_{n}T_{\ell-1}^{\prime}$ and the $L_{p}$-boundedness of $U_{n}$, respectively:

[TABLE]

where the implied constants only depend on $k$ and $k^{\prime}$. ∎

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

[TABLE]

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)

$g_{n}\leq g_{n+1}$, 2. (2)

$X_{n}^{1/2}\leq g_{n}$** 3. (3)

$\mathbb{E}g_{n}\lesssim\mathbb{E}X_{n}^{1/2}$, where the implied constant depends on $k$ and on $\sup_{m\leq n}\gamma_{k}(\mathscr{F}_{m})$.

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 $\ell:\{1,\ldots,N\}\to\{1,\ldots,n\}$ 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

[TABLE]

Then, there exists a set-valued mapping $\varphi$ 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 $i\neq j$.*

Proof.

Without restriction, we assume that the sequence $(A_{j})$ is enumerated such that $\ell(j+1)\leq\ell(j)$ for all $1\leq j\leq N-1$. 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 $1\leq j\leq j_{0}$, 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 $\Gamma=\{i:\ell(i)\geq\ell(j_{0}+1),D_{i}\subseteq D_{j_{0}+1}\}$ and $\Lambda=\{i:i\leq j_{0}+1,D_{i}\subseteq D_{j_{0}+1}\}$. 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 $i\leq j_{0}+1$ that either $D_{i}\subset D_{j_{0}+1}$ or $|D_{i}\cap D_{j_{0}+1}|=0$. This implies

[TABLE]

Therefore, we can choose $\varphi(j_{0}+1)\subseteq B_{j_{0}+1}$ that is disjoint to $\varphi(i)$ for any $i\leq j_{0}$ and $|\varphi(j_{0}+1)|=c_{1}|A_{j_{0}+1}|$ which completes the proof. ∎

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 $E_{n,j}=\operatorname{supp}N_{n,j}$ and $a_{n,j}:=\max_{\ell\leq n}\max_{r:E_{\ell,r}\supset E_{n,j}}\|X_{\ell}\|_{L_{\infty}(E_{\ell,r})}^{1/2}$ and we define $\ell(j)\leq n$ and $r(j)$ so that $E_{\ell(j),r(j)}\supseteq E_{n,j}$ and $a_{n,j}=\|X_{\ell(j)}\|_{L_{\infty}(E_{\ell(j),r(j)})}^{1/2}$. Set

[TABLE]

and it will be proved subsequently that this $g_{n}$ has the desired properties.

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

[TABLE]

where $\beta_{n,j}$ is a convex combination of those $a_{n,r}$ with $E_{n+1,j}\subseteq E_{n,r}$, and thus

[TABLE]

By the very definition of $a_{n+1,j}$, we have

[TABLE]

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

Property (2): Now we show that $X_{n}^{1/2}\leq g_{n}$. Indeed, for any $x\in[0,1]$,

[TABLE]

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 $\sup_{m\leq n}\gamma_{k}(\mathscr{F}_{m})$. By B-spline stability (Theorem 2.3), we estimate the integral of $g_{n}$ by

[TABLE]

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

[TABLE]

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 $m\leq\ell$)

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

In order to see assumption (3.8) of Lemma 3.5, fix the index $j$ and let $i$ be such that $\ell(i)\geq\ell(j)$. 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 $\sup_{u\leq\ell(j)}\gamma_{k}(\mathscr{F}_{u})\leq\sup_{u\leq n}\gamma_{k}(\mathscr{F}_{u})$ so that

[TABLE]

which, since $|B_{j}|\geq|D_{j}|/2$ by Corollary 2.2, shows the assumption of Lemma 3.5 and we get a set-valued 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 $i,j$. Using these properties of $\varphi$, we continue the estimate in (3.12) and write

[TABLE]

with constants depending only on $k$ and $\sup_{u\leq n}\gamma_{k}(\mathscr{F}_{u})$. ∎

Employing this construction of $g_{n}$, we now give the following duality estimate for spline projections (for the martingale case, see for instance [5]). The martingale version of this result is the essential estimate in the proof of both Lépingle’s inequality (1.1) and the $H^{1}$-$\operatorname{BMO}$ duality.

Theorem 3.6**.**

Let $(\mathscr{F}_{n})$ be such that $\gamma:=\sup_{n}\gamma_{k}(\mathscr{F}_{n})<\infty$ and $(f_{n})_{n\geq 1}$ 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 $N$,

[TABLE]

where the implied constant depends only on $k$ and $\gamma$.

Proof.

Let $X_{n}:=\sum_{\ell\leq n}f_{\ell}^{2}$ and $f_{\ell}\equiv 0$ for $\ell>N$ and $\ell\leq 0$. 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 $X_{n}^{1/2}\leq g_{n}$ 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

[TABLE]

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

[TABLE]

The first factor is estimated by the pointwise inequality $X_{n}^{1/2}\leq g_{n}$:

[TABLE]

We continue with $\Sigma_{2}$:

[TABLE]

where the last inequality follows from $g_{n}\geq g_{n-1}$. 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. ∎

4. Applications

We give two applications of Theorem 3.6, (i) D. Lépingle’s inequality and (ii) an analogue of C. Fefferman’s $H_{1}$-$\operatorname{BMO}$ duality in the setting of splines. Once the results from Section 3 are known, the proofs of the subsequent results proceed similarly to their martingale counterparts in [9] and [5] by using spline properties instead of martingale properties.

4.1. Lépingle’s inequality for splines

Theorem 4.1**.**

Let $k,k^{\prime}$ 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 $P_{n}^{\prime}$ be the orthogonal projection operator on $\mathscr{S}_{k^{\prime}}(\mathscr{F}_{n})$. Then,

[TABLE]

where the implied constant depends only on $k$, $k^{\prime}$ and $\sup_{n}\gamma_{k}(\mathscr{F}_{n})$.

Proof.

We first assume that $f_{n}=0$ for $n>N$ and begin by using duality

[TABLE]

where sup is taken over all $(H_{n})\in L_{\infty}(\ell_{2})$ with $\|(H_{n})\|_{L_{\infty}(\ell_{2})}=1$. By the self-adjointness of $P_{n-1}^{\prime}$,

[TABLE]

Then we apply Theorem 3.6 for $f_{n}$ and $h_{n}=P_{n-1}^{\prime}H_{n}$ to obtain (denoting by $P_{n}$ the orthogonal projection operator onto $\mathscr{S}_{k}(\mathscr{F}_{n})$)

[TABLE]

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

[TABLE]

Therefore, (4.1) implies

[TABLE]

with a constant depending only on $k$,$k^{\prime}$ and $\sup_{n\leq N}\gamma_{k}(\mathscr{F}_{n})$. Letting $N$ tend to infinity, we obtain the conclusion. ∎

4.2. $H_{1}$-$\operatorname{BMO}$ 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

[TABLE]

Observe that for $\ell<n$ and $f,g\in L_{1}$,

[TABLE]

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 $P_{n}f\to f$ in $L_{1}$ for $f\in V$. Next, set

[TABLE]

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

[TABLE]

and the corresponding quasinorm

[TABLE]

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 $H_{1}$-$\operatorname{BMO}$-duality (cf. [4, 7, 5]), $\operatorname{BMO}_{k}$ 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 $h\in\operatorname{BMO}_{k}$,

[TABLE]

This estimate also implies that the limit \lim_{n}\mathbb{E}\big{[}(P_{n}f)\cdot(P_{n}h)\big{]} exists and satisfies

[TABLE]

On the other hand, similarly to the martingale case (see [5]), 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 Section 2.4, has the form

[TABLE]

for some $\sigma\in L_{\infty}(\ell_{2})$. The $k$-martingale spline sequence $h_{n}=\sum_{\ell\leq n}\Delta_{\ell}\sigma_{\ell}$ is bounded in $L_{2}$ and therefore, by the spline convergence theorem ((v) on page v), has a limit $h\in L_{2}$ with $P_{n}h=h_{n}$ and which is also contained in $\operatorname{BMO}_{k}$. Indeed, by using Corollary 3.3, we obtain $\|h\|_{\operatorname{BMO}_{k}}\lesssim\|\sigma\|_{L_{\infty}(\ell_{2})}=\|\Lambda\|=\|L\|$ 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 $H_{1,k}$,

[TABLE]

This yields the following

Theorem 4.2**.**

If $\sup_{n}\gamma_{k}(\mathscr{F}_{n})<\infty$, the linear mapping

[TABLE]

is bijective and satisfies

[TABLE]

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, $\|Sf\|_{1}\lesssim\sup_{\varepsilon\in\{-1,1\}^{\mathbb{Z}}}\|\sum_{n}\varepsilon_{n}\Delta_{n}f\|_{1}$. 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

[TABLE]

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 $p>1$

[TABLE]

This implies $L_{p}\subset H_{1,k}$ for all $p>1$ and, by duality, $\operatorname{BMO}_{k}\subset L_{p}$ 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 [6] that

[TABLE]

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

Acknowledgments

The author is supported by the Austrian Science Fund (FWF), Project F5509-N26, which is a part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”.