Unconditional Bases for Homogeneous $\alpha$-Modulation Type Spaces

Abstract

We construct orthonormal bases compatible with bi-variate homogeneous $\alpha$-modulation spaces and the associated spaces of Triebel–Lizorkin type. The construction is based on generating a separable $\alpha$-covering and using carefully selected tensor products of univariate brushlet functions with regard to this covering. We show that the associated systems form an unconditional bases for the homogeneous $\alpha$-spaces of Triebel–Lizorkin type.

Appendix A: Some Technical Results

This appendix contains some results on vector-valued maximal functions needed for the analysis of the $\alpha$-TL spaces.

For $0<r<\infty$, the Hardy–Littlewood maximal function is defined by

$$\begin{aligned} M_r u(x):=\sup _{t>0}\bigg (\frac{1}{|B(x,t)|} \int _{{B}(x,t)} |u(y)|^r \mathrm{d}y\bigg )^{1/r},\qquad u\in L_{r,\text {loc}}(\mathbb {R}^2). \end{aligned}$$

For $0<p,q\le \infty$, and a sequence $f=\{f_j\}_{j\in \mathbb {N}}$ of $L_p(\mathbb {R}^2)$ functions, we define the norm

$$\begin{aligned} \Vert f\Vert _{L_p(\ell _q)}:=\big \Vert \big (\sum _{j\in \mathbb {N}} |f_j|^q\big )^{1/q}\big \Vert _{L_p(\mathbb {R}^2)}, \end{aligned}$$

Where there is no risk of ambiguity we will abuse notation slightly and write $\Vert f_k\Vert _{L_p(\ell _q)}$ instead of $\Vert \{f_k\}_k\Vert _{L_p(\ell _q)}$.

The vector-valued Fefferman–Stein maximal inequality gives the estimate (see [26, Chapters I&II])

$$\begin{aligned} \Vert \{M_rf_j\}\Vert _{L_p(\ell _q)}\le C_B\Vert \{f_j\}\Vert _{L_p(\ell _q)} \end{aligned}$$

(27)

for $r<q\le \infty$ and $r<p<\infty$, $C_B:=C_B(r,p,q)$.

For $\Omega =\{\Omega _n\}$ a sequence of compact subsets of $\mathbb {R}^2$, we let

$$\begin{aligned} L_p^\Omega (\ell _q):=\{\{f_n\}_{n\in \mathbb {N}}\in L_p(\ell _q)\,|\,{\text {supp}}(\hat{f}_n)\subseteq \Omega _n,\,\forall n\}. \end{aligned}$$

For $\xi \in \mathbb {R}^2$, we let $\langle \xi \rangle :=(1+|\xi |^2)^{1/2}$. Let u(x) be a continuous function on $\mathbb {R}^2$. We define, for $a,R>0$,

$$\begin{aligned} u^*(a,R;x):=\sup _{y\in \mathbb {R}^2} \langle y\rangle ^{-a}|u(x-y/R)|,\qquad x\in \mathbb {R}^2. \end{aligned}$$

The following is a variation on Peetre’s maximal estimate in a vector-valued setting.

Proposition 3.7

Suppose $0<p<\infty$ and $0<q\le \infty$, and let $\Omega =\{T_k \mathcal {C}\}_{k\in \mathbb {N}}$ be a sequence of compact subsets of $\mathbb {R}^2$ generated by a family $\{T_k={t_k}Id \cdot +\xi _k\}_{k\in \mathbb {N}}$ of invertible affine transformations on $\mathbb {R}^2$, with $\mathcal {C}$ a fixed compact subset of $\mathbb {R}^2$. If $0<r<\min (p,q)$, then there exists a constant K such that

$$\begin{aligned} \bigg \Vert \big \{ (f_k)^*(2/r,t_k;\cdot ) \big \} \bigg \Vert _{L_p(\ell _q)}\le K\Vert \{f_k\}\Vert _{L_p(\ell _q)}, \end{aligned}$$

(28)

for all $f\in L_p^\Omega (\ell _q)$, where $f=\{f_k\}_{k\in \mathbb {N}}.$

Finally, we need the following vector-valued multiplier result. For $s\in \mathbb {R}_+$, we let

$$\begin{aligned} \Vert f\Vert _{H_2^s}:=\biggl ( \int |\mathcal {F}^{-1}f(x)|^2 \langle x\rangle ^{2s} \mathrm{d}x\biggr )^{1/2} \end{aligned}$$

denote the Sobolev norm.

Proposition 3.8

Suppose $0<p<\infty$ and $0<q\le \infty$, and let $\Omega =\{T_k \mathcal {C}\}_{k\in \mathbb {N}}$ be a sequence of compact subsets of $\mathbb {R}^2$ generated by a family $\{T_k={t_k}Id \cdot +\xi _k\}_{k\in \mathbb {N}}$ of invertible affine transformations on $\mathbb {R}^2$, with $\mathcal {C}$ a fixed compact subset of $\mathbb {R}^2$. Assume $\{\psi _j\}_{j\in \mathbb {N}}$ is a sequence of functions satisfying $\psi _j\in H^s_2$ for some $s>\frac{\nu }{2}+\frac{\nu }{\min (p,q)}$. Then there exists a constant $C<\infty$ such that

$$\begin{aligned} \Vert \{\psi _k(D)f_k\}\Vert _{L_p(\ell _q)} \le C\sup _j\Vert \psi _j(T_j\cdot )\Vert _{H^s_2}\cdot \Vert \{f_k\}\Vert _{L_p(\ell _q)} \end{aligned}$$

for all $\{f_k\}_{k\in \mathbb {N}} \in L_p^\Omega (\ell _q)$.

The following Lemma was used in the proof of Lemma 3.5.

Lemma 3.9

Let $0<r\le 1$. There exists a constant C such that for any sequence $\{s_{Q,{n}}\}_{Q,{n}}$ we have

$$\begin{aligned} \sum _{{n}} |s_{Q,{n}}||w_{n,Q}|(x)\le C|Q|^{1/2} \sum _{\ell =1}^4 M_r\Bigl (\sum _{{n}} |s_{Q,{n}}| \chi _{U(Q,{n})}\Bigr )(R_\ell x). \end{aligned}$$

Proof

From (9) we have that

$$\begin{aligned} |w_{n,Q}(x)|\le C_N|Q|^{1/2}\sum _{\ell =1}^4\big (1+\big | R_\ell \delta _{Q}x-\pi ({n}+{a})\big |\big )^{-N}, \end{aligned}$$

(29)

for any $N>0$, with $C_N$ independent of Q, where we use the same notation as in the proof of Lemma 3.5. Fix $N>2/r$. We can, without loss of generality, suppose $x\in U(Q,\mathbf {0})$.

For $j\in \mathbb {N}$, we let $A_j=\{ {n}\in \mathbb {N}_0^2:2^{j-1}<| \pi ({n}+{a})|\le 2^j\}$. Notice that $\cup _{{n}\in A_j} U(Q,{n})$ is a bounded set contained in the ball ${B}(0,c2^{j+1}|Q|^{-1/2})$. Now,

$$\begin{aligned} \sum _{{n}\in A_j}&|s_{Q,{n}}|\big (1+\big | \delta _{Q}x-\pi ({n}+{a})\big |\big )^{-N} \\ {}&\le C2^{-jN} \sum _{{n}\in A_j} |s_{Q,{n}}|\\&\le C2^{-jN} \Bigl ( \sum _{{n}\in A_j} |s_{Q,{n}}|^r\Bigr )^{1/r}\\&\le C2^{-jN} |Q|^{1/r} \biggl ( \int \sum _{{n}\in A_j} |s_{Q,{n}}|^r\chi _{U(Q,{n})}(y)\, \mathrm{d}y \biggr )^{1/r}\\&\le CL^{1-r}2^{-jN} |Q|^{1/r} \biggl ( \int _{{B}(0,c2^{j+1}|Q|^{-1/2})} \Bigl (\sum _{{n}\in A_j} |s_{Q,{n}}|\chi _{U(Q,{n})}(y)\Bigr )^r \, \mathrm{d}y\biggr )^{1/r}\\&\le C'2^{-j(N-2/r)} M_r \Bigl (\sum _{{n}\in \mathbb {N}_0^2} |s_{Q,{n}}|\chi _{U(Q,{n})} \Bigr )(x). \end{aligned}$$

We now perform the summation over $j\in \mathbb {N}_0$ to obtain

$$\begin{aligned} \sum _{{n}\in \mathbb {N}_0^2} |s_{Q,{n}}|\big (1+\big | \delta _{Q}x-\pi ({n}+{a})\big |\big )^{-N} \le CM_r \Bigl (\sum _{{n}\in \mathbb {N}_0^2} |s_{Q,{n}}|\chi _{U(Q,{n})} \Bigr )(x). \end{aligned}$$

We then use the substitutions $x=R_\ell z$, $\ell =1,\ldots ,4$, to cover all four terms on the RHS of (29), where we use the fact that $R_{\ell }$ and $\delta _Q$ commute. $\square$