The Haar System as a Schauder Basis in Spaces of Hardy–Sobolev Type

Abstract

We show that, for suitable enumerations, the Haar system is a Schauder basis in the classical Sobolev spaces in ${\mathbb R}^d$ with integrability $1<p<\infty$ and smoothness $1/p-1<s<1/p$. This complements earlier work by the last two authors on the unconditionality of the Haar system and implies that it is a conditional Schauder basis for a nonempty open subset of the (1 / p, s)-diagram. The results extend to (quasi-)Banach spaces of Hardy–Sobolev and Triebel–Lizorkin type in the range of parameters $\frac{d}{d+1}<p<\infty$ and $\max \{d(1/p-1),1/p-1\}<s<\min \{1,1/p\}$, which is optimal except perhaps at the end-points.

1 Introduction

We recall the definition of the (inhomogeneous) Haar system in ${\mathbb R^d}$. Consider the 1-variable functions

$$\begin{aligned} h^{(0)}={\mathbb 1}_{[0,1)}{\quad \text{ and }\quad }h^{(1)}={\mathbb 1}_{[0,1/2)}-\;{\mathbb 1}_{[1/2,1)}. \end{aligned}$$

For every ${\varvec{{\varepsilon }}}=({\varepsilon }_1,\ldots ,{\varepsilon }_d)\in \{0,1\}^d$ one defines

$$\begin{aligned} h^{({\varvec{{\varepsilon }}})}(x_1,\ldots ,x_d)\,=\,h^{({\varepsilon }_1)}(x_1)\cdots h^{({\varepsilon }_d)}(x_d). \end{aligned}$$

Finally, one sets

$$\begin{aligned} h^{({\varvec{{\varepsilon }}})}_{k,\ell }(x)= h^{({\varvec{{\varepsilon }}})}(2^kx-\ell ),\quad k\in {\mathbb {\mathbb N}_0},\;\ell \in {\mathbb Z^d}, \end{aligned}$$

Denoting $\Upsilon =\{0,1\}^d\setminus \{\vec {0}\}$, the Haar system is then given by

$$\begin{aligned} {\mathscr {H}}_d=\Big \{h^{(\vec {0})}_{0,\ell }\Big \}_{\ell \in {\mathbb Z^d}}\cup \Big \{h^{({\varvec{{\varepsilon }}})}_{k,\ell }\mid k\in {\mathbb {\mathbb N}_0},\;\ell \in {\mathbb Z^d},\;{\varvec{{\varepsilon }}}\in \Upsilon \Big \}. \end{aligned}$$

Observe that ${\mathrm{supp}}\ h^{({\varvec{{\varepsilon }}})}_{k,\ell }$ is the dyadic cube $I_{k,\ell }:=2^{-k}(\ell +[0,1]^d)$.

In this paper we consider basis properties of ${\mathscr {H}}_d$ in Besov spaces $B^s_{p,q}$, and Triebel–Lizorkin spaces $F^s_{p,q}$ in ${\mathbb R^d}$. We refer to [9, 10] for definitions and properties of these spaces, and to [1] for terminology and general facts about bases in Banach spaces.

In the 1970s, Triebel [7, 8] proved that the Haar system ${\mathscr {H}}_d$ is a Schauder basis on $B^s_{p,q}({\mathbb R^d})$ if

$$\begin{aligned} \tfrac{d}{d+1}<p<\infty ,\quad 0<q<\infty ,\quad \max \big \{d(\tfrac{1}{p}-1),\tfrac{1}{p}-1\big \}<s< \min \big \{1,\tfrac{1}{p}\big \}, \end{aligned}$$

(1)

and that this range is maximal, except perhaps at the endpoints. Moreover, the basis is unconditional when (1) holds; see [11, Theorem 2.21]. Concerning $F^s_{p,q}$ spaces, however, in [11] it is only shown that ${\mathscr {H}}_d$ is an unconditional basis for $F^s_{p,q}({\mathbb R^d})$ when, besides (1), the additional assumption

$$\begin{aligned} \max \big \{d(\tfrac{1}{q}-1),\tfrac{1}{q}-1\big \}<s<\tfrac{1}{q} \end{aligned}$$

(2)

is satisfied. Recently, two of the authors showed in [5, 6] that the additional restriction (2) is in fact necessary, at least when $d=1$. It was left open whether suitable enumerations of the Haar system can form a Schauder basis in $F^s_{p,q}$ in the larger range (1). We shall answer this question affirmatively.

Given an enumeration $\{u_1,u_2,\ldots \}$ of the system ${\mathscr {H}}_d$, we let $P_N$ be the orthogonal projection onto the subspace spanned by $u_1,\dots , u_N$, i.e.

$$\begin{aligned} P_N f= \sum _{n=1}^N \Vert u_n\Vert _2^{-2} \langle f,u_n\rangle u_n \,. \end{aligned}$$

(3)

The sequence $\{u_n\}_{n=1}^\infty$ is a Schauder basis on $F^s_{p,q}$ if

$$\begin{aligned} \lim _{N\rightarrow \infty } \Vert P_Nf-f\Vert _{F^s_{p,q}} =0,\quad \text{ for } \text{ all } f\in F^s_{p,q}. \end{aligned}$$

(4)

In view of the uniform boundedness principle, density theorems and the result for Besov spaces, (4) follows if we can show that the operators $P_N$ have uniform $F^s_{p,q}\rightarrow F^s_{p,q}$ operator norms. Note, that the condition $s<1/p$ is necessary since the Haar functions need to belong to $F^s_{p,q}$. By duality, if $1<p<\infty$, the condition $s>1/p-1$ becomes also necessary, so the range in (1) is optimal in this case. If $p\le 1$, then an interpolation argument shows that (1) is also a maximal range, except perhaps at the end-points; see Sect. 4 below.

Definition An enumeration ${\mathcal {U}}=\{u_1, u_2, ...\}$ of the Haar system ${\mathscr {H}}_d$ is admissible if the following condition holds for each cube $I_{\nu } = \nu +[0,1]^d, \nu \in {\mathbb {Z}}^d$. If $u_n$ and $u_{n'}$ are both supported in $I_\nu$ and $|{\mathrm{supp}}(u_n)|>|{\mathrm{supp}}(u_{n'})|$, then necessarily $n<n'$ .

Fig. 1

An admissible enumeration of ${\mathscr {H}}_d$

The table in Fig. 1 shows how to obtain an admissible (natural) enumeration of ${\mathscr {H}}_d$ via a diagonalization of the intervals $I_{\nu }$ versus the levels k. We first label the set ${\mathbb Z^d}=\{\nu _1,\nu _2,\ldots \}$. Then, we follow the order indicated by the table, where being at position $(\nu _i,k)$ means to pick all the Haar functions with support contained in $I_{\nu _i}$ and size $2^{-kd}$, arbitrarily enumerated, before going to the subsequent table entry.

Our main result reads as follows.

Theorem 1.1

Let ${\mathcal {U}}=\{u_n\}_{n=1}^\infty$ be an admissible enumeration of the Haar system ${\mathscr {H}}_d$. Assume that

• (i) $\frac{d}{d+1}<p<\infty$,

• (ii) $0<q <\infty$,

• (iii) $\max \{d(\frac{1}{p}-1),\frac{1}{p}-1\}<s< \min \{1,\frac{1}{p}\}$.

Then ${\mathcal {U}}$ is a Schauder basis in $F^s_{p,q}({\mathbb R^d})$.

Fig. 2

Unconditionality of the Haar system in Hardy–Sobolev spaces in ${\mathbb R}$ and ${\mathbb R^d}$

In the left part of Fig. 2, the trapezoid is the parameter domain for which the Haar system is a Schauder basis in the Hardy–Sobolev space $H^s_p(\mathbb {R})$ ($= F^s_{p,2} ({\mathbb {R}})$) while the shaded part represents the parameter domain for which the Haar system is an unconditional basis in $H^s_p(\mathbb {R})$. The right figure shows the respective parameter domain for $H^s_p(\mathbb {R}^d)$.

The heart of the matter is a boundedness result for the dyadic averaging operators ${\mathbb {E}}_N$ given by

$$\begin{aligned} {\mathbb {E}}_N f(x)= \sum _{\mu \in {\mathbb {Z}}^d} {\mathbb 1}_{I_{N,\mu }} (x) \, 2^N \int _{I_{N,\mu }} f(t) dt\, \end{aligned}$$

(5)

with

$$\begin{aligned} I_{N,\mu }=2^{-N}(\mu +[0,1)^d),\quad \mu \in {\mathbb Z^d},\;N=0,1,2,\ldots \end{aligned}$$

Note that ${\mathbb {E}}_N f$ is just the conditional expectation of f with respect to the $\sigma$-algebra generated by the set ${\mathscr {D}}_N$ of all dyadic cubes of side length $2^{-N}$. There is a well known relation between the Haar system and the dyadic averaging operators, namely for $N=0,1,2,\dots$,

$$\begin{aligned} {\mathbb {E}}_{N+1} f-{\mathbb {E}}_Nf = \sum _{{\varvec{{\varepsilon }}}\in \Upsilon }\sum _{\mu \in {\mathbb {Z}}^d} 2^{Nd} \langle f, h^{({\varvec{{\varepsilon }}})}_{N,\mu } \rangle h^{({\varvec{{\varepsilon }}})}_{N,\mu }, \end{aligned}$$

(6)

i.e. ${\mathbb {E}}_{N+1} -{\mathbb {E}}_N$ is the orthogonal projection onto the space generated by the Haar functions with Haar frequency $2^N$.

Now let $\eta _0$ be a Schwartz function on ${\mathbb {R}}^d$, supported in $\{|\xi |<3/8\}$ and so that $\eta _0(\xi )=1$ for $|\xi |\le 1/4$. Let $\varPi _N$ be defined by

$$\begin{aligned} \widehat{ \varPi _N f}(\xi )= \eta _0(2^{-N}\xi )\widehat{f}(\xi ). \end{aligned}$$

(7)

There is a basic standard inequality (almost immediate from the definition of Triebel–Lizorkin spaces)

$$\begin{aligned} \sup _N\Vert \varPi _N f\Vert _{F^s_{p,q}}\le C(p,q,s)\Vert f\Vert _{F^s_{p,q}} \end{aligned}$$

(8)

which is valid for all $s\in {\mathbb {R}}$ and for $0<p<\infty$, $0<q\le \infty$. Moreover, (8) and the fact that $\Vert \varPi _Ng-g\Vert _{F^s_{p,q}}\rightarrow 0$ for Schwartz functions g gives

$$\begin{aligned} \lim _{N\rightarrow \infty } \Vert \varPi _Nf-f\Vert _{F^s_{p,q}} = 0 \end{aligned}$$

(9)

if $f\in F^s_{p,q}$ and $0<p,q<\infty$. The main tool in proving Theorem 1.1 is a similar bound for the operators ${\mathbb {E}}_N$ which of course follows from the corresponding bound for ${\mathbb {E}}_N-\varPi _N$. It turns out that the operators ${\mathbb {E}}_N-\varPi _N$ enjoy better mapping properties in Besov spaces.

Similar bounds are also satisfied by projection operators into sets of Haar functions with fixed Haar frequency. Namely, for $N\in {\mathbb {N}}$ and functions $a\in \ell ^\infty ({\mathbb {Z}}^d\times \Upsilon )$, we define

$$\begin{aligned} T_{N}[f,a] = \sum _{{\varvec{{\varepsilon }}}\in \Upsilon }\sum _{\mu \in {\mathbb {Z}}^d} a_{\mu ,{\varvec{{\varepsilon }}}} 2^{Nd}\langle f,h^{({\varvec{{\varepsilon }}})}_{N,\mu }\rangle h^{({\varvec{{\varepsilon }}})}_{N,\mu }. \end{aligned}$$

(10)

Observe that the choice $a_{\mu ,{\varvec{{\varepsilon }}}}\equiv 1$ recovers the operator ${\mathbb {E}}_{N+1}-{\mathbb {E}}_N$. Then, we shall prove the following.

Theorem 1.2

Let ${d}/{(d+1)}<p\le \infty$, $0<r\le \infty$, and

$$\begin{aligned} \max \{d(1/p-1),1/p-1\}<s< \min \{1,1/p\}. \end{aligned}$$

(11)

Then there is a constant $C:=C(p,r,s)>0$ such that for all $f\in B^s_{p,\infty }$

$$\begin{aligned} \sup _N \Vert {\mathbb {E}}_Nf -\varPi _Nf \Vert _{B^s_{p,r}} \le C\Vert f\Vert _{B^s_{p,\infty }}. \end{aligned}$$

(12)

Moreover,

$$\begin{aligned} \sup _N \Vert T_N[f,a]\Vert _{B^s_{p,r}}\lesssim \Vert a\Vert _\infty \Vert f\Vert _{B^s_{p,\infty }}\,. \end{aligned}$$

(13)

We have the embedding $F^s_{p,q}\subset F^s_{p,\infty }\subset B^s_{p,\infty }$ which we use on the function side. For $r\le p$ we have the embedding $B^s_{p,r}\subset F^s_{p,r}$ (by Minkowski’s inequality in $L^{p/r}$) and if also $r<q$ we have $F^s_{p,r}\subset F^s_{p,q}$; these two are used for ${\mathbb {E}}_N f-\varPi _N f$, or $T_N[f,a]$. In particular we conclude from Theorem 1.2 that ${\mathbb {E}}_N-\varPi _N$ is bounded on $F^s_{p,q}$, uniformly in N. Hence

Corollary 1.3

Let p, s be as in (11) and $0<q\le \infty$. Then

$$\begin{aligned} \sup _N \Vert {\mathbb {E}}_Nf \Vert _{F^s_{p,q}}+ \sup _N\sup _{\Vert a\Vert _{\ell ^\infty }\le 1} \Vert T_N[f,a]\Vert _{F^s_{p,q}} \lesssim \Vert f\Vert _{F^s_{p,q}}. \end{aligned}$$

(14)

The proofs in this paper use basic principles in the theory of function spaces, such as $L^p$ inequalities for the Peetre maximal functions. A different approach to Corollary 1.3 via wavelet theory is presented in the subsequent paper [2]. The main arguments and the proof of Theorem 1.2 are contained in Sect. 2. In Sect. 3 we show how estimates in the proof of Theorem 1.2 are used to deduce Theorem 1.1. Finally, in Sect. 4 we discuss the optimality of the results.

2 Proof of Theorem 1.2

We start with some preliminaries on convolution kernels which are used in Littlewood-Paley type decompositions. Let $\beta _0, \beta$ be Schwartz functions on ${\mathbb R^d}$, compactly supported in $(-1/2,1/2)^d$ such that $|\widehat{\beta }_0(\xi )|>0$ when $|\xi |\le 1$ and $|\widehat{\beta }(\xi )|>0$ when $1/8\le |\xi |\le 1$. Moreover assume $\beta$ has vanishing moments up to a large order

$$\begin{aligned} M> \frac{d}{p} +|s|, \end{aligned}$$

(15)

that is,

$$\begin{aligned} \int _{{\mathbb R^d}} \beta (x)\,x_1^{m_1}\cdots x_d^{m_d}\,dx = 0\quad \text{ when }\quad m_1+\ldots +m_d < M\,. \end{aligned}$$

(16)

For $k=1,2,\dots$ let $\beta _k:=2^{kd}\beta (2^k\cdot )$ and $L_k f=\beta _k*f$. We shall use the inequality

$$\begin{aligned} \Vert g\Vert _{B^s_{p,r}}\lesssim \Big (\sum _{k=0}^\infty 2^{ksr}\Vert L_k g\Vert _p^r\Big )^{1/r} \end{aligned}$$

(17)

and apply it to $g={\mathbb {E}}_Nf-\varPi _N f$. Inequality (17) is of course just one part of a characterization of $B^s_{p,r}$ spaces by sequences of compactly supported kernels (or ‘local means’), with sufficient cancellation assumptions, see for example [10, Sect. 2.5.3].

Let $\eta _0\in C^\infty _c({\mathbb R^d})$ be as in (7), that is, supported on $\{|\xi |<3/8\}$ and such that $\eta _0(\xi )=1$ when $|\xi |\le 1/4$. Define $\Lambda _0$, and $\Lambda _k$ for $k\ge 1$ by

$$\begin{aligned} \widehat{\Lambda _0 f}(\xi )&=\frac{\eta _0(\xi )}{\widehat{\beta }_0(\xi )}\widehat{f}(\xi ) \\ \widehat{\Lambda _k f}(\xi )&=\frac{\eta _0(2^{-k}\xi ) -\eta _0(2^{-k+1}\xi )}{\widehat{\beta }(2^{-k}\xi )}\widehat{f}(\xi ), \quad k\ge 1. \end{aligned}$$

Then $\sum _{j=0}^\infty L_j \Lambda _j=\text { Id}$ with convergence in ${\mathcal {S}}'$, and

$$\begin{aligned} \sup _{j\ge 0}2^{js}\Vert \Lambda _j f\Vert _p\lesssim \Vert f\Vert _{B^s_{p,\infty }}\,. \end{aligned}$$

Moreover $\varPi _N = \sum _{j=0}^N L_j\Lambda _j$, and therefore

$$\begin{aligned} {{\mathbb {E}}}_{N} f-\varPi _{N} f= \sum _{j=0}^N( {{\mathbb {E}}}_{N} L_{j}\Lambda _{j} f - L_{j}\Lambda _{j} f) +\sum _{j={N+1}}^\infty {\mathbb {E}}_N L_j\Lambda _j f. \end{aligned}$$

(18)

If we use the convenient notation

$$\begin{aligned} {{\mathbb {E}}_N^\perp }:= I-{\mathbb {E}}_N, \end{aligned}$$

then the asserted estimate (12) will follow from

$$\begin{aligned} \Big (\sum _{k=0}^\infty 2^{ksr} \Big \Vert \sum _{j=N+1}^\infty L_k {\mathbb {E}}_N L_j\Lambda _j f \Big \Vert _p^r\Big )^{1/r} \lesssim \sup _j 2^{js}\Vert \Lambda _j f\Vert _p\,. \end{aligned}$$

(19)

and

$$\begin{aligned} \Big (\sum _{k=0}^\infty 2^{ksr} \Big \Vert \sum _{j=0}^N L_k {{\mathbb {E}}_N^\perp }L_j\Lambda _j f \Big \Vert _p^r\Big )^{1/r} \lesssim \sup _j 2^{js}\Vert \Lambda _j f\Vert _p\,. \end{aligned}$$

(20)

Below we shall use variants of the Peetre maximal functions, which are a standard tool in the study of Besov and Triebel–izorkin spaces. We define

$$\begin{aligned} {\mathfrak {M}}_j g(x)&= \sup _{|h|_\infty \le 2^{-j+1}} |g(x+h)|\,, \end{aligned}$$

(21a)

$$\begin{aligned} {\mathfrak {M}}_j^* g(x)&= \sup _{|h|_\infty \le 2^{-j+5}} |g(x+h)|\,, \end{aligned}$$

(21b)

$$\begin{aligned} {\mathfrak {M}}_{A,j}^{**} g(x)&= \sup _{h\in {\mathbb R^d}}\frac{ |g(x+h)|}{(1+2^j|h|)^A}\,, \end{aligned}$$

(21c)

where $|h|_\infty =\max \{|h_1|,\ldots ,|h_d|\}$, $h=(h_1,\ldots ,h_d)\in {\mathbb R^d}$. These different versions are introduced for technical purposes in the proofs. They satisfy obvious pointwise inequalities,

$$\begin{aligned} {\mathfrak {M}}_j g(x)\le {\mathfrak {M}}_j^* g(x) \le C_A {\mathfrak {M}}_{A,j}^{**} g(x), \end{aligned}$$

and

$$\begin{aligned}&{\mathfrak {M}}_j g(x) \le \inf _{|h|_\infty \le 2^{-j+4}} {\mathfrak {M}}_j^* g(x+h) \nonumber \\&\le \Big (2^{(j-4)d} \int _{|h|_\infty \le 2^{-j+4}} [{\mathfrak {M}}_j ^*g(x+h)]^r dh\Big )^{1/r},\quad 0<r\le \infty . \end{aligned}$$

(22)

Below we shall use Peetre’s inequality ([3], see also [9, Sect. 1.3.1])

$$\begin{aligned} \Vert {\mathfrak {M}}_{A,j}^{**} f\Vert _p\le C_{p,A} \Vert f\Vert _p , {\quad 0<p\le \infty , \quad A>d/p, } \end{aligned}$$

(23)

for $f\in {\mathcal {S}}'({\mathbb {R}}^d)$ satisfying

$$\begin{aligned} {\mathrm{supp}}(\widehat{f}) \subset \{\xi : |\xi |\le 2^{j+1}\} . \end{aligned}$$

(24)

Throughout we shall assume that $M\gg A$; we require specifically

$$\begin{aligned} d/p<A< M-|s| \,. \end{aligned}$$

The main estimates needed in the proof of (19) and (20) are summarized in

Proposition 2.1

Let $0<p\le \infty$ and

$$\begin{aligned} B(j,k,N)= {\left\{ \begin{array}{ll} 2^{N-j}\,2^{\frac{j-k}{p}}\, 2^{(j-N)(d-1)(\frac{1}{p}-1)_+} &{}\text { if } j,k\ge N+1, \\ 2^{\frac{N-k}{p}} 2^{j-N} &{}\text { if } j\le N, \,\, k\ge N+1, \\ 2^{k-N} 2^{j-N}2^{(N-k)d(\frac{1}{p}-1)_+} &{}\text { if } 0\le j,k\le N, \\ 2^{k-j+\frac{j-N}{p}+[N-k+(j-k)(d-1)](\frac{1}{p}-1)_+} &{}\text { if } j\ge N+1, \,\, k\le N. \end{array}\right. } \end{aligned}$$

(25)

Then the following inequalities hold for all $f\in {\mathcal {S}}'({\mathbb {R}}^d)$ whose Fourier transform is supported in $\{|\xi |\le 2^{j+1}\}$.

• (i) For $j\ge N+1$,

  $$\begin{aligned} \Vert L_k{\mathbb {E}}_N[L_j f]\Vert _p \lesssim \left\{ \begin{array}{ll} B(j,k,N) \Vert f\Vert _p &{}\text { if } k\ge N+1, \\ \,[B(j,k,N)+ 2^{-|j-k|(M-A)}]\Vert f\Vert _p &{}\text { if } 0\le k\le N. \end{array}\right. \end{aligned}$$

  (26)

• (ii) For $0\le j\le N$,

  $$\begin{aligned} \Vert L_k{{\mathbb {E}}_N^\perp }[L_j f]\Vert _p \lesssim \left\{ \begin{array}{ll} \big [B(j,k,N)+ 2^{-|j-k|(M-A)}\big ] \Vert f\Vert _p &{}\text { if } k\ge N+1, \\ B(j,k,N) \Vert f\Vert _p &{}\text { if } 0\le k\le N. \end{array}\right. \end{aligned}$$

  (27)

• (iii) The same bounds hold if the operators ${\mathbb {E}}_N$ in (i) and ${\mathbb {E}}_N^\perp$ in (ii) are replaced by $T_N[\cdot ,a]$, uniformly in $\Vert a\Vert _\infty \le 1$.

We begin with two preliminary lemmata, the first a straightforward estimate for $L_kL_j$.

Lemma 2.2

Let $k,j\ge 0$ and suppose that f is locally integrable. Let M be as in (16) with $M>A>d/p$. Then

$$\begin{aligned} |L_k L_j f(x) |\lesssim 2^{-|k-j|(M-A)} {\mathfrak {M}}_{A,\max \{j,k\}}^{**} f(x). \end{aligned}$$

(28)

If $f\in {\mathcal {S}}'({\mathbb R^d})$ with $\widehat{f}(\xi )=0$ for $|\xi |\ge 2^{j+1}$ then

$$\begin{aligned} \Vert L_kL_j f\Vert _p \lesssim 2^{-|k-j|(M-A)} \Vert f\Vert _p\,. \end{aligned}$$

Proof

The second assertion is an immediate consequence of (28), by (23). We have $L_kL_jf=\gamma _{j,k}*f$ where $\gamma _{j,k}=\beta _k*\beta _j$. By symmetry we may assume $k\le j$. Using the cancellation assumption (16) on the $\beta _j$ we get

$$\begin{aligned}&|\gamma _{j,k}(x)|= \Big |\int 2^{kd} \Big [ \beta (2^k(x-y) -\sum _{m=0}^{M-1} \frac{1}{m!}\langle -2^ky,\nabla \rangle ^m \!\beta (2^kx)\Big ] 2^{jd} \beta (2^jy) dy\Big | \\&= \Big |\int 2^{kd} \int _0^1 \frac{(1-s)^{M-1}}{(M-1)!} \langle -2^ky,\nabla \rangle ^M\!\beta (2^kx-s2^ky)\, ds\, 2^{jd} \beta (2^jy) dy\Big | \\ {}&\lesssim 2^{(k-j)M}\,2^{kd} \,{\mathbb 1}_{[-2^{-k},2^{-k}]^d}(x), \end{aligned}$$

and thus

$$\begin{aligned}&2^{(j-k)M} |\gamma _{j,k}*f(x) |\lesssim 2^{kd} \int _{|h|_\infty \le 2^{-{k}}}|f(x-h)|\, dh \\&\quad \lesssim 2^{kd}\int _{|h|_\infty \le 2^{-k}} \frac{2^{(j-k)A}|f(x-h)|}{(1+2^j|h|)^A} dh \lesssim 2^{(j-k)A}\,{\mathfrak {M}}_{A,j}^{**} f(x). \end{aligned}$$

Hence (28) holds. $\square$

2.1 Some Notation

• (i) Below, when $j>N$ we use the notation

  $$\begin{aligned} {\mathcal {U}}_{N,j} =\Big \{(y_1,\ldots ,y_d)\in {\mathbb R^d}\mid \min _{1\le i\le d}{\mathrm{dist}}(y_i, 2^{-N}{\mathbb {Z}})\le 2^{-j-1}\Big \}. \end{aligned}$$

  That is, ${\mathcal {U}}_{N,j}$ is a $2^{-j-1}$-neighborhood of the set $\cup _{I\in {\mathscr {D}}_N}\;\partial I$.

• (ii) For a dyadic cube I of side length $2^{-N}$ and $l>N$ we shall denote by ${\mathscr {D}}_l[\partial I]$ the set of dyadic cubes $J\in {\mathscr {D}}_{l}$ such that ${\bar{J}}\cap \partial I\not =\emptyset$.

• (iii) For a dyadic cube I of side length $2^{-N}$ denote by ${\mathscr {D}}_N(I)$ the neighboring cubes of I, that is, the cubes $I'\in {\mathscr {D}}_N$ with ${\bar{I}}\cap {\bar{I'}}\not =\emptyset$.

Lemma 2.3

• (i) Let $k>N\ge 1$ and g be locally integrable. Then

  $$\begin{aligned} L_k ({\mathbb {E}}_N g)(x)=0, \quad \text { for all } x\in {\mathcal {U}}_{N,k}^\complement ={\mathbb {R}}^d\setminus {\mathcal {U}}_{N,k}\,. \end{aligned}$$

  (29)

• (ii) Let $j>N\ge 1$, and f locally integrable. Then

  $$\begin{aligned} {\mathbb {E}}_N [L_jf]={\mathbb {E}}_N[L_j({\mathbb 1}_{{\mathcal {U}}_{N,j}} f)]. \end{aligned}$$

  (30)

  Moreover,

  $$\begin{aligned} \big |{\mathbb {E}}_N(L_jf)\big |\lesssim 2^{(N-j)d}\sum _{I\in {\mathscr {D}}_N} \sum _{J\in {\mathscr {D}}_{j+1}[\partial I]}\Vert f\Vert _{L^\infty (J)}\,{\mathbb 1}_I. \end{aligned}$$

  (31)

Proof

• (i) We use the support and cancellation properties of $\beta _k$. Note that

  $$\begin{aligned} L_k({\mathbb {E}}_N g)(x)=\int \beta _k(x-y)\,{\mathbb {E}}_Ng(y)\,dy, \end{aligned}$$

  and ${\mathrm{supp}}\beta _k(x-\cdot )\subset x+2^{-k}[-1/2,1/2]^d$. So, if $I\in {\mathscr {D}}_N$ and $x\in I\cap {\mathcal {U}}_{N,k}^\complement$, then ${\mathrm{supp}}\beta _k(x-\cdot )\subset I$, and hence

  $$\begin{aligned} L_k({\mathbb {E}}_N g)(x)=({\mathbb {E}}_Ng)_{|_I}(x)\,\int _I \beta _k(x-y)\,dy\,=\,0. \end{aligned}$$

• (ii) One argues similarly. First note that, changing the order of integration,

  $$\begin{aligned} {\mathbb {E}}_N(L_jf)=\sum _{I\in {\mathscr {D}}_N}\int _{{\mathbb R^d}}f(y)\Big [\int _I\beta _j(x-y)\,dx\Big ]\;dy\;\frac{{\mathbb 1}_I}{|I|}. \end{aligned}$$

  (32)

  Now if $J\in {\mathscr {D}}_N$ and $y\in J\cap {\mathcal {U}}_{N,k}^\complement$ then ${\mathrm{supp}}\beta _j(\cdot -y) \subset J$, and hence $\int _I\beta _j(x-y)\,dx=0$. Thus ${\mathbb {E}}_N[L_j({\mathbb 1}_{{\mathcal {U}}_{N,j}^\complement }f)]=0$. Finally, to prove (31) note that, if $I\in {\mathscr {D}}_N$ and $x\in I$, then from (32) it follows

  $$\begin{aligned} \big |{\mathbb {E}}_N(L_jf)(x)\big |= & {} |I|^{-1}\;\Big |\sum _{J\in {\mathscr {D}}_{j+1}[\partial I]} \int _{J}f(y)\Big [\int _I\beta _j(x-y)\,dx\Big ]\;dy\Big |\;\\\le & {} 2^{Nd}\sum _{J\in {\mathscr {D}}_{j+1}[\partial I]}\Vert f\Vert _{L^\infty (J)} 2^{-(j+1)d}\Vert \beta _j\Vert _1, \end{aligned}$$

  which gives the asserted (31).

$\square$

2.2 Proof of Proposition 2.1

2.2.1 Proof of (26) in the Case $j,k\ge N+1$

By Lemma 2.3. i, $L_k{\mathbb {E}}_N[L_j f](x)=0$ if $x\in {\mathcal {U}}_{N,K}^\complement$, so we assume that $x\in {\mathcal {U}}_{N,k}\cap I$, for some $I\in {\mathscr {D}}_N$. Recall that ${\mathscr {D}}_N(I)$ consists of the neighboring cubes of I. Then (31) and the support property of $\beta _k$ give

$$\begin{aligned} |L_k{\mathbb {E}}_N[L_j f](x)|\le & {} \int |\beta _k(x-y)|\,\big |{\mathbb {E}}_N(L_jf)(y)\big |\,dy\\\lesssim & {} 2^{(N-j)d} \sum _{I'\in {\mathscr {D}}_N(I)}\sum _{J\in {\mathscr {D}}_{j+1}[\partial I']} \Vert f\Vert _{L^\infty (J)}\,\Vert \beta _k\Vert _{1}. \end{aligned}$$

Hence

$$\begin{aligned} \Vert L_k{\mathbb {E}}_N[L_j f]\Vert _p= & {} \Big [\sum _{I\in {\mathscr {D}}_N}\int _{I\cap {\mathcal {U}}_{N,k}}|L_k({\mathbb {E}}_N L_jf)|^p\,dx\Big ]^{\frac{1}{p}}\nonumber \\\lesssim & {} 2^{(N-j)d}\Big [\sum _{I\in {\mathscr {D}}_N} \Big (\sum _{J\in {\mathscr {D}}_{j+1}[\partial I]}\Vert f\Vert _{L^\infty (J)}\Big )^p \,|I\cap {\mathcal {U}}_{N,k}|\Big ]^\frac{1}{p}. \end{aligned}$$

(33)

Now, $|I\cap {\mathcal {U}}_{N,k}|\approx 2^{-k}2^{-N(d-1)}$, and $\mathrm{card}\,{\mathscr {D}}_{j+1}[\partial I]\approx 2^{(j-N)(d-1)}$. Also, if we write $J=2^{-j-1}(\ell _J+[0,1]^d)$, then

$$\begin{aligned} \Vert f\Vert _{L^\infty (J)}\le \inf _{|h|_\infty \le 2^{-j-1}}{\mathfrak {M}}^*_jf(\ell _J+h) \le \Big [2^{jd}\int _{|h|_\infty \le 2^{-j-1}}{\mathfrak {M}}^*_jf(\ell _J+h)^p\,dh\Big ]^{\frac{1}{p}}. \end{aligned}$$

Therefore, using either Hölder’s inequality (if $p>1$), or the embedding $\ell ^p\hookrightarrow \ell ^1$ (if $p\le 1$), we have

$$\begin{aligned}&\Big [\sum _{I\in {\mathscr {D}}_N} \Big (\!\!\!\sum _{J\in {\mathscr {D}}_{j+1}[\partial I]}\Vert f\Vert _{L^\infty (J)}\Big )^p \Big ]^\frac{1}{p} \nonumber \\&\lesssim 2^{(j-N)(d-1)(1-\frac{1}{p})_+}\Big [\sum _{I\in {\mathscr {D}}_N}\sum _{J\in {\mathscr {D}}_{j+1}[\partial I]} \Vert f\Vert _{L^\infty (J)}^p\Big ]^\frac{1}{p}\nonumber \\&\lesssim 2^{(j-N)(d-1)(1-\frac{1}{p})_+}\,\Big [\sum _{J\in {\mathscr {D}}_{j+1}} 2^{jd}\int _{|h|_\infty \le 2^{-j-1}}{\mathfrak {M}}^*_jf(\ell _J+h)^p\,dh\Big ]^\frac{1}{p}\nonumber \\&\lesssim 2^{(j-N)(d-1)(1-\frac{1}{p})_+}\, 2^{\frac{jd}{p}}\,\big \Vert {\mathfrak {M}}^*_jf\big \Vert _{L^p({\mathbb R^d})}\,. \end{aligned}$$

(34)

Finally, inserting (34) into (33), and using (23), yields

$$\begin{aligned} \Vert L_k{\mathbb {E}}_N[L_j f]\Vert _p\lesssim & {} 2^{(N-j)d} 2^{(j-N)(d-1)(1-\frac{1}{p})_+}\, 2^{\frac{jd}{p}}\,\Vert f\Vert _p\, 2^{-\frac{k}{p}}2^{-\frac{N(d-1)}{p}}\\= & {} 2^{N-j}\,2^{\frac{j-k}{p}}\, 2^{(j-N)(d-1)(\frac{1}{p}-1)_+}\,\Vert f\Vert _p\,, \end{aligned}$$

using in the last step the trivial identity $(1-\frac{1}{p})_+=(\frac{1}{p}-1)_+-(\frac{1}{p}-1)$. This establishes (26) for $j,k\ge N+1$. $\square$

2.2.2 Proof of (27) in the Case $j\le N$, $k\ge N+1$

For $w\in I$ with $I\in {\mathscr {D}}_N$ we have

$$\begin{aligned}&\big |{\mathbb {E}}_N^\perp (L_jf)(w)\big |=\big |{\mathbb {E}}_N[L_j f](w)-L_jf(w)\big |\\&\quad =2^{Nd}\Big |\int _{I} \int _{{\mathbb R^d}} 2^{jd}\big [\beta (2^j(v-y))-\beta (2^j(w-y))\big ] f(y) dy\, dv\Big | \\&\quad = 2^{(N+j)d} \Big | \int _{I} \int _{{\mathbb R^d}} \int _0^1 \nabla \beta (2^j[(1-t)w+tv-y])\cdot 2^j(v-w)\,dt\,f(y)\, dy \, dv \Big | \\&\quad \le 2^{(N+j)d} 2^{j-N} \,\int _I\,\int _0^1\,\int _{{\mathbb R^d}}|f(y)|\, |\nabla \beta (2^j[(1-t)w+tv-y])|\,dy\,dt\,dv \\&\quad \lesssim 2^{j-N} \,{\mathfrak {M}}_jf(w) , \end{aligned}$$

since for fixed w, v, t the expression involving $\nabla \beta$ is supported in the set $\{y\mid |y-w|_\infty \le 2^{-j-1}+2^{-N}\}$. Moreover, since $k>N$, when $w\in I_{N,\mu }$ and $|z|_\infty \le 2^{-k-1}$ we have

$$\begin{aligned} \big |{\mathbb {E}}_N^\perp [L_jf](w-z)\big | \lesssim 2^{j-N} \inf _{|h|_\infty \le 2^{-j}} {\mathfrak {M}}_j^* f(2^{-N}\mu +h)\,, \end{aligned}$$

(35)

and therefore,

(36)

Now Lemma 2.3. i gives

$$\begin{aligned}&\big \Vert L_k\big ({\mathbb {E}}_N^\perp [L_jf]\big )\big \Vert _p\nonumber \\&\lesssim \Vert L_kL_jf\Vert _{L^p({\mathcal {U}}_{N,k}^\complement )}+ \Big [\sum _{\mu \in {\mathbb Z^d}} \Vert L_k\big ({\mathbb {E}}_N^\perp [L_jf]\big )\big \Vert _{L^p({\mathcal {U}}_{N,k}\cap I_{N,\mu })}^p \Big ]^{\frac{1}{p}}. \end{aligned}$$

(37)

Using (36), the last term is controlled by

Finally, the first term in (37) is controlled by Lemma 2.2, so overall one obtains

$$\begin{aligned} \big \Vert L_k\big ({\mathbb {E}}_N^\perp [L_jf]\big )\big \Vert _p\lesssim \left[ 2^{-(M-A)|k-j|}+2^{j-N}\,2^{\frac{N-k}{p}}\right] \,\Vert f\Vert _p, \end{aligned}$$

establishing (27) in the case $j\le N$, $k\ge N+1$. $\square$

2.2.3 Proof of (27) in the Case $0\le j,k\le N$

We use

$$\begin{aligned} \int _{I} {\mathbb {E}}_N^\perp [ L_jf](y) \,dy=0, \quad I\in {\mathscr {D}}_N, \end{aligned}$$

to write

$$\begin{aligned} L_k\big ( {\mathbb {E}}_N^\perp [ L_jf]\big )(x)=\sum _\mu \int _{I_{N,\mu } } \big (\beta _k(x-y) -\beta _k(x-2^{-N}\mu )\big )\, {\mathbb {E}}_N^\perp [ L_jf](y)\, dy\,. \end{aligned}$$

For fixed x, we say that

$$\begin{aligned} \mu \in \Lambda (x) \ \text {if}\ |x-2^{-N}\mu |_\infty \le 2^{-N}+2^{-k-1}. \end{aligned}$$

(38)

Observe that only these $\mu$’s contribute to the above sum. Notice also that

$$\begin{aligned} |\beta _k(x-y) -\beta _k(x-2^{-N}\mu )|\lesssim 2^{kd}\,2^{k-N},\quad \text {if}\, y\in I_{N,\mu }, \end{aligned}$$

and since $j\le N$, the estimate in (35) gives

$$\begin{aligned} \big |{\mathbb {E}}_N^\perp [L_jf](y)\big | \lesssim 2^{j-N} \inf _{|h|_\infty \le 2^{-j}} {\mathfrak {M}}_j^* f(2^{-N}\mu +h)\,,\quad y\in I_{N,\mu }. \end{aligned}$$

Combining all these bounds we obtain

using in the last step Hölder’s inequality (or $\ell ^p\hookrightarrow \ell ^1$ if $p\le 1$) and the fact that $\mathrm{card}\,\Lambda (x)\approx 2^{(N-k)d}$. Observe also that the $L^p$-quasinorm of the last bracketed expression satisfies

Thus, overall we obtain

$$\begin{aligned} \big \Vert L_k{\mathbb {E}}_N^\perp [ L_jf]\big \Vert _p\lesssim & {} 2^{(k-N)(d+1)} 2^{j-N}2^{(N-k)d(1-\frac{1}{p})_+}\,2^{(N-k)d/p}\,\Vert f\Vert _p\\= & {} 2^{k-N} 2^{j-N}2^{(N-k)d(\frac{1}{p}-1)_+}\,\Vert f\Vert _p, \end{aligned}$$

after simplifying the indices in the last step. This establishes (27) in the case $0\le j,k\le N$. $\square$

2.2.4 Proof of (26) in the Case $j\ge N+1$, $k\le N$

This condition and (30) in Lemma 2.3 imply that ${\mathbb {E}}_N[L_jf]={\mathbb {E}}_N[L_j(f{\mathbb 1}_{{\mathcal {U}}_{N,j}})]$. For simplicity, we denote ${\widetilde{f}}= f{\mathbb 1}_{{\mathcal {U}}_{N,j}}$, and write

$$\begin{aligned} L_k{\mathbb {E}}_N[L_j f] = L_k({\mathbb {E}}_N[L_j {\widetilde{f}}] -L_j {\widetilde{f}}) + L_kL_j {\widetilde{f}}. \end{aligned}$$

(39)

Observe that, by Lemma 2.2,

$$\begin{aligned} \Vert L_kL_j{\widetilde{f}}\Vert _p\lesssim 2^{-(M-A)|j-k|}\,\big \Vert {\mathfrak {M}}^{**}_{A,j}f(x)\big \Vert _p\lesssim 2^{-(M-A)|j-k|}\,\Vert f\Vert _p. \end{aligned}$$

So, we only need to estimate $\Vert L_k{\mathbb {E}}_N^\perp [ L_j{\widetilde{f}}]\Vert _p$. Proceeding as in the proof of the case $j,k\le N$, we write (with $\Lambda (x)$ as in (38))

$$\begin{aligned}&|L_k\big ( {\mathbb {E}}_N^\perp [ L_j{\widetilde{f}}]\big )(x)| \nonumber \\&\quad \le \sum _{\mu \in \Lambda (x)} \int _{I_{N,\mu } } \big |\beta _k(x-y) -\beta _k(x-2^{-N}\mu )\big |\, \big |{\mathbb {E}}_N^\perp [ L_j{\widetilde{f}}](y)\big |\, dy\,\nonumber \\&\quad \lesssim 2^{kd}2^{k-N}\sum _{\mu \in \Lambda (x)} \int _{I_{N,\mu } }\Big (|{\mathbb {E}}_N[L_j{\widetilde{f}}]|\,+\;|L_j({\widetilde{f}})|\Big )\nonumber \\&\quad = {\mathcal {A}}_1(x)+{\mathcal {A}}_2(x). \end{aligned}$$

(40)

Now, using again (31), we have

$$\begin{aligned} |{\mathcal {A}}_1(x)|\lesssim & {} 2^{(k-N)(d+1)}2^{(N-j)d}\sum _{\mu \in \Lambda (x)} \sum _{J\in {\mathscr {D}}_{j+1}[\partial I_{N,\mu }]}\Vert f\Vert _{L^\infty (J)}\nonumber \\\lesssim & {} 2^{k-N}2^{(k-j)d}\,2^{(N-k)d(1-\frac{1}{p})_+}\,\Big [\sum _{\mu \in \Lambda (x)} \big (\sum _{J\in {\mathscr {D}}_{j+1}[\partial I_{N,\mu }]}\Vert f\Vert _{L^\infty (J)}\big )^p\Big ]^\frac{1}{p},\nonumber \\ \end{aligned}$$

(41)

since $\mathrm{card}\,\Lambda (x)\approx 2^{(N-k)d}$. Taking the $L^p$-quasinorm of the last bracketed expression gives

$$\begin{aligned}&\Big [\int _{x\in {\mathbb R^d}}\sum _{\mu \in \Lambda (x)} \big (\!\!\!\sum _{J\in {\mathscr {D}}_{j+1}[\partial I_{N,\mu }]}\!\!\Vert f\Vert _{L^\infty (J)}\big )^p\,dx\Big ]^\frac{1}{p} \nonumber \\&\quad \quad \lesssim \Big [\sum _{I\in {\mathscr {D}}_N} 2^{-kd}\big (\!\!\sum _{J\in {\mathscr {D}}_{j+1}[\partial I]}\Vert f\Vert _{L^\infty (J)}\big )^p\Big ]^\frac{1}{p} \nonumber \\&\quad \quad \lesssim \; 2^{\frac{(j-k)d}{p}}\,2^{(j-N)(d-1)(1-\frac{1}{p})_+}\, \big \Vert {\mathfrak {M}}^*_jf\big \Vert _{L^p({\mathbb R^d})}\, \quad {\mathrm{by \, (34)}}. \end{aligned}$$

(42)

Therefore, combining exponents in (41) and (42) one obtains

$$\begin{aligned} \Vert {\mathcal {A}}_1\Vert _p\lesssim & {} 2^{k-N}2^{(k-j)d}\,2^{(N-k)d(1-\frac{1}{p})_+}\, 2^{\frac{(j-k)d}{p}}\,2^{(j-N)(d-1)(1-\frac{1}{p})_+}\,\Vert f\Vert _p\nonumber \\= & {} 2^{k-j}\,2^{\frac{j-N}{p}}\,2^{(N-k)(\frac{1}{p}-1)_+}\,2^{(j-k)(d-1)(\frac{1}{p}-1)_+}\,\Vert f\Vert _p. \end{aligned}$$

(43)

Finally, we estimate the term ${\mathcal {A}}_2(x)$ in (40). First notice that

$$\begin{aligned} |L_j({\widetilde{f}})(y)|\le \int _{{\mathcal {U}}_{N,{j}}}|\beta _j(y-z)||f(z)|\,dz=0, \quad \text {if } y\in {\mathcal {U}}_{N,{j-1}}^\complement , \end{aligned}$$

since ${\mathrm{supp}}\beta _j(y-\cdot )\subset y+2^{-j}[-\frac{1}{2},\frac{1}{2}]^d \subset {\mathcal {U}}_{N,{j}}^\complement$. Moreover, if $I\in {\mathscr {D}}_N$, then for every cube $J\in {\mathscr {D}}_j$ such that $J\subset I\cap {\mathcal {U}}_{N,j-1}$ we have

$$\begin{aligned} |L_j({\widetilde{f}})(y)|\le \int |\beta _j(z)||f(y-z)|\,dz\lesssim \Vert f\Vert _{L^\infty (J^*)}, \quad \text {if } y\in J \end{aligned}$$

where $J^*=J+2^{-j}[-\frac{1}{2},\frac{1}{2}]^d$. Therefore,

$$\begin{aligned} \int _I|L_j({\widetilde{f}})(y)|\lesssim \sum _{J\in {\mathscr {D}}_{j}[\partial I]}\Vert f\Vert _{L^\infty (J^*)}|J|, \end{aligned}$$

and overall we obtain

$$\begin{aligned} |{\mathcal {A}}_2(x)|\lesssim 2^{(k-j)d}2^{k-N}\,\sum _{\mu \in \Lambda (x)} \sum _{J\in {\mathscr {D}}_{j-1}[\partial I_{N,\mu }]}\Vert f\Vert _{L^\infty (J)}. \end{aligned}$$

But this is essentially the same expression we obtained in (41) for the term $|{\mathcal {A}}_1(x)|$, so the same argument will give an estimate of $\Vert {\mathcal {A}}_2\Vert _p$ in terms of the quantity in (43). This concludes the proof of (26) in the case $j\ge N+1$, $k\le N$.

Finally, concerning (iii) in Proposition 2.1, we remark that the previous proofs can easily be adapted replacing the operators ${\mathbb {E}}_N$ and ${\mathbb {E}}_N^\perp$ by $T_N[\cdot ,a]$, keeping in mind that $T_N[g,a]$ is now constant in cubes $I\in {\mathscr {D}}_{N+1}$, and enjoys an additional cancellation, $\int _{I_{N,\mu }}T_N[g,a](x) dx=0$, which simplifies some of the previous steps. $\square$

2.3 Proof of Theorem 1.2, Conclusion

It remains to prove inequalities (19) and (20). By the embedding properties for the sequence spaces $\ell ^r$ it suffices to verify both inequalities for very small r, say

$$\begin{aligned} r\le \min \{p,1\}. \end{aligned}$$

(44)

In view of the embedding $\ell ^r\hookrightarrow \ell ^1$ and Minkowski’s inequality (in $L^{p/r}$) it suffices then to prove

$$\begin{aligned} \sup _N\Big (\sum _{k=0}^\infty 2^{ksr} \sum _{j=N+1}^\infty \big \Vert L_k {\mathbb {E}}_N L_j\Lambda _j f \big \Vert _p^r\Big )^{1/r} \lesssim \sup _{j}2^{js} \Vert \Lambda _j f\Vert _p \end{aligned}$$

(45)

and

$$\begin{aligned} \sup _N \Big (\sum _{k=0}^\infty 2^{ksr} \sum _{j=0}^N \big \Vert L_k ({\mathbb {E}}_N^\perp L_j\Lambda _j f )\big \Vert _p^r\Big )^{1/r} \lesssim \sup _{j}2^{js} \Vert \Lambda _j f\Vert _p\,. \end{aligned}$$

(46)

If we apply Proposition 2.1 to each of the functions $\Lambda _jf$, we reduce matters to observe that

$$\begin{aligned} \sup _N \sum _{k=0}^\infty 2^{ksr} \sum _{j=0}^\infty \big [2^{-js}B(j,k,N)\big ]^r< \infty , \end{aligned}$$

(47)

with B(j, k, N) as in (25), and that

$$\begin{aligned} \Big (\sum _{j=N+1}^\infty \sum _{k=0}^N + \sum _{k=N+1}^\infty \sum _{j=0}^N\Big ) 2^{-|j-k|(M-A)} <\infty \end{aligned}$$

which is trivial. The verification of (47) under the assumptions in (11) is also elementary, but we carry out some details to clarify how the conditions on p and s are used.

When $j,k>N$, we have $B(j,k,N)=2^{N-j}\,2^{\frac{j-k}{p}}\, 2^{(j-N)(d-1)(\frac{1}{p}-1)_+}$ and thus

$$\begin{aligned} \nonumber&\sum _{k>N}2^{ksr} \sum _{j>N}\big [2^{-js}B(j,k,N)\big ]^r \\&\quad = \Big (\sum _{k>N}2^{-kr(\frac{1}{p}-s)}\Big )\Big (\sum _{j>N}2^{-rj[s+1-\frac{1}{p}-(d-1)(\frac{1}{p}-1)_+]}\Big )\,2^{Nr[1-(d-1)(\frac{1}{p}-1)_+]}, \end{aligned}$$

(48)

and the series converge provided $s<1/p$ and

$$\begin{aligned} s>\tfrac{1}{p}-1 + (d-1)(\tfrac{1}{p}-1)_+\,=\, \max \Big \{d(\tfrac{1}{p}-1),\tfrac{1}{p}-1\Big \}. \end{aligned}$$

(49)

Further, being geometric sums, the final outcome in (48) is bounded uniformly in N.

Next assume $j\le N<k$, then $B(j,k,N)=2^{\frac{N-k}{p}} 2^{j-N}$ and hence

$$\begin{aligned} \sum _{k>N}2^{ksr} \sum _{j\le N}\big [2^{-js}B(j,k,N)\big ]^r = \Big (\sum _{k>N}2^{-kr(\frac{1}{p}-s)}\Big )\Big (\sum _{j\le N}2^{rj(1-s)}\Big )\,2^{Nr(\frac{1}{p}-1)}, \end{aligned}$$

which are finite expressions provided $s<\min \{1,1/p\}$.

Consider $j,k\le N$, with $B(j,k,N)=2^{k-N} 2^{j-N}2^{(N-k)d(\frac{1}{p}-1)_+}$. Then

$$\begin{aligned} \sum _{k\le N}2^{ksr} \sum _{j\le N}\big [2^{-js}B(j,k,N)\big ]^r=\end{aligned}$$

$$\begin{aligned} = \Big (\sum _{k\le N}2^{kr[s+1-d(\frac{1}{p}-1)_+]}\Big )\Big (\sum _{j\le N}2^{rj(1-s)}\Big )\,2^{-Nr[2-d(\frac{1}{p}-1)_+]}, \end{aligned}$$

which leads to uniform expressions in N under the assumptions $s<1$ and

$$\begin{aligned} s>d(\tfrac{1}{p}-1)_+-1, \end{aligned}$$

(50)

the latter being weaker than (49).

When $k\le N<j$ we have $B(j,k,N)=2^{k-j+\frac{j-N}{p}+[N-k+(j-k)(d-1)](\frac{1}{p}-1)_+}$ and

$$\begin{aligned} \sum _{k\le N}2^{ksr} \sum _{j> N}\big [2^{-js}B(j,k,N)\big ]^r=\end{aligned}$$

$$\begin{aligned} = \Big (\sum _{k\le N}2^{kr[s+1-d(\frac{1}{p}-1)_+]}\Big )\Big (\sum _{j> N} 2^{-rj[s+1-\frac{1}{p}-(d-1)(\frac{1}{p}-1)_+]}\Big )\,2^{-Nr[\frac{1}{p}-(\frac{1}{p}-1)_+]}, \end{aligned}$$

where in the first series we would use (50) and in the second series (49). We have verified (47) in all cases. This finishes the proof of Theorem 1.2. $\square$

3 Schauder Bases

Let $P_N$ be defined as in (3). For the proof of Theorem 1.1 we need to prove that $\Vert P_Nf-f\Vert _{F^s_{p,q}}\rightarrow 0$ for every $f\in F^s_{p,q}$, with (p, s) as in (11) and $0<q<\infty$. We first discuss some preliminaries about localization and pointwise multiplication by characteristic functions of cubes, then prove uniform bounds for the $F^s_{p,q}\rightarrow F^s_{p,q}$ operator norms of the $P_N$ and then establish the asserted limiting property.

3.1 Preliminaries

For $\nu \in {\mathbb {Z}}^d$ let $\chi _\nu$ be the characteristic function of $\nu +[0,1)^d$.

Lemma 3.1

Assume that

$$\begin{aligned} \tfrac{d-1}{d}<p<\infty ,\quad 0<q\le \infty ,{\quad \text{ and }\quad }\max \{d(\tfrac{1}{p}-1),\tfrac{1}{p}-1\}<s<\tfrac{1}{p}. \end{aligned}$$

(51)

Then, the following holds for all $g_\nu$ and $f\in F^s_{p,q}$:

$$\begin{aligned} \Big \Vert \sum _{\nu \in {\mathbb {Z}}^d} \chi _\nu g_\nu \Big \Vert _{F^s_{p,q}} \lesssim \Big (\sum _{\nu \in {\mathbb {Z}}^d} \big \Vert g_\nu \big \Vert _{F^s_{p,q}}^p\Big )^{1/p} \end{aligned}$$

and

$$\begin{aligned} \Big (\sum _{\nu \in {\mathbb {Z}}^d} \big \Vert f\chi _\nu \big \Vert _{F^s_{p,q}}^p\Big )^{1/p}\lesssim \Vert f\Vert _{F^s_{p,q}}\,. \end{aligned}$$

Proof

Let $\varsigma \in C^\infty _c({\mathbb {R}}^d)$ so that ${\mathrm{supp}}(\varsigma )\subset (-1,1)^d$ and $\sum _{\nu \in {\mathbb Z^d}} \varsigma (x-\nu )=1$ for all $x\in {\mathbb {R}}^d$. Let $\varsigma _\nu =\varsigma (\cdot -\nu )$. We have, for all $s\in {\mathbb {R}}$,

$$\begin{aligned} \Vert g\Vert _{F^s_{p,q}} \asymp \Big (\sum _\nu \big \Vert \varsigma _\nu g\big \Vert _{F^s_{p,q}}^p\Big )^{1/p}\,; \end{aligned}$$

(52)

see [10, 2.4.7]. Hence

$$\begin{aligned}&\Big \Vert \sum _{\nu \in {\mathbb Z^d}} \chi _\nu g_\nu \Big \Vert _{F^s_{p,q}} = \Big \Vert \sum _{\nu '}\varsigma _{\nu '} \sum _\nu \chi _\nu g_\nu \Big \Vert _{F^s_{p,q}} \lesssim \Big (\sum _{\nu '} \Big \Vert \varsigma _{\nu '}\!\!\sum _{|\nu -\nu '|_\infty \le 1} \chi _\nu g_\nu \Big \Vert _{F^s_{p,q}}^p \Big )^{1/p} \\&\quad \lesssim \Big (\sum _{\nu '} \sum _{|\nu -\nu '|_\infty \le 1} \Vert g_\nu \Vert _{F^s_{p,q}}^p \Big )^{1/p}\lesssim \Big ( \sum _{\nu } \Vert g_\nu \Vert _{F^s_{p,q}}^p \Big )^{1/p}. \end{aligned}$$

Here we have used that $\varsigma _{\nu '} \chi _\nu$ are pointwise multipliers of $F^s_{p,q}$, with uniform bounds in $(\nu , \nu ')$, in the range given by (51); see [4, Thm. 4.6.3/1]. This proves the first inequality.

For the second inequality we first observe that, by (52),

$$\begin{aligned} \Vert f\chi _{\nu }\Vert _{F^s_{p,q}} \lesssim \Big (\sum \limits _{\nu '} \Vert f\chi _{\nu } \varsigma _{\nu '}\Vert ^p_{F^s_{p,q}}\Big )^{1/p}\,,\quad \nu \in {\mathbb {Z}}^d, \end{aligned}$$

which yields

$$\begin{aligned} \Big (\sum \limits _{\nu }\Vert f\chi _{\nu }\Vert _{F^s_{p,q}}^p\Big )^{1/p}\lesssim & {} \Big (\sum \limits _{\nu }\sum \limits _{\nu '} \Vert f\chi _{\nu } \varsigma _{\nu '}\Vert ^p_{F^s_{p,q}}\Big )^{1/p}\\\lesssim & {} \Big (\sum \limits _{\nu '}\sum \limits _{|\nu -\nu '|_\infty \le 1} \Vert f\chi _{\nu } \varsigma _{\nu '}\Vert ^p_{F^s_{p,q}}\Big )^{1/p}\\\lesssim & {} \Big (\sum \limits _{\nu '}\Vert f\varsigma _{\nu '}\Vert ^p_{F^s_{p,q}}\Big )^{1/p}\lesssim \Vert f\Vert _{F^s_{p,q}}\,, \end{aligned}$$

where we have used the pointwise multiplier assertion [4, Thm. 4.6.3/1] and then again (52) in the last step. $\square$

3.2 Uniform Boundedness of the $P_N$

Observe that by the localization property of the Haar functions we have $P_Nf = \sum _{\nu \in {\mathbb Z^d}} \chi _\nu P_N f= \sum _{\nu } \chi _\nu P_N [f\chi _\nu ].$ Thus by Lemma 3.1

$$\begin{aligned} \Vert P_N f\Vert _{F^s_{p,q}} \lesssim \Big (\sum _\nu \big \Vert P_N[f\chi _\nu ]\big \Vert _{F^s_{p,q}}^p\Big )^{1/p}\,. \end{aligned}$$

Since the enumeration of the Haar system is assumed to be admissible we have

$$\begin{aligned} P_N[f\chi _\nu ]= {\mathbb {E}}_{N_\nu } [f\chi _\nu ]+ T_{N_\nu }[f\chi _\nu , a^{N,\nu }] \end{aligned}$$

(53)

for some $N_\nu \in {\mathbb {N}}$, with $N_\nu \le N$ and appropriate sequences $a^{N,\nu }$ assuming only the values 1 and 0. We remark that for each $\nu$, $N_\nu =N_\nu (N)$ with

$$\begin{aligned} \lim _{N\rightarrow \infty } N_\nu (N)=\infty \,. \end{aligned}$$

(54)

By Theorem 1.2

$$\begin{aligned}&\Big (\sum _\nu \big \Vert P_N[f\chi _\nu ]\big \Vert _{F^s_{p,q}}^p\Big )^{1/p} \\&\quad \lesssim \Big (\sum _\nu \big \Vert {\mathbb {E}}_{N_\nu } [f\chi _\nu ] \big \Vert _{F^s_{p,q}}^p\Big )^{1/p} +\Big (\sum _\nu \big \Vert T_{N_\nu }[f\chi _\nu , a^{N,\nu }] \big \Vert _{F^s_{p,q}}^p\Big )^{1/p}\\&\quad \lesssim \Big (\sum _\nu \big \Vert f\chi _\nu \big \Vert _{F^s_{p,q}}^p\Big )^{1/p} \,\lesssim \, \Vert f\Vert _{F^s_{p,q}}\,, \end{aligned}$$

where for the last inequality we have used Lemma 3.1 again.

Proof (Proof of Theorem 1.1, Conclusion)

[Proof of Theorem 1.1, Conclusion] Let $f\in F^s_{p,q}$, with (p, s) as in (11) and $0<q<\infty$. Let $C=\max \{ 1, \sup _N\Vert P_N\Vert _{F^s_{p,q}\rightarrow F^s_{p,q}} \}.$ Since Schwartz functions are dense in $F^s_{p,q}$ when $0<p,q<\infty$ there is ${\tilde{f}}\in {\mathcal {S}}({\mathbb {R}})$ such that $\Vert f-{\tilde{f}}\Vert _{F^s_{p,q}}<(3C)^{-1}\epsilon$ and hence $\Vert P_N f-P_N {\tilde{f}}\Vert _{F^s_{p,q}}< \epsilon /3$. Choose $s_1$ so that $s<s_1<\max \{1/p,1\}$ then ${\tilde{f}} \in B^{s_1}_{p,q} \hookrightarrow F^s_{p,q}$. Since the Haar system is an unconditional basis on $B^{s_1}_{p,q}$ ([11]) we have $\lim _{N\rightarrow \infty } \Vert P_N{\tilde{f}}-{\tilde{f}}\Vert _{B^{s_1}_{p,q}}=0$ and therefore $\lim _{N\rightarrow \infty } \Vert P_N{\tilde{f}}-{\tilde{f}}\Vert _{F^{s}_{p,q}}=0$. Combining these facts we get $\Vert P_Nf -f\Vert _{F^s_{p,q}}<\epsilon$ for sufficiently large N which shows that $P_N f\rightarrow f$ in $F^s_{p,q}$ . $\square$

4 Optimality Away from the End-Points

Proposition 4.1

Let $0<q<\infty$. Then, the Haar system ${\mathscr {H}}_d$ is not a Schauder basis of $F^s_{p,q}({\mathbb R^d})$ in each of the following cases:

• (i) if $1<p<\infty$ and $s\ge 1/p$ or $s< 1/p-1$,

• (ii) if $d/(d+1)\le p\le 1$ and $s>1$ or $s< d(1/p-1)$,

• (iii) if $0<p<d/(d+1)$ and $s\in {\mathbb R}$.

The same result for the spaces $B^s_{p,q}({\mathbb R^d})$ was proved by Triebel in [8]; see also [11, Proposition 2.24]. Proposition 4.1 can be obtained from this and Theorem 1.1 by suitable interpolation.

Indeed, assertion (i) was already discussed in the paragraph following (4), so we restrict to $p\le 1$. Assume next that ${\mathscr {H}}_d$ is a basis for $F^s_{p,q}$ for some $d/(d+1)<p<1$ and $s>1$ or $s<d(1/p-1)$. By Theorem 1.1, ${\mathscr {H}}_d$ is also a basis for $F^{s_0}_{p,q}$, for any $d(1/p-1)<s_0<1$. By real interpolation, see e.g. [9, Theorem 2.4.2(ii)], for all $0<\theta <1$, the system ${\mathscr {H}}_d$ will then be a basis of

$$\begin{aligned} \big (F^{s_0}_{p,q},F^{s}_{p,q}\big )_{\theta ,q}=B^{s_\theta }_{p,q},\quad \text{ with } s_\theta =(1-\theta )s_0+\theta s. \end{aligned}$$

But when $\theta$ is close to 1 this would contradict Triebel’s result. The remaining cases, $p=1$ and $p\ge d/(d+1)$ can be proved similarly using complex interpolation of F-spaces; see [10, 1.6.7].

We remark that, in the paper [8], the failure of the Schauder basis property in the B-spaces is sometimes due to the fact that span$\,{\mathscr {H}}_d$ fails to be dense in $B^s_{p,q}$. This is the case, for instance, in the region

$$\begin{aligned} (d-1)/d<p<1{\quad \text{ and }\quad }\max \big \{1,d(1/p-1)\big \}<s<1/p; \end{aligned}$$

(55)

see [8, Corollary 2]. Here we show that also a quantitative bound holds, therefore ruling out the possibility that ${\mathscr {H}}_d$ could be a basic sequence.

Proposition 4.2

Let $0<q\le \infty$, and (p, s) be as in (55). Then,

$$\begin{aligned} \Vert {\mathbb {E}}_N\Vert _{B^s_{p,q} \rightarrow B^s_{p,q}}\gtrsim 2^{(s-1)N}. \end{aligned}$$

Proof

Let $\eta \in C^\infty _c({\mathbb R}^d)$ such that $\eta \equiv 1$ on $[-2,2]^d$, and consider the Schwartz function $f(x)=x_1\,\eta (x)$. It suffices to show that

$$\begin{aligned} \big \Vert {\mathbb {E}}_Nf\big \Vert _{B^s_{p,q}}\gtrsim 2^{(s-1)N}. \end{aligned}$$

(56)

Under (55) we have $s>\sigma _p:=d(1/p-1)_+$. Assume first that $s<2$ (which is always the case if $d>1$). Then we can use the equivalence of quasi-norms

$$\begin{aligned} \Vert g\Vert _{{B^s_{p,q}(\mathbb {R}^d)}}\approx \Vert g\Vert _p+\sum _{j=1}^d\Big (\int _0^1\frac{\Vert \Delta ^2_{he_j}g\Vert ^q_p}{h^{sq}}\,\frac{dh}{h}\Big )^{1/q}\,, \end{aligned}$$

with the usual modification in the case $q=\infty$, see [10, 2.6.1]. In particular

$$\begin{aligned} \big \Vert {\mathbb {E}}_Nf\big \Vert _{B^s_{p,q}}\gtrsim \Bigg (\int _0^{2^{-N-1}}\frac{\big \Vert \Delta ^2_{he_1}\big ({\mathbb {E}}_Nf\big )\big \Vert ^q_{L^p([0,1]^d)}}{h^{sq}}\,\frac{dh}{h}\Bigg )^{1/q}. \end{aligned}$$

(57)

Now, it is easily checked that, when $x\in [0,1)^d$, one has

$$\begin{aligned} {\mathbb {E}}_Nf=\sum _{0\le k<2^N}\tfrac{k+1/2}{2^N}{\mathbb 1}_{[\frac{k}{2^N},\frac{k+1}{2^N})\times [0,1)^{d-1}}, \end{aligned}$$

and likewise, if we additionally assume $0<h<2^{-N-1}$, then

$$\begin{aligned} \Delta _{he_1}\big ({\mathbb {E}}_Nf\big )=2^{-N-1}\sum _{k=1}^{2^N}{\mathbb 1}_{[\frac{k}{2^N}-h,\frac{k}{2^N})\times [0,1)^{d-1}}. \end{aligned}$$

and

$$\begin{aligned} \Delta ^2_{he_1}\big ({\mathbb {E}}_Nf\big )=2^{-N-1}\sum _{k=1}^{2^N}\Big [{\mathbb 1}_{[\frac{k}{2^N}-2h,\frac{k}{2^N}-h)\times [0,1)^{d-1}}-{\mathbb 1}_{[\frac{k}{2^N}-h,\frac{k}{2^N})\times [0,1)^{d-1}}\Big ]. \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert \Delta ^2_{he_1}{\mathbb {E}}_Nf\Vert _{L^p([0,1]^d)}= 2^{(N+1)(1/p-1)}\,h^{1/p}, \end{aligned}$$

which, inserted into (57), gives (56). If $d=1$ and $s\ge 2$, one applies a similar argument to the functions $\Delta ^L_{he_1}({\mathbb {E}}_Nf)$ with $L=\lfloor s\rfloor +1$ and $h<2^{-N}/L$. $\square$

By interpolation one obtains as well a quantitative bound for the relevant cases in Proposition 4.1(ii).

Corollary 4.3

Let $0<q\le \infty$, $d/(d+1)<p<1$ and $1<s<1/p$. Then, for all ${\varepsilon }>0$,

$$\begin{aligned} \Vert {\mathbb {E}}_N\Vert _{F^s_{p,q} \rightarrow F^s_{p,q}}\gtrsim c_{\varepsilon }\,2^{(s-1-{\varepsilon })N}. \end{aligned}$$

(58)

Proof

If $d(1/p-1)<s_0<1$ and $\theta \in (0,1)$, then the real interpolation inequalities give

$$\begin{aligned} \big \Vert {\mathbb {E}}_N\big \Vert _{F^{s_0}_{p,q} \rightarrow F^{s_0}_{p,q}}^{1-\theta }\,\big \Vert {\mathbb {E}}_N\big \Vert _{F^s_{p,q} \rightarrow F^s_{p,q}}^{\theta }\, \ge \,c_\theta \, \big \Vert {\mathbb {E}}_N\big \Vert _{B^{s_{\theta }}_{p,q} \rightarrow B^{s_{\theta }}_{p,q}}, \end{aligned}$$

with $s_\theta =(1-\theta )s_0+\theta s$. By Proposition 4.2 the right hand side is larger than a constant times $2^{N(s_\theta -1)}$, while by Corollary 1.3 we have $\big \Vert {\mathbb {E}}_N\big \Vert _{F^{s_0}_{p,q} \rightarrow F^{s_0}_{p,q}}\approx 1$. Choosing $\theta$ sufficiently close to 1 one derives (58). $\square$