2.1 Bases in quasi-Banach spaces

Throughout this paper, a basis of a quasi-Banach space ${\mathbb {X}}$ over the real or complex field ${\mathbb {F}}$ will be a norm-bounded countable family ${\mathcal {X}}=({\boldsymbol {x}}_n)_{n\in {\mathcal {N}}}$ which generates the entire space ${\mathbb {X}}$, and for which there is a (unique) norm-bounded family ${\mathcal {X}}^*=({\boldsymbol {x}}_n^*)_{n\in {\mathcal {N}}}$ in the dual space ${\mathbb {X}}^*$ such that $({\boldsymbol {x}}_n, {\boldsymbol {x}}_n^*)_{n\in {\mathcal {N}}}$ is a biorthogonal system. A basic sequence will be a sequence in ${\mathbb {X}}$ which is a basis of its closed linear span. If ${\mathcal {X}}=({\boldsymbol {x}}_n)_{n\in {\mathcal {N}}}$ is a basis, then it is semi-normalized, i.e.,

\[ 0<\inf_{n\in {\mathcal{N}}} \left\lVert {\boldsymbol{x}}_n\right\rVert\le \sup_{n\in {\mathcal{N}}} \left\lVert {\boldsymbol{x}}_n\right\rVert<\infty, \]

and ${\mathcal {X}}^*$ is a basic sequence called the dual basis of ${\mathcal {X}}$. Note that semi-normalized Schauder bases are a particular case of bases.

Given a linearly independent family of vectors ${\mathcal {X}}=({\boldsymbol {x}}_n)_ {n\in {\mathcal {N}}}$ in ${\mathbb {X}}$ and scalars $\gamma =(\gamma _n)_{n\in {\mathcal {N}}}\in {\mathbb {F}}^{\mathcal {N}}$, we consider the map

\[ S_\gamma=S_\gamma[{\mathcal{X}},{\mathbb{X}}]\colon \operatorname{span}( {\boldsymbol{x}}_n \colon n\in{\mathbb{N}}) \to {\mathbb{X}}, \quad \sum_{n\in {\mathcal{N}}} a_n\, {\boldsymbol{x}}_n \mapsto \sum_{n\in {\mathcal{N}}} \gamma_n\, a_n \, {\boldsymbol{x}}_n. \]

The family ${\mathcal {X}}$ is an unconditional basis of ${\mathbb {X}}$ if and only if it generates the whole space ${\mathbb {X}}$ and $S_\gamma$ is well-defined and bounded on ${\mathbb {X}}$ for all $\gamma \in \ell _\infty$, in which case the uniform boundedness principle yields

(2.1)\begin{equation} K_{u}=K_{u}[{\mathcal{X}},{\mathbb{X}}]:=\sup_{\left\lVert \gamma\right\rVert_\infty\le 1} \left\lVert S_\gamma\right\rVert<\infty. \end{equation}

If ${\mathcal {X}}$ is an unconditional basis, $K_u$ is called its unconditional basis constant. Now, given $A\subseteq {\mathbb {N}}$, we define the coordinate projection onto $A$ (with respect to the sequence ${\mathcal {X}}$) as

\[ S_A=S_{\gamma_A}[{\mathcal{X}},{\mathbb{X}}], \]

where $\gamma _A=(\gamma _n)_{n\in {\mathcal {N}}}$ is the family defined by $\gamma _n=1$ if $n\in A$ and $\gamma _n=0$ otherwise. It is known (see, e.g., [Reference Albiac, Ansorena, Berná and Wojtaszczyk5, Theorem 2.10]) that ${\mathcal {X}}$ is an unconditional basis if and only if it generates ${\mathbb {X}}$ and it is suppression unconditional, i.e.,

\[ \sup \{\left\lVert S_A\right\rVert\colon A\subseteq{\mathbb{N}} \mbox{ finite}\}<\infty. \]

Unconditional bases (indexed on the set ${\mathbb {N}}$ of natural numbers) are a particular case of Schauder bases, and so semi-normalized unconditional bases are a particular case of bases.

2.2 Quasi-greedy bases

Given a basis ${\mathcal {X}}=({\boldsymbol {x}}_n)_{n\in {\mathcal {N}}}$ of a quasi-Banach space ${\mathbb {X}}$ with dual basis ${\mathcal {X}}^*=({\boldsymbol {x}}_n^*)_{n\in {\mathcal {N}}}$, the coefficient transform

\[ {\mathcal{F}}\colon {\mathbb{X}} \to {\mathbb{F}}^{\mathcal{N}} , \quad f\mapsto ({\boldsymbol{x}}_n^*(f))_{n\in {\mathcal{N}}} \]

is a bounded linear operator from ${\mathbb {X}}$ into $c_0$. Thus, for each $m\in {\mathbb {N}}$ there is a unique set $A=A_m(f)\subseteq {\mathbb {N}}$ of cardinality $\left \lvert A\right \rvert =m$ such that whenever $i\in A$ and $j \in {\mathbb {N}}\setminus A$, either $\left \lvert a_i\right \rvert >\left \lvert a_j\right \rvert$ or $\left \lvert a_i\right \rvert =\left \lvert a_j\right \rvert$ and $i< j$. The $m$th greedy approximation to $f\in {\mathbb {X}}$ with respect to the basis ${\mathcal {X}}$ is

\[ {\mathcal{G}}_m(f)={\mathcal{G}}_m[{\mathcal{X}},{\mathbb{X}}](f):=S_{A_m(f)}(f). \]

Note that the operators $({\mathcal {G}}_m)_{m=1}^{\infty }$ defining the greedy algorithm on ${\mathbb {X}}$ with respect to ${\mathcal {X}}$ are not linear nor continuous. The basis ${\mathcal {X}}$ is said to be quasi-greedy if there is a constant $C\ge 1$ such that

\[ \left\lVert {\mathcal{G}}_m(f)\right\rVert\le C\left\lVert f\right\rVert, \quad f\in{\mathbb{X}}, \, m\in{\mathbb{N}}. \]

Equivalently, by [Reference Wojtaszczyk42, Theorem 1] (cf. [Reference Albiac, Ansorena, Berná and Wojtaszczyk5, Theorem 4.1]), these are precisely the bases for which the greedy algorithm converges, i.e.,

\[ \lim_{m\to\infty} {\mathcal{G}}_{m}(f)=f, \quad f\in {\mathbb{X}}. \]

2.3 Truncation quasi-greedy bases

Another family of nonlinear operators of key relevance in the study of the greedy algorithm in a quasi-Banach space ${\mathbb {X}}$ with respect to a basis ${\mathcal {X}}=({\boldsymbol {x}}_n)_{n\in {\mathcal {N}}}$ is the sequence $({\mathcal {R}}_m)_{m=1}^{\infty }$ of the so-called restricted truncation operators, whose definition we recall next. Let

\[ {\mathbb{E}}=\{\lambda\in{\mathbb{F}} \colon \left\lvert \lambda\right\rvert=1\}. \]

Given $A\subset {\mathcal {N}}$ finite and $\varepsilon =(\varepsilon _n)_{n\in A}\in {\mathbb {E}}^A$ we set

\[ {{\mathbb 1}}_{\varepsilon,A}= \sum_{n\in A} \varepsilon_n \,{\boldsymbol{x}}_n, \]

and given $f\in {\mathbb {X}}$ we define $\varepsilon (f)\in {\mathbb {E}}^{{\mathcal {N}}}$ by

\[ \varepsilon(f)=(\operatorname{sign}({\boldsymbol{x}}_n^*(f)))_{n\in{\mathcal{N}}}, \]

where, as is customary, $\operatorname {sign}(\cdot )$ denotes the sign function, i.e., $\operatorname {sign}(0)=1$ and $\operatorname {sign}(a)=a/\left \lvert a\right \rvert$ if $a\in {\mathbb {F}}\setminus \{0\}$. For $m\in {\mathbb {N}}$, the $m$th-restricted truncation operator ${\mathcal {R}}_m\colon {\mathbb {X}} \to {\mathbb {X}}$ is the map

\[ {\mathcal{R}}_m(f)=\min_{n\in A_m(f)} \left\lvert {\boldsymbol{x}}_n^*(f)\right\rvert {{\mathbb 1}}_{\varepsilon(f),A_m(f)}, \quad f\in{\mathbb{X}}, \]

and the basis ${\mathcal {X}}$ is said to be truncation quasi-greedy if

(2.2)\begin{equation} C_r:=\sup_{m\in{\mathbb{N}}}\left\lVert {\mathcal{R}}_m\right\rVert<\infty. \end{equation}

For the sake of generality, most of our results below will be stated and proved for truncation quasi-greedy bases (or even for bases fulfilling weaker unconditionality conditions such as being UCC). However, the uneasy reader can safely replace ‘truncation quasi-greedy basis’ with ‘quasi-greedy basis’ and can rest assured of the validity of the corresponding statements thanks to the following theorem.

Semi-normalized unconditional bases are a special kind of quasi-greedy bases, and although the converse is not true in general, quasi-greedy bases always retain in a certain sense a flavour of unconditionality. For example, truncation quasi-greedy bases of quasi-Banach spaces are unconditional for constant coefficients (UCC, for short) [Reference Albiac, Ansorena, Berná and Wojtaszczyk5, Proposition 4.16]. This means that there is a constant $C\ge 1$ such that $\left \lVert {\mathbb 1}_{\varepsilon,A}\right \rVert \le \left \lVert {\mathbb 1}_{\varepsilon,B}\right \rVert$ whenever $A$, $B$ are finite subsets of ${\mathbb {N}}$ with $A\subseteq B$ and $\varepsilon \in {\mathbb {E}}^B$. If a basis ${\mathcal {X}}$ is UCC then there is another constant $C_u\ge 1$ such that

(2.3)\begin{equation} \left\lVert {{\mathbb 1}}_{\delta,A}\right\rVert\le C_u\left\lVert {{\mathbb 1}}_{\varepsilon,A}\right\rVert \end{equation}

for all finite subsets $A$ of ${\mathbb {N}}$ and all choices of signs $\delta$ and $\varepsilon \in {\mathbb {E}}^A$ (see [Reference Albiac, Ansorena, Berná and Wojtaszczyk5, Lemma 3.2]; for a detailed discussion on the unconditionality-related properties enjoyed by truncation quasi-greedy bases in quasi-Banach spaces we refer to [Reference Albiac, Ansorena, Berná and Wojtaszczyk5, Section 3]).

2.4 Democracy functions

Given a basis ${\mathcal {X}}=({\boldsymbol {x}}_n)_{n\in {\mathcal {N}}}$ of a quasi-Banach space ${\mathbb {X}}$ and $A\subseteq {\mathcal {N}}$ finite, we set ${\mathbb 1}_A={\mathbb 1}_{\varepsilon,A}$, where $\varepsilon =1$ on $A$. The basis ${\mathcal {X}}$ is said to be democratic if there is a constant $D\ge 1$ such that

\[ \left\lVert {{\mathbb 1}}_A\right\rVert\le D \left\lVert {{\mathbb 1}}_B\right\rVert \]

for any two finite subsets $A$ and $B$ of ${\mathcal {N}}$ with $\left \lvert A\right \rvert \le \left \lvert B\right \rvert$. The lack of democracy of a basis ${\mathcal {X}}$ exhibits some sort of asymmetry. To measure how much a basis ${\mathcal {X}}$ deviates from being democratic, we consider its upper democracy function, also known as its fundamental function,

\[ {\boldsymbol{\varphi_u}}[{\mathcal{X}}, {\mathbb{X}}](m): = {\boldsymbol{\varphi_u}}(m)=\sup_{\left\lvert A\right\rvert\le m}\left\lVert {{\mathbb 1}}_A\right\rVert,\quad m\in{\mathbb{N}}, \]

and its lower democracy function,

\[ {\boldsymbol{\varphi_l}}[{\mathcal{X}}, {\mathbb{X}}](m):= {\boldsymbol{\varphi_l}}(m)=\inf_{\left\lvert A\right\rvert\ge m}\left\lVert {{\mathbb 1}}_A\right\rVert, \quad m\in{\mathbb{N}}. \]

Notice that given a basis ${\mathcal {X}}$ there is a constant $C_s$, depending only on the modulus of concavity of the space ${\mathbb {X}}$, such that

(2.4)\begin{equation} \left\lVert \sum_{n\in A} a_n\, {\boldsymbol{x}}_n\right\rVert \le C_s {\boldsymbol{\varphi_u}}[{\mathcal{X}},{\mathbb{X}}](m),\quad \left\lvert A\right\rvert\le m,\; \left\lvert a_n\right\rvert\le 1. \end{equation}

In the particular case that the basis ${\mathcal {X}}$ is UCC the following hold:

1. (i) ${\boldsymbol {\varphi _l}}(m)\lesssim {\boldsymbol {\varphi _u}}(m)$ for $m\in {\mathbb {N}}$;

2. (ii) $\inf _{\left \lvert A\right \rvert = m}\left \lVert {\mathbb 1}_A\right \rVert \lesssim {\boldsymbol {\varphi _l}}[{\mathcal {X}}, {\mathbb {X}}](m)$ for $m\in {\mathbb {N}}$; and

3. (iii) ${\mathcal {X}}$ is democratic if and only ${\boldsymbol {\varphi _u}}(m)\lesssim {\boldsymbol {\varphi _l}}(m)$ for $m\in {\mathbb {N}}$, in which case it is super-democratic, i.e., there is a constant $D\ge 1$ such that

   \[ \left\lVert {{\mathbb 1}}_{\varepsilon,A}\right\rVert \le D \left\lVert {{\mathbb 1}}_{\delta,B}\right\rVert \]
   for any two finite subsets $A$, $B$ of ${\mathbb {N}}$ with $\left \lvert A\right \rvert \le \left \lvert B\right \rvert$, any $\varepsilon \in {\mathcal {E}}^A$ and any $\delta \in {\mathcal {E}}^B$.

Here and throughout this paper, the symbol $\alpha _j\lesssim \beta _j$ for $j\in {\mathcal {N}}$ means that there is a positive constant $C$ such that the families of non-negative real numbers $(\alpha _j)_{j\in {\mathcal {N}}}$ and $(\beta _j)_{j\in {\mathcal {N}}}$ are related by the inequality $\alpha _j\le C\beta _j$ for all $j\in {\mathcal {N}}$. If $\alpha _j\lesssim \beta _j$ and $\beta _j\lesssim \alpha _j$ for $j\in {\mathcal {N}}$ we say $(\alpha _j)_{j\in {\mathcal {N}}}$ are $(\beta _j)_{j\in {\mathcal {N}}}$ are equivalent, and we write $\alpha _j\approx \beta _j$ for $j\in {\mathcal {N}}$.

2.5 Quasi-Banach lattices

Let $0< r\le \infty$. A quasi-Banach space ${\mathbb {X}}$ is (topologically) $r$-convex or $r$-normable if there is a constant $C\ge 1$ such that

(2.5)\begin{equation} \left\lVert \sum_{i\in A} f_i\right\rVert\le C \left(\sum_{i\in A}\left\lVert f_i\right\rVert^r\right)^{1/r} ,\quad \left\lvert A\right\rvert<\infty,\ f_i\in{\mathbb{X}}. \end{equation}

If a quasi-Banach space is $r$-convex then $r\le 1$. By the Aoki–Rolewicz theorem any quasi-Banach space is $r$-convex for some $r\in (0,1]$. In turn, any $r$-convex quasi-Banach space ${\mathbb {X}}$ becomes an $r$-Banach space under a suitable renorming, i.e., ${\mathbb {X}}$ can be endowed with an equivalent quasi-norm satisfying (2.5) with $C=1$.

The existence of a lattice structure in ${\mathbb {X}}$ leads to a (related but) different notion of convexity. A quasi-Banach lattice ${\mathbb {X}}$ is said to be $r$-convex ($0< r<\infty$) if there is a constant $C$ such that

(2.6)\begin{equation} \left\lVert \left(\sum_{i\in A}\left\lvert f_i\right\rvert^r\right)^{1/r}\right\rVert\le C \left(\sum_{i\in A}\left\lVert f_i\right\rVert^r\right)^{1/r}, \quad \left\lvert A\right\rvert<\infty,\ f_i\in{\mathbb{X}}, \end{equation}

where the lattice $r$-sum $(\sum _{i\in A}\left \lvert f_i\right \rvert ^r)^{1/r}\in {\mathbb {X}}$ is defined unambiguously exactly as for the case of Banach lattices (cf. [Reference Lindenstrauss and Tzafriri33, pp. 40–41] and [Reference Popa36]). If ${\mathbb {X}}$ is an $r$-convex quasi-Banach lattice, we will denote by $M^{(r)}({\mathbb {X}})$ the smallest constant $C$ such that (2.6) holds.

We will also consider lattice averages and use them to reformulate lattice convexity in those terms. Given $0< r\le \infty$, the $r$-average of a finite family $(f_i)_{i\in A}$ in a quasi-Banach lattice ${\mathbb {X}}$ is defined as

\[ \left(Ave_{i\in A} \left\lvert f_i\right\rvert^r\right)^{1/r}=\left(\frac{1}{\left\lvert A\right\rvert}\sum_{i\in A} \left\lvert f_i\right\rvert^r \right)^{1/r}=\left\lvert A\right\rvert^{{-}1/r} \left(\sum_{i\in A}\left\lvert f_i\right\rvert^r\right)^{1/r}. \]

This way, a quasi-Banach lattice ${\mathbb {X}}$ is $r$-convex with $M^{(r)}({\mathbb {X}})\le C<\infty$ if and only if

\[ \left\lVert \left(Ave_{i\in A} (\left\lvert f_i\right\rvert^r\right)^{1/r}\right\rVert \le C \left(Ave_{i\in A} (\left\lVert f_i\right\rVert^r\right)^{1/r}, \quad \left\lvert A\right\rvert<\infty,\; f_i\in{\mathbb{X}}. \]

Defining $r$-sums and $r$-averages in quasi-Banach lattices allows us to state a lattice-valued version of Khintchine's inequalities:

Theorem 2.2 Let ${\mathbb {X}}$ be a quasi-Banach lattice. For each $0< r<\infty$ there are constants $T_r$ and $C_r$ such that for any finite family $(f_i)_{i\in A}$ in ${\mathbb {X}},$

\[ \frac{1}{C_r} \left(Ave_{\varepsilon_i={\pm} 1} \left\lvert \sum_{i\in A} \varepsilon_i\, f_i\right\rvert^r\right)^{1/r} \le \left(\sum_{i\in A} \left\lvert f_i\right\rvert^2\right)^{1/2} \le T_r \left(Ave_{\varepsilon_i={\pm} 1} \left\lvert \sum_{i\in A} \varepsilon_i\, f_i\right\rvert^r\right)^{1/r}. \]

Proof. Just apply the functional calculus $\tau$ described in [Reference Lindenstrauss and Tzafriri33, Theorem 1.d.1] to the functions $f$, $g\colon {\mathbb {R}}^n \to {\mathbb {R}}$ given by

\[ f((x_i)_{i=1}^n)=\left(\sum_{i=1}^n \left\lvert x_i\right\rvert^2\right)^{1/2}, \quad g((x_i)_{i=1}^n)= \left(Ave_{\varepsilon_i={\pm} 1} \left\lvert \sum_{i=1}^n \varepsilon_i\, x_i\right\rvert^{r}\right)^{1/r}, \]

and use Khintchine's inequalities [Reference Khintchine31].

If ${\mathbb {X}}$ is a $r$-convex quasi-Banach lattice then ${\mathbb {X}}$ is a $\min \{r,1\}$-convex quasi-Banach space. The converse does not hold, to the extent that there are quasi-Banach lattices that are not $r$-convex for any $r>0$ (see [Reference Kalton28, Example 2.4]). Theorem 2.2 from the same paper characterizes quasi-Banach lattices with some nontrivial convexity as those that are $L$-convex. A quasi-Banach lattice ${\mathbb {X}}$ is said to be $L$-convex if there is $\varepsilon >0$ so that whenever $f$ and $(f_i)_{i\in A}$ in ${\mathbb {X}}$ satisfy $0\le f_i\le f$ for every $i\in A$, and $(1-\varepsilon )\left \lvert A\right \rvert f\le \sum _{i\in A} f_i$ we have $\varepsilon \left \lVert f \right \rVert \le \max _{i\in A} \left \lVert f_i\right \rVert$.

The aforementioned Theorem 2.2 from [Reference Kalton28] also gives the following.

Quoting Kalton from [Reference Kalton28], $L$-convex lattices behave similarly to Banach lattices in many respects. The following result, which generalizes to quasi-Banach lattices [Reference Maurey34, Lemme 5], is in this spirit.

Lemma 2.4 Let ${\mathbb {X}}$ be an $L$-convex quasi-Banach lattice. Then for every $0< r<\infty$ there is a constant $C$ such that

\[ \left\lVert \left(\sum_{i\in A} \left\lvert f_i\right\rvert^2\right)^{1/2}\right\rVert \le C\left(Ave_{\varepsilon_i={\pm} 1} \left\lVert \sum_{i\in A} \varepsilon_i\, f_i\right\rVert^r\right)^{1/r},\quad \left\lvert A\right\rvert<\infty,\; f_i\in{\mathbb{X}}. \]

Proof. Since that map

\[ r\mapsto \left(Ave_{\varepsilon_i={\pm} 1} \left\lVert \sum_{i\in A} \varepsilon_i\, f_i\right\rVert^r\right)^{1/r} \]

is increasing, by Theorem 2.3 we can assume that  ${\mathbb {X}}$ is an $r$-convex Banach lattice. Then, by Theorem 2.2,

\begin{align*} \left\lVert \left(\sum_{i\in A} \left\lvert f_i\right\rvert^2\right)^{1/2}\right\rVert & \le T_r \left\lVert \left(Ave_{\varepsilon_i={\pm} 1}\left\lvert \sum_{i\in A} \varepsilon_i\, f_i\right\rvert^r\right)^{1/r}\right\rVert\\ & \le T_r M^{(r)}({\mathbb{X}}) \left(Ave_{\varepsilon_i={\pm} 1} \left\lVert \sum_{i\in A} \varepsilon_i\, f_i\right\rVert^r\right)^{1/r} \end{align*}

for every finite family $(f_i)_{i\in A}$ in ${\mathbb {X}}$.

We are interested in quasi-Banach lattices of functions defined on a countable set ${\mathcal {N}}$. The term sequence space will apply to a quasi-Banach space ${\mathbb {B}}\subseteq {\mathbb {F}}^{\mathcal {N}}$ such that:

• • ${\boldsymbol {e}}_n:=(\delta _{i,n})_{i\in {\mathcal {N}}}$ is a norm-one vector in ${\mathbb {B}}$ for all $n\in {\mathcal {N}}$; and

• • if $f\in {\mathbb {X}}$ and $g\in {\mathbb {F}}^{\mathcal {N}}$ satisfy $\left \lvert g\right \rvert \le \left \lvert f\right \rvert$, then $g\in {\mathbb {X}}$ and $\left \lVert g\right \rVert \le \left \lVert f\right \rVert$.

That way, ${\mathbb {B}}$ becomes a quasi-Banach lattice with the natural order. In this particular case, $r$-sums take a more workable form: given $f_i=(a_{i,n})_{n\in {\mathcal {N}}}$ for $i\in A$,

\[ \left(\sum_{i\in A}\left\lvert f_i\right\rvert^r\right)^{1/r} = \left(\left( \sum_{i\in A} \left\lvert a_{i,n}\right\rvert^r\right)^{1/r}\right)_{n\in {\mathcal{N}}}. \]

There is a close relation between sequence spaces and unconditional bases. If ${\mathbb {B}}$ is a sequence space the unit vector system ${\mathcal {E}}_{\mathcal {N}}=({\boldsymbol {e}}_n)_{n\in {\mathcal {N}}}$ is a $1$-unconditional basic sequence of ${\mathbb {B}}$ whose coordinate functionals are the restrictions to ${\mathbb {B}}$ of the coordinate functionals $({\boldsymbol {e}}_n^*)_{n\in {\mathcal {N}}}$ defined as

\[ {\boldsymbol{e}}_n^*(f)=a_n, \quad f=(a_n)_{n\in{\mathcal{N}}}\in{\mathbb{F}}^{\mathcal{N}}. \]

And, conversely, every semi-normalized unconditional basis ${\mathcal {X}}$ of a quasi-Banach space ${\mathbb {X}}$ becomes normalized and $1$-unconditional after a suitable renorming of ${\mathbb {X}}$; this way we can associate a sequence space with ${\mathcal {X}}$.

2.6 Embeddings via bases and squeeze-symmetric bases

Let ${\mathbb {X}}$ be a quasi-Banach space with a basis ${\mathcal {X}}=({\boldsymbol {x}}_n)_{n\in {\mathcal {N}}}$ and let $({\mathbb {B}},\left \lVert \cdot \right \rVert _{\mathbb {B}})$ be a sequence space on ${\mathcal {N}}$. Let us recall the following terminology, which we borrow from [Reference Albiac and Ansorena1].

1. (a) We say that ${\mathbb {B}}$ embeds in ${\mathbb {X}}$ via ${\mathcal {X}}$, and put ${\mathbb {B}}\stackrel {{\mathcal {X}}}\hookrightarrow {\mathbb {X}}$, if there is a constant $C$ such that for every $g\in {\mathbb {B}}$ there is $f\in {\mathbb {X}}$ such that ${\mathcal {F}}(f)=g$, and $\left \lVert f\right \rVert \le C\left \lVert g\right \rVert _{{\mathbb {B}}}$.

2. (b) We say that ${\mathbb {X}}$ embeds in ${\mathbb {B}}$ via ${\mathcal {X}}$, and put ${\mathbb {X}}\stackrel {{\mathcal {X}}}\hookrightarrow {\mathbb {B}}$, if there is a constant $C$ such that ${\mathcal {F}}(f)\in {\mathbb {B}}$ with $\left \lVert {\mathcal {F}}(f)\right \rVert _{{\mathbb {B}}}\le C \left \lVert f\right \rVert$ for all $f\in {\mathbb {X}}$.

The sequence space ${\mathbb {B}}$ is said to be symmetric if $f_\pi :=(a_{\pi (n)})_{n\in {\mathcal {N}}}\in {\mathbb {B}}$ and $\left \lVert f_\pi \right \rVert _{\mathbb {B}}=\left \lVert f\right \rVert _{\mathbb {B}}$ for all $f=(a_n)_{n\in {\mathcal {N}}}\in {\mathbb {B}}$ and every permutation $\pi$ of ${\mathcal {N}}$.

Loosely speaking, by squeezing the space ${\mathbb {X}}$ between two symmetric sequence spaces we obtain qualitative estimates on the symmetry of the basis in ${\mathbb {X}}$. Thus, a basis ${\mathcal {X}}$ is said to be squeeze-symmetric if there are symmetric sequence spaces ${\mathbb {S}}_1$ and ${\mathbb {S}}_2$ on ${\mathcal {N}}$, which are close to each other in the sense that

\[ \left\lVert {{\mathbb 1}}_A[{\mathcal{E}}_{\mathcal{N}},{\mathbb{S}}_1]\right\rVert \approx \left\lVert {{\mathbb 1}}_A[{\mathcal{E}}_{\mathcal{N}},{\mathbb{S}}_2]\right\rVert, \quad \left\lvert A\right\rvert<\infty, \]

such that ${\mathbb {S}}_1\stackrel {{\mathcal {X}}}\hookrightarrow {\mathbb {X}} \stackrel {{\mathcal {X}}}\hookrightarrow {\mathbb {S}}_2$. A basis of a quasi-Banach space is squeeze-symmetric if and only if it is truncation quasi-greedy and democratic (see [Reference Albiac, Ansorena, Berná and Wojtaszczyk5, Lemma 9.3 and Corollary 9.15]). In particular, almost greedy bases are squeeze-symmetric.

Embeddings involving weighted Lorentz sequence spaces play an important role in greedy approximation theory using bases. For our purposes here, it will be sufficient to deal with weak Lorentz spaces.

Let ${\boldsymbol {w}}=(w_n)_{n=1}^\infty$ be a weight, i.e., a sequence of nonnegative numbers with $w_1>0$, and let ${\boldsymbol {s}}=(s_m)_{m=1}^\infty$ be its primitive weight defined by $s_m=\sum _{n=1}^m w_n$. The weighted Lorentz sequence space (on the countable set ${\mathcal {N}}$) $d_{1,\infty }({\boldsymbol {w}})$ consists of all functions $f\in c_0({\mathcal {N}})$ whose non-increasing rearrangement $(a_m)_{m=1}^\infty$ satisfies

\[ \left\lVert f\right\rVert_{\infty,{\boldsymbol{w}}}:=\sup_{m\in{\mathbb{N}}} s_m\, a_m<\infty. \]

We must pay attention to whether ${\boldsymbol {s}}$ is doubling, i.e., whether ${\boldsymbol {s}}$ satisfies the condition

\[ s_{2m}\lesssim s_m, \quad m\in{\mathbb{N}}. \]

The $L$-convexity of the space will play a key role as well.

Suppose ${\mathcal {X}}$ is a truncation quasi-greedy basis of a quasi-Banach space ${\mathbb {X}}$. Then, regardless of whether ${\mathcal {X}}$ is democratic or not, the mere definition of lower democracy function yields a constant $C$ such that

(2.7)\begin{equation} \sup_m a_m\, {\boldsymbol{\varphi_l}}[{\mathcal{X}},{\mathbb{X}}](m) \le C \left\lVert f\right\rVert, \quad f\in{\mathbb{X}}, \end{equation}

where $(a_m)_{m=1}^\infty$ is the non-increasing rearrangement of $f$.

We point out that since ${\boldsymbol {\varphi _l}}[{\mathcal {X}},{\mathbb {X}}]$ is not necessarily doubling (see [Reference Wojtaszczyk43]), inequality (2.7) might not hold for an embedding of ${\mathbb {X}}$ into a sequence space; thus we will need to appeal to the following consequence of (2.7).

2.7 Banach envelopes

When dealing with a quasi-Banach space ${\mathbb {X}}$ it is often convenient to know which is the ‘smallest’ Banach space containing ${\mathbb {X}}$. Formally, the Banach envelope of a quasi-Banach space ${\mathbb {X}}$ consists of a Banach space $\widehat {{\mathbb {X}}}$ together with a linear contraction $J_{\mathbb {X}}\colon {\mathbb {X}} \to \widehat {{\mathbb {X}}}$ satisfying the following universal property: for every Banach space ${\mathbb {Y}}$ and every linear contraction $T\colon {\mathbb {X}} \to {\mathbb {Y}}$ there is a unique linear contraction $\widehat {T}\colon \widehat {{\mathbb {X}}}\to {\mathbb {Y}}$ such that $\widehat {T}\circ J_{\mathbb {X}}=T$. Pictorially we have

If a Banach space ${\mathbb {V}}$ and a bounded linear map $J\colon {\mathbb {X}}\to {\mathbb {V}}$ are such that $\widehat {J}\colon \widehat {{\mathbb {X}}}\to {\mathbb {V}}$ is an isomorphism, we say that ${\mathbb {V}}$ is an isomorphic representation of the Banach envelope of ${\mathbb {X}}$ via $J$. For instance, given $0< p<1$, the Bergman space $A^1_{1-p/2}$ is an isomorphic representation of the Banach envelope of $H_p({\mathbb {D}})$ via the inclusion map (see [Reference Duren, Romberg and Shields19, Reference Shapiro37]).

If a quasi-Banach space ${\mathbb {X}}$ has a basis ${\mathcal {X}}$ and ${\mathbb {X}}\stackrel {{\mathcal {X}}}\hookrightarrow \ell _1$, then $\ell _1$ is an isomorphic representation of the Banach envelope of ${\mathbb {X}}$ via the coefficient transform ([Reference Albiac and Ansorena2, Proposition 2.10]). The following lemma is an immediate consequence of this and so we leave its verification to the reader.

2.8 The Marriage Lemma

A classical problem in combinatorics is to determine whether a given family $({\mathcal {N}}_i)_{i\in I}$ of subsets of a set ${\mathcal {N}}$ admits a one-to-one map $\nu \colon I\to {\mathcal {N}}$ such that $\nu (i)\in {\mathcal {N}}_i$ for all $i\in I$. A necessary condition for the existence of such a map is

(2.8)\begin{equation} \left\lvert F\right\rvert\le\left\lvert \bigcup_{i\in F} {\mathcal{N}}_i\right\rvert, \quad F\subseteq I,\; \left\lvert F\right\rvert<\infty. \end{equation}

P. Hall proved in [Reference Hall23] that (2.8) is also sufficient provided the set $I$ of indices and all the sets ${\mathcal {N}}_i$ are finite. Subsequently, M. Hall [Reference Hall22] extended this result to the case when $I$ is not necessarily finite. Here, we will use a generalization by Wojtaszczyk [Reference Wojtaszczyk41] of the latter result which was effectively used in his study of the uniqueness of unconditional bases in quasi-Banach spaces.

Theorem 2.9 see [Reference Wojtaszczyk41, Corollary 3.1]

Let ${\mathcal {N}}$ be a set and $({\mathcal {N}}_i)_{i\in I}$ be a family of finite subsets of ${\mathcal {N}}$. Let $K\in {\mathbb {N}}$. Suppose that

\[ \left\lvert F\right\rvert\le K\left\lvert \bigcup_{i\in F} {\mathcal{N}}_i\right\rvert \]

for every $F\subseteq I$ finite. Then there is a partition $(I_j)_{j=1}^K$ of $I$ and one-to-one maps $\nu _j\colon I_j\to {\mathcal {N}}$ for $j=1,\ldots,K$ such that $\nu _j(i)\in {\mathcal {N}}_i$ for each $i\in I$.

3.1 Lower estimates for democracy functions

A subtle, yet important, obstruction to apply Lemma 3.5 to basic sequences ${\mathcal {Y}}$ in a quasi-Banach space ${\mathbb {X}}$ is that the coordinate functionals associated with ${\mathcal {Y}}$ are not defined on ${\mathbb {X}}$ but on the closed subspace of ${\mathbb {X}}$ generated by ${\mathcal {Y}}$, denoted by ${\mathbb {Y}}$. If ${\mathbb {X}}$ is locally convex, the Hahn-Banach theorem comes to our aid: any bounded linear functional on ${\mathbb {Y}}$ extends to a bounded linear functional on ${\mathbb {X}}$ without increasing its norm. However, there are important spaces, such as Hardy spaces, that are not locally convex and so this extension cannot be taken for granted. This situation motivated Day [Reference Day14] to define the Hahn-Banach Extension Property (HBEP for short). We say that ${\mathbb {Y}}$ has the HBEP in ${\mathbb {X}}$ if there is a constant $C$ such that for every $f^*\in {\mathbb {Y}}^*$ there is $g^*\in {\mathbb {X}}^*$ such that $g^*|_{\mathbb {Y}}=f^*$ and $\left \lVert g^*\right \rVert \le C\left \lVert f^*\right \rVert$. Needless to say, ${\mathbb {X}}$ has the HBEP in ${\mathbb {X}}$. For the purposes of this paper it suffices to keep in mind that any complemented subspace of a quasi-Banach space ${\mathbb {X}}$ has the HBEP in ${\mathbb {X}}$. We say that the basic sequence ${\mathcal {Y}}$ has the HBEP in ${\mathbb {X}}$ if ${\mathbb {Y}}$ does.

The results in this section rely on the following lemma.

Lemma 3.7 Let ${\mathbb {X}}$ be a quasi-Banach space with a basis ${\mathcal {X}}=({\boldsymbol {x}}_n)_{n\in {\mathcal {N}}}$. Suppose that ${\mathbb {X}}$ embeds in an $L$-convex sequence space ${\mathbb {B}}$ via ${\mathcal {X}}$ and that the unit vector system of ${\mathbb {B}}$ is strongly absolute. Then, for every UCC basic sequence ${\mathcal {Y}}=({\boldsymbol {y}}_i)_{i\in {\mathcal {M}}}$ with the HBEP in ${\mathbb {X}}$ there is a constant $c$ such that

\[ {\boldsymbol{\varphi_l}}[{\mathcal{Y}},{\mathbb{X}}](m)\gtrsim {\boldsymbol{\varphi_l}}[{\mathcal{E}}_{\mathcal{N}},{\mathbb{B}}](\lceil cm\rceil), \quad m\in{\mathbb{N}}. \]

Proof. Choose for each $i\in {\mathcal {M}}$ an extension ${\boldsymbol {z}}_i^*$ of the coordinate functional ${\boldsymbol {y}}_i^*$ in such a way that $a=\sup _{i\in {\mathcal {M}}} \left \lVert {\boldsymbol {y}}_i\right \rVert \left \lVert {\boldsymbol {z}}_i^*\right \rVert <\infty$, and pick $K$ and $\delta$ as in Lemma 3.5. For each $i\in {\mathcal {M}}$, let $f_i\in {\mathbb {F}}^{\mathcal {N}}$ be given by

\[ f_i=\left( {\boldsymbol{z}}_i^*({\boldsymbol{x}}_n) \, {\boldsymbol{x}}_n^*({\boldsymbol{y}}_i)\right)_{n\in{\mathcal{N}}}. \]

Notice that for each $A\subseteq {\mathcal {M}}$ finite, $\max _{i\in A} \left \lvert f_i\right \rvert \ge \delta$ on the set

\[ \Omega_A=\{n \in {\mathcal{N}} \colon \left\lvert {\boldsymbol{z}}_i^*({\boldsymbol{x}}_n) \, {\boldsymbol{x}}_n^*({\boldsymbol{y}}_i)\right\rvert\ge \delta \mbox{ for some } i\in A\}. \]

Fix $m\in {\mathbb {N}}$. Let $A\subseteq {\mathcal {M}}$ be a finite set with $\left \lvert A\right \rvert \ge m$. Using Lemma 2.4 we obtain

\begin{align*} \left\lVert {{\mathbb 1}}_A[{\mathcal{Y}},{\mathbb{X}}]\right\rVert & \approx Ave_{\varepsilon_i=\pm1} \left\lVert \sum_{i\in A}\varepsilon_i\, {\boldsymbol{y}}_i\right\rVert\\ & \gtrsim Ave_{\varepsilon_i=\pm1}\left\lVert \sum_{i\in A}\varepsilon_i\, {\mathcal{F}}({\boldsymbol{y}}_i)\right\rVert_{\mathbb{B}}\\ & \gtrsim \left\lVert \left(\sum_{i\in A}\left\lvert {\mathcal{F}}({\boldsymbol{y}}_i)\right\rvert^2\right)^{1/2}\right\rVert_{\mathbb{B}}\\ & \gtrsim \left\lVert \left(\sum_{i\in A}\left\lvert f_i\right\rvert^2\right)^{1/2}\right\rVert_{\mathbb{B}}\\ & \gtrsim \left\lVert \max_{i\in A}\left\lvert f_i\right\rvert\right\rVert_{\mathbb{B}}\\ & \gtrsim \left\lVert {{\mathbb 1}}_{\Omega_A}[{\mathcal{E}}_{\mathcal{N}}, {\mathbb{B}}]\right\rVert. \end{align*}

Now the statement follows since $\left \lvert \Omega _A\right \rvert \ge \lceil m/K \rceil$.

Theorem 3.8 Let ${\mathbb {X}}$ be a quasi-Banach space with a strongly absolute semi-normalized unconditional basis ${\mathcal {X}}$ which induces an $L$-convex lattice structure on ${\mathbb {X}}$. Suppose that ${\mathcal {Y}}$ is a UCC basic sequence with the HBEP in ${\mathbb {X}}$. Then there is a constant $c>0$ such that

\[ {\boldsymbol{\varphi_l}}[{\mathcal{Y}},{\mathbb{X}}](m) \gtrsim {\boldsymbol{\varphi_l}}[{\mathcal{X}},{\mathbb{X}}](\lceil cm\rceil), \quad m\in{\mathbb{N}}. \]

Theorem 3.9 Let ${\mathbb {X}}$ be a quasi-Banach space with a truncation quasi-greedy basis ${\mathcal {X}}$. Let ${\boldsymbol {s}}=(s_m)_{m=1}^\infty$ be a non-decreasing doubling sequence of positive scalars such that ${\boldsymbol {\varphi _l}}[{\mathcal {X}},{\mathbb {X}}] \gtrsim {\boldsymbol {s}}$ and

\[ \sum_{m=1}^\infty \frac{1}{s_m}<\infty. \]

Suppose that ${\mathcal {Y}}$ is a UCC basic sequence with the HBEP in ${\mathbb {X}}$. Then

\[ {\boldsymbol{\varphi_l}}[{\mathcal{Y}},{\mathbb{X}}](m) \gtrsim s_m,\quad m\in{\mathbb{N}}. \]

Corollary 3.10 Let ${\mathcal {X}}$ be a squeeze-symmetric basis of a quasi-Banach space ${\mathbb {X}}$. Suppose that

\[ \sum_{m=1}^\infty \frac{1}{{\boldsymbol{\varphi_u}}[{\mathcal{X}},{\mathbb{X}}](m)}<\infty. \]

Let ${\mathcal {Y}}$ be a UCC basic sequence with the HBEP in ${\mathbb {X}}$. Then

\[ {\boldsymbol{\varphi_l}}[{\mathcal{Y}},{\mathbb{X}}]\gtrsim {\boldsymbol{\varphi_u}}[{\mathcal{X}},{\mathbb{X}}]\approx {\boldsymbol{\varphi_l}}[{\mathcal{X}},{\mathbb{X}}]. \]

3.2 Upper estimates for democracy functions

Our results here heavily depend on the complementability of the closed subspaces of ${\mathbb {X}}$ generated by the basic sequences we tackle. A basic sequence in ${\mathbb {X}}$ that spans a complemented subspace is said to be a complemented basic sequence of ${\mathbb {X}}$.

Lemma 3.11 Suppose that a quasi-Banach space ${\mathbb {X}}$ embeds via a basis ${\mathcal {X}}=({\boldsymbol {x}}_n)_{n\in {\mathcal {N}}}$ in a sequence space ${\mathbb {B}}$ whose unit vector system is strongly absolute. Let ${\mathcal {Y}}=({\boldsymbol {y}}_i)_{i\in {\mathcal {M}}}$ be a complemented basic sequence of ${\mathbb {X}}$. If ${\mathcal {Y}}$ is truncation quasi-greedy then

\[ {\boldsymbol{\varphi_u}}[{\mathcal{Y}},{\mathbb{X}}]\lesssim {\boldsymbol{\varphi_u}}[{\mathcal{X}},{\mathbb{X}}]. \]

Proof. Let $P\colon {\mathbb {X}}\to {\mathbb {Y}}$ be a bounded linear projection, where ${\mathbb {Y}}$ is the closed subspace of ${\mathbb {X}}$ generated by ${\mathcal {Y}}$. Put

\[ {\boldsymbol{z}}_i^*={\boldsymbol{y}}_i^*\circ P, \quad i\in I, \]

where $({\boldsymbol {y}}_i^*)_{i\in {\mathcal {M}}}$ in ${\mathbb {Y}}^*$ are the coordinate functionals of ${\mathcal {Y}}$. We have ${\boldsymbol {z}}_i^*({\boldsymbol {y}}_i)={\boldsymbol {y}}_i^*({\boldsymbol {y}}_i)$ and $\left \lVert {\boldsymbol {z}}_i^*\right \rVert \left \lVert {\boldsymbol {y}}_i \right \rVert \le \left \lVert P\right \rVert \left \lVert {\boldsymbol {y}}_i^*\right \rVert \left \lVert {\boldsymbol {y}}_i \right \rVert$ for every $i\in I$. Hence, by Lemma 3.5, there are $K\in {\mathbb {N}}$ and $\delta >0$ such that, if we put

\[ \Omega_i=\{n \in {\mathcal{N}} \colon \left\lvert {\boldsymbol{z}}_i^*({\boldsymbol{x}}_n) \, {\boldsymbol{x}}_n^*({\boldsymbol{y}}_i)\right\rvert\ge \delta\}, \quad i\in {\mathcal{M}}, \]

then $\left \lvert A\right \rvert \le K \left \lvert \cup _{i\in A} \Omega _i\right \rvert$ for all $A\subseteq {\mathcal {M}}$ finite. Moreover, by Lemma 3.6, $\left \lvert \Omega _i\right \rvert <\infty$ for all $i\in {\mathcal {M}}$. Thus, by Theorem 2.9, there are a partition $({\mathcal {M}}_j)_{j=1}^K$ of ${\mathcal {M}}$ and one-to-one maps $\nu _j\colon {\mathcal {M}}_j\to {\mathcal {N}}$ such that

\[ \left\lvert {\boldsymbol{z}}_i^*({\boldsymbol{x}}_{\nu_j(i)}) \, {\boldsymbol{x}}_{\nu_j(i)}^*({\boldsymbol{y}}_i)\right\rvert\ge \delta, \quad 1\le j \le K, \; i\in {\mathcal{M}}_j. \]

Pick $m\in {\mathbb {N}}$, and let $A\subseteq {\mathcal {M}}$ be such that $\left \lvert A\right \rvert \le m$. Put $A_j=A\cap {\mathcal {M}}_j$ for $j=1$, …, $K$. Set also

\[ a_{i,n}= \left\lvert {\boldsymbol{x}}_n^*({\boldsymbol{y}}_i)\right\rvert \operatorname{sign}({\boldsymbol{z}}_i^*({\boldsymbol{x}}_n)), \quad i\in {\mathcal{M}}, \; n\in{\mathcal{N}}. \]

For each $l\in A_j$ we have

\begin{align*} {\boldsymbol{y}}_l^* \left( P\left( \sum_{i\in A_j} a_{l,\nu_j(i)} {\boldsymbol{x}}_{\nu_j(i)}\right)\right) & =\sum_{i\in A_j} \left\lvert {\boldsymbol{z}}_l^*({\boldsymbol{x}}_{\nu_j(i)}) \, {\boldsymbol{x}}_{\nu_j(i)}^*({\boldsymbol{y}}_l)\right\rvert\\ & \ge \left\lvert {\boldsymbol{z}}_l^*({\boldsymbol{x}}_{\nu_j(l)}) \, {\boldsymbol{x}}_{\nu_j(l)}^*({\boldsymbol{y}}_l)\right\rvert\\ & \ge \delta. \end{align*}

Consequently, if $C_u$ is as in (2.3) and $C_r$ is the truncation quasi-greedy constant of ${\mathcal {Y}}$ (see (2.2)), we have

\[ \left\lVert {{\mathbb 1}}_{A_j}[{\mathcal{Y}},{\mathbb{Y}}] \right\rVert \le \frac{C_u C_r}{\delta} \left\lVert P\left( \sum_{i\in A_j} \varepsilon_{l,\nu_j(i)} {\boldsymbol{x}}_{\nu_j(i)}\right)\right\rVert, \quad j=1,\dots, K. \]

If we put $a=\sup _{n\in {\mathcal {N}}} \left \lVert {\boldsymbol {x}}_n^*\right \rVert$, $b=\sup _{i\in {\mathcal {M}}} \left \lVert {\boldsymbol {y}}_i\right \rVert$, and let $\kappa$ be the optimal constant such that

\[ \left\lVert \sum_{j=1}^K y_j\right\rVert \le \kappa \sum_{j=1}^K \left\lVert y_j\right\rVert, \quad y_j\in{\mathbb{Y}}, \]

and $C_s$ be as in (2.4), we obtain

\begin{align*} \left\lVert {{\mathbb 1}}_{A}[{\mathcal{Y}},{\mathbb{Y}}] \right\rVert & =\left\lVert \sum_{j=1}^{K}{{\mathbb 1}}_{A_j}[{\mathcal{Y}},{\mathbb{Y}}]\right\rVert\\ & \le\kappa\sum_{j=1}^{K}\left\lVert {{\mathbb 1}}_{A_j}[{\mathcal{Y}},{\mathbb{Y}}]\right\rVert\\ & \le \kappa K\frac{C_uC_r}{\delta}\left\lVert P\right\rVert abC_s{\boldsymbol{\varphi_u}}[{\mathcal{X}},{\mathbb{X}}](m).\end{align*}

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Theorem 3.12 Let ${\mathbb {X}}$ be a quasi-Banach space with a basis ${\mathcal {X}}$. Assume that:

1. (a) Either ${\mathcal {X}}$ is unconditional and strongly absolute, or

2. (b) there is a non-decreasing doubling sequence ${\boldsymbol {s}}=(s_m)_{m=1}^\infty$ such that ${\boldsymbol {\varphi _l}}[{\mathcal {X}},{\mathbb {X}}] \gtrsim {\boldsymbol {s}}$ and

   \[ \sum_{m=1}^\infty \frac{1}{s_m}<\infty. \]

Suppose that ${\mathcal {Y}}$ is a complemented truncation quasi-greedy basic sequence of ${\mathbb {X}}$. Then

\[ {\boldsymbol{\varphi_u}}[{\mathcal{Y}},{\mathbb{X}}]\lesssim {\boldsymbol{\varphi_u}}[{\mathcal{X}},{\mathbb{X}}]. \]