2. Preliminary results

For broader applicability, we will consider problem 1.2 within a setting more general than that of Schauder bases. A biorthogonal system in a Banach space $\mathbb {X}$ over the real or complex field $\mathbb {F}$ is a sequence $\mathcal {O}=(\boldsymbol {x}_n,\boldsymbol {x}_n^{*})_{n=1}^{\infty }$ in $\mathbb {X}\times \mathbb {X}^{*}$ with $\boldsymbol {x}_n^{*}(\boldsymbol {x}_k)=\delta _{n,k}$ for all $(n,k)\in \mathbb {N}^{2}$. The Besselian and Hilbertian intervals of the biorthogonal system $\mathcal {O}$ are defined analogously to those of a Schauder basis. The interval $J_B[\mathcal {O},\mathbb {X}]$ is nonempty, that is, $\infty \in J_B[\mathcal {O},\mathbb {X}]$, if and only if $(\boldsymbol {x}_n^{*})_{n=1}^{\infty }$ is norm-bounded. Similarly, $1$ belongs to $J_H[\mathcal {O},\mathbb {X}]$, so that $J_H[\mathcal {O},\mathbb {X}]\not =\emptyset$, if and only if $(\boldsymbol {x}_n)_{n=1}^{\infty }$ is norm-bounded. Notice that $(\boldsymbol {x}_n)_{n=1}^{\infty }$ and $(\boldsymbol {x}_n^{*})_{n=1}^{\infty }$ are simultaneously norm-bounded if and only if $(\boldsymbol {x}_n)_{n=1}^{\infty }$ is semi-normalized and

(2.1)\begin{equation} \sup_n \Vert{\boldsymbol{x}_n}\Vert \Vert{\boldsymbol{x}_n^{*}}\Vert<\infty. \end{equation}

If (2.1) holds, we say that the biorthogonal system $\mathcal {O}$ is $M$-bounded.

If $\mathcal {X}=(\boldsymbol {x}_n)_{n=1}^{\infty }$ is a Schauder basis with dual basis $(\boldsymbol {x}_n^{*})_{n=1}^{\infty }$ then, since the partial sum projections with respect to $\mathcal {X}$ are uniformly bounded, $(\boldsymbol {x}_n,\boldsymbol {x}_n^{*})_{n=1}^{\infty }$ is an $M$-bounded biorthogonal system.

Given a biorthogonal system $\mathcal {O}=(\boldsymbol {x}_n,\boldsymbol {x}_n^{*})_{n=1}^{\infty }$, the set

\[ J[\mathcal{O},\mathbb{X}]:=J_H[\mathcal{O},\mathbb{X}]\cap J_B[\mathcal{O},\mathbb{X}] \]

is either empty or a singleton. In fact, if $r\in J[\mathcal {O},\mathbb {X}]$, then $(\boldsymbol {x}_n)_{n=1}^{\infty }$ is equivalent to the unit vector system of $\ell _r$. The Radamacher type and cotype of the space enables us to obtain more significant information.

Proof. Suppose that $\mathcal {O}$ is $r$-Besselian. Then,

\begin{align*} m^{1/r}& =\mathop{\mathrm{Ave}}\limits_{\varepsilon_n={\pm} 1}\left( \sum_{k=1}^{\infty} \vert{\boldsymbol{x}_k^{*}\left( \sum_{n=1}^{m} \varepsilon_n\,\boldsymbol{x}_n \right)}\vert^{r} \right)^{1/r}\\ & \lesssim \mathop{\mathrm{Ave}}\limits_{\varepsilon_n={\pm} 1} \left\|\sum_{n=1}^{m} \varepsilon_n\,\boldsymbol{x}_n\right\|\lesssim \left(\sum_{n=1}^{m} \Vert{\boldsymbol{x}_n}\Vert^{r_0}\right)^{1/r_0}\approx m^{1/r_0}. \end{align*}

for $m\in \mathbb {N}$. Consequently, $r\ge r_0$. Similarly, if $\mathcal {O}$ is $q$-Hilbertian,

\begin{align*} m^{1/q_0} \approx \left(\sum_{n=1}^{m}\Vert{\boldsymbol{x}_n}\Vert^{q_0}\right)^{1/q_0}\lesssim \mathop{\mathrm{Ave}}\limits_{\varepsilon_n={\pm} 1}\left\|\sum_{n=1}^{m} \varepsilon_n\,\boldsymbol{x}_n\right\| \lesssim m^{1/q} \end{align*}

for $m\in \mathbb {N}$. Hence, $q\le q_0$.

In the case when $\mathbb {X}=\ell _p$, $1\le p\le \infty$, proposition 2.1 gives that the Hilbertian interval of the biorthogonal system $\mathcal {O}$ is contained in $[1,\max \{2,p\}]$, and its Besselian interval is contained in $[\min \{2,p\},\infty ]$. If $p=2$, this information allows us to draw apart the Hilbertian and Besselian intervals of any biorthogonal system of $\mathbb {X}$. If $p\not =2$, to obtain a similar result we must take advantage of the concavity and convexity properties of the lattice structure induced on $\mathbb {X}$ by its unit vector system.

Following [Reference Lindenstrauss and Tzafriri25], we say that a quasi-Banach lattice $\mathbb {X}$ satisfies an upper (resp., lower) $r$-estimate, where $1\le r\le \infty$, if there is a constant $C>0$ such that for every choice of finitely many disjoint elements $(f_j)_{j=1}^{m}$ in $\mathbb {X}$ we have $\Vert {f}\Vert \le C N$ (resp. $N\le C\Vert {f}\Vert$), where

\begin{align*} f=\sum_{j=1}^{m} f_j\ \text{and}\ N=\left(\sum_{j=1}^{m}\Vert{f_j}\Vert^{r}\right)^{1/r}. \end{align*}

It is clear that if $1\le r\le 2$ (resp., $2\le r \le \infty$) and the Banach lattice $\mathbb {X}$ has Radamacher type (resp., cotype) $r$, then it satisfies an upper (resp., lower) $r$-estimate.

Given a set $A\subseteq [1,\infty ]$, we put

\[ A'=\{q':=q/(q-1) \colon q\in A\}, \]

and we say that $A$ and $A'$ are conjugate sets.

Proof. Let $\mathcal {U}^{*}=({{\boldsymbol {u}}}_j^{*})_{j=1}^{\infty }$ be the dual basis of $\mathcal {U}$. Suppose that $\mathcal {O}$ is $q$-Hilbertian, $1\le q \le \infty$. By the Cantor diagonal technique, passing to a subsequence we can suppose that $({{\boldsymbol {u}}}_j^{*}(\boldsymbol {x}_n))_{n=1}^{\infty }$ converges for every $j\in \mathbb {N}$. Then, replacing $\boldsymbol {x}_n$ with $\boldsymbol {x}_{2n}-\boldsymbol {x}_{2n-1}$ and $\boldsymbol {x}_n^{*}$ with $(\boldsymbol {x}_{2n}^{*}-\boldsymbol {x}_{2n-1}^{*})/2$ we can suppose that $\lim _n {{\boldsymbol {u}}}_j^{*}(\boldsymbol {x}_n)=0$ for every $j\in \mathbb {N}$. By the Bessaga-Pełczyński Selection Principle (see, e.g., [Reference Albiac and Kalton7, proposition 1.3.10]), passing to a subsequence we can suppose that $(\boldsymbol {x}_n)_{n=1}^{\infty }$ is equivalent to a block basic sequence with respect to $\mathcal {U}$, say $\mathcal {Y}=(\boldsymbol {y}_n)_{n=1}^{\infty }$. On the one hand, the mapping

\[ (a_n)_{n=1}^{\infty}\mapsto \sum_{n=1}^{\infty} a_n\,\boldsymbol{y}_n \]

defines a bounded operator from $\ell _q$ into $\mathbb {X}$. On the other hand, since $\mathcal {Y}$ is semi-normalized, there is a constant $C$ such that

\begin{align*} \Vert{f}\Vert_{r_0} \le C \left\|\sum_{n=1}^{\infty} a_n\,\boldsymbol{y}_n\right\|,\quad f=(a_n)_{n=1}^{\infty}\in c_{00}. \end{align*}

We infer that $q\le r_0$.

The above argument proves the Hilbertian part. In order to prove the Besselian part, we assume that $q_0>1$; otherwise there is nothing to prove. Since every semi-normalized block basic sequence $(\boldsymbol {z}_n)_{n=1}^{\infty }$ with respect to $\mathcal {U}$ satisfies the estimate

\begin{align*} \left\|\sum_{n=1}^{\infty} a_n\,\boldsymbol{z}_n\right\|\le C \Vert{f}\Vert_{q_0}, \quad f=(a_n)_{n=1}^{\infty}\in c_{00}, \end{align*}

for some constant $C$, $\mathcal {U}$ does not have any block basic sequence equivalent to the unit vector system of $\ell _1$. By [Reference Albiac and Kalton7, theorem 3.3.1], $\mathcal {U}^{*}$ is a basis of $\mathbb {X}^{*}$. By [Reference Lindenstrauss and Tzafriri25, proposition 1.f.5], $\mathcal {U}^{*}$ induces on $\mathbb {X}^{*}$ a lattice structure which satisfies a lower $q_0'$-estimate. Let $\mathcal {O}^{*}$ denote the biorthogonal system $(\boldsymbol {x}_n^{*},h_\mathbb {X}(\boldsymbol {x}_n))_{n=1}^{\infty }$, where $h_\mathbb {X}\colon \mathbb {X}\to \mathbb {X}^{**}$ is the bidual map. By the already proved Hilbertian part, $J_H[\mathcal {O}^{*},\mathbb {X}^{*}]\subseteq [1,q_0']$. By duality, $J_B[\mathcal {O},\mathbb {X}]\subseteq (J_H[\mathcal {O}^{*},\mathbb {X}^{*}])'$. Consequently, $J_B[\mathcal {O},\mathbb {X}]\subseteq [1,q_0']'=[q_0,\infty ]$.

We conclude the proof by noticing that $\ell _p$ satisfies both a lower and an upper $p$-estimate.

To establish our results as accurately as possible, we will need notions more general than $p$-Besselian and $p$-Hilbertian bases. A sequence space will be a Banach space or, more generally, a quasi-Banach space $\mathbb {U}\subseteq \mathbb {F}^{\mathbb {N}}$ for which the unit vector system

\[ \mathcal{E}=(\text{e}_n)_{n=1}^{\infty} \]

is a normalized $1$-unconditional basis. We say that a biorthogonal system $\mathcal {O}$ in a Banach space $\mathbb {X}$ is $\mathbb {U}$-Besselian (resp., $\mathbb {U}$-Hilbertian) if the coefficient transform (resp., the series transform) is a bounded operator from $\mathbb {X}$ into $\mathbb {U}$ (resp., from $\mathbb {U}$ into $\mathbb {X}$).

Given sequences $\mathcal {X}=(\boldsymbol {x}_n)_{n=1}^{\infty }$ and $\mathcal {Y}=(\boldsymbol {y}_n)_{n=1}^{\infty }$ in Banach spaces $\mathbb {X}$ and $\mathbb {Y}$, respectively, their direct sum is the sequence $\mathcal {X}\oplus \mathcal {Y}=({{\boldsymbol {u}}}_n)_{n=1}^{\infty }$ in $\mathbb {X}\oplus \mathbb {Y}$ given by

\[ {{\boldsymbol{u}}}_{2n-1}=(\boldsymbol{x}_n,0), \quad {{\boldsymbol{u}}}_{2n}=(0,\boldsymbol{y}_n), \quad n\in\mathbb{N}. \]

We put $\mathcal {X}^{2}=\mathcal {X}\oplus \mathcal {X}$, and we call $\mathcal {X}^{2}$ the square of $\mathcal {X}$. If $\mathcal {X}$ is a semi-normalized Schauder basis of $\mathbb {X}$, and $\mathcal {Y}$ is a semi-normalized Schauder basis of $\mathbb {Y}$, then $\mathcal {X}\oplus \mathcal {Y}$ is a seminormalized Schauder basis of $\mathbb {X}\oplus \mathbb {Y}$ whose dual basis is, modulus the natural identification of $(\mathbb {X}\oplus \mathbb {Y})^{*}$ with $\mathbb {X}^{*}\oplus \mathbb {Y}^{*}$, $\mathcal {X}^{*}\oplus \mathcal {Y}^{*}$. Given biorthogonal systems $\mathcal {O}=(\boldsymbol {x}_n,\boldsymbol {x}_n^{*})_{n=1}^{\infty }$ and $\mathcal {R}=(\boldsymbol {y}_n,\boldsymbol {y}_n^{*})_{n=1}^{\infty }$ in Banach spaces $\mathbb {X}$ and $\mathbb {Y}$, respectively, their direct sum is the biorthogonal system $\mathcal {O}\oplus \mathcal {R}$ in $\mathbb {X}\oplus \mathbb {Y}$ whose first and second components are $(\boldsymbol {x}_n)_{n=1}^{\infty }\oplus (\boldsymbol {y}_n)_{n=1}^{\infty }$ and $(\boldsymbol {x}^{*}_n)_{n=1}^{\infty }\oplus (\boldsymbol {y}^{*}_n)_{n=1}^{\infty }$, respectively.

The rotation of a sequence $\mathcal {X}=(\boldsymbol {x}_n)_{n=1}^{\infty }$ in a Banach space $\mathbb {X}$ will be the sequence $\mathcal {X}_\diamond =(\boldsymbol {y}_n)_{n=1}^{\infty }$ given by

\[ \boldsymbol{y}_{2n-1}=\frac{\boldsymbol{x}_{2n-1}-\boldsymbol{x}_{2n}}{\sqrt{2}},\quad \boldsymbol{y}_{2n}=\frac{\boldsymbol{x}_{2n-1}+\boldsymbol{x}_{2n}}{\sqrt{2}}, \quad n\in\mathbb{N}. \]

If $\mathcal {X}$ is Schauder basis, then $\mathcal {X}_\diamond$ is a Schauder basis with dual basis $(\mathcal {X}^{*})_\diamond$. Given a biorthogonal system $\mathcal {O}=(\boldsymbol {x}_n,\boldsymbol {x}_n^{*})_{n=1}^{\infty }$ in $\mathbb {X}$, its rotation is the biorthogonal system $\mathcal {O}_\diamond$ in $\mathbb {X}$ whose first and second components are the rotations of $(\boldsymbol {x}_n)_{n=1}^{\infty }$ and $(\boldsymbol {x}^{*}_n)_{n=1}^{\infty }$, respectively.

We say that a sequence space $\mathbb {U}$ is lattice isomorphic to its square if the unit vector system of $\mathbb {U}$ is equivalent to its square.

Lemma 2.3 below, whose straightforward proof we omit, gathers some properties of these notions that we will need.

The authors of [Reference Albiac, Ansorena and Wojtaszczyk5] introduced an alternative quantitative measure of the unconditionality of a biorthogonal system $\mathcal {O}=(\boldsymbol {x}_n,\boldsymbol {x}_n^{*})_{n=1}^{\infty }$ which is more accurate in some situations. Set

\[ \mathbb{N}[m]=\{n\in\mathbb{Z} \colon 1\le n\le m\} \]

and define

\[ \boldsymbol{\tilde{k}}_m=\boldsymbol{\tilde{k}}_m[\mathcal{O},\mathbb{X}] :=\sup\{ \Vert{S_A(f)}\Vert \colon \Vert{f}\Vert\le 1,\; \mathop{\mathrm{supp}}\nolimits(f) \subseteq\mathbb{N}[m],\; A\subseteq\mathbb{N}\}. \]

We have $\boldsymbol {\tilde {k}}_m\le \boldsymbol {k}_m$ for all $m\in \mathbb {N}$, and $\sup _m\boldsymbol {\tilde {k}}_m=\sup _m\boldsymbol {k}_m$.

If $\mathcal {O}$ and $\mathcal {R}$ are biorthogonal systems in Banach spaces $\mathbb {X}$ and $\mathbb {Y}$, then

(2.2)\begin{align} \boldsymbol{\tilde{k}}_m[\mathcal{O}\oplus \mathcal{R},\mathbb{X}\oplus\mathbb{Y}]& = \max\{\boldsymbol{\tilde{k}}_{\lceil m/2\rceil}[\mathcal{O},\mathbb{X}], \boldsymbol{\tilde{k}}_{\lfloor m/2\rfloor}[\mathcal{R},\mathbb{Y}]\}, \end{align}

(2.3)\begin{align} \boldsymbol{k}_m[\mathcal{O}\oplus \mathcal{R},\mathbb{X}\oplus\mathbb{Y}]& =\max\{\boldsymbol{k}_m[\mathcal{O},\mathbb{X}], \boldsymbol{k}_m[\mathcal{R},\mathbb{Y}]\}. \end{align}

Loosely speaking, we could say that $\mathcal {O}\oplus \mathcal {R}$ inherits naturally the properties of $\mathcal {O}$ and $\mathcal {R}$. In contrast, ‘rotating’ $\mathcal {O}\oplus \mathcal {R}$ gives rise to more interesting situations. Set $\mathcal {O}\diamond \mathcal {R}:=(\mathcal {O}\oplus \mathcal {R})_\diamond$ and define

\begin{align*} \boldsymbol{\tilde{\delta}}_m[\mathcal{O},\mathbb{X},\mathcal{R},\mathbb{Y}]& =\sup_{\substack{A\subseteq\mathbb{N}[m]\\ (a_n)_{n\in A}\in\mathbb{F}^{A}\setminus\{0\}}} \frac{\Vert{\sum_{n\in A} a_n \,\boldsymbol{x}_n}\Vert}{\Vert{\sum_{n\in A} a_n \, \boldsymbol{y}_n}\Vert},\quad m\in\mathbb{N},\\ \boldsymbol{\delta}_m[\mathcal{O},\mathbb{X},\mathcal{R},\mathbb{Y}]& =\sup_{\substack{\vert{A}\vert\le m\\ (a_n)_{n\in A}\in\mathbb{F}^{A}\setminus\{0\}}} \frac{\Vert{\sum_{n\in A} a_n\, \boldsymbol{x}_n}\Vert}{\Vert{\sum_{n\in A} a_n\, \boldsymbol{y}_n}\Vert}, \quad m\in\mathbb{N}. \end{align*}

Lemma 2.4 Let $\mathcal {O}$ and $\mathcal {R}$ be biorthogonal systems in Banach spaces $\mathbb {X}$ and $\mathbb {Y}$ respectively. Then

\begin{align*} \boldsymbol{k}_{2m}[\mathcal{O}\diamond \mathcal{R},\mathbb{X}\oplus\mathbb{Y}]& \ge \frac{1}{2}\max\{\boldsymbol{\delta}_{m}[\mathcal{O},\mathbb{X},\mathcal{R},\mathbb{Y}], \boldsymbol{\delta}_{m}[\mathcal{R},\mathbb{Y},\mathcal{O},\mathbb{X}]\},\ \text{and}\\ \boldsymbol{\tilde{k}}_{2m}[\mathcal{O}\diamond \mathcal{R},\mathbb{X}\oplus\mathbb{Y}]& \ge \frac{1}{2}\max\{\boldsymbol{\tilde{\delta}}_{m}[\mathcal{O},\mathbb{X},\mathcal{R},\mathbb{Y}], \boldsymbol{\tilde{\delta}}_{m}[\mathcal{R},\mathbb{Y},\mathcal{O},\mathbb{X}]\}. \end{align*}

The parameters $(\boldsymbol {\delta }_m)_{m=1}^{\infty }$ also serve to estimate the conditionality parameters of Schauder bases, or biorthogonal systems, squeezed between two sequence spaces. If $\mathbb {U}_1$ and $\mathbb {U}_2$ are sequence spaces, we set

\[ \boldsymbol{\delta}_m[\mathbb{U}_1,\mathbb{U}_2]=\boldsymbol{\delta}_m[\mathcal{E},\mathbb{U}_1,\mathcal{E},\mathbb{U}_2], \quad m\in\mathbb{N}. \]

Lemma 2.5 Let $\mathcal {X}$ be a biorthogonal system of the Banach space $\mathbb {X}$. Suppose that $\mathcal {X}$ $\mathbb {U}_1$-Hilbertian and $\mathbb {U}_2$-Besselian. Then,

\[ \boldsymbol{k}_m[\mathcal{X},\mathbb{X}]\lesssim \boldsymbol{\delta}_m[\mathbb{U}_1,\mathbb{U}_2]. \]

Proof. Let $C_1$ be the norm of the series transform, and $C_2$ be the norm of the coefficient transform. Pick $f\in \mathbb {X}$ and $A\subseteq \mathbb {N}$ with $\vert {A}\vert \le m$. We have

\begin{align*} \Vert{S_A(f)}\Vert& \le C_1 \Vert{(\boldsymbol{x}_n^{*}(f) \mathbf{1}_A(n))_{n=1}^{\infty}}\Vert_{\mathbb{U}_1}\\ & \le C_1 \boldsymbol{\delta}_m[\mathbb{U}_1,\mathbb{U}_2] \Vert{(\boldsymbol{x}_n^{*}(f)\mathbf{1}_A(n))_{n=1}^{\infty}}\Vert_{\mathbb{U}_2}\\ & \le C_1 \boldsymbol{\delta}_m[\mathbb{U}_1,\mathbb{U}_2] \Vert{(\boldsymbol{x}_n^{*}(f))_{n=1}^{\infty}}\Vert_{\mathbb{U}_2}\\ & \le C_1 C_2 \boldsymbol{\delta}_m[\mathbb{U}_1,\mathbb{U}_2] \Vert{f}\Vert. \end{align*}

The following consequence of lemma 2.5, whose straightforward proof we omit, single out an argument that we will use several times.

For further reference, we record the value of the parameters $\boldsymbol {\delta }_m$ in some important cases. Given a sequence $f=(a_n)_{n=1}^{\infty }\in \mathbb {F}^{\mathbb{N}}$ and an increasing map $\pi \colon \mathbb {N}\to \mathbb {N}$, let $f_\pi =(b_n)_{n=1}^{\infty }\in \mathbb {F}^{\mathbb{N}}$ be the sequence defined by $b_n=a_k$ if $n=\pi (k)$ for some $k\in \mathbb {N}$, and $b_n=0$ otherwise. A sequence space $\mathbb {S}$ is said to be subsymmetric if $f_\pi \in \mathbb {S}$ for every $f\in \mathbb {S}$ and every increasing map $\pi \colon \mathbb {N}\to \mathbb {N}$, and we have $\Vert {f_\pi }\Vert _\mathbb {S}=\Vert {f}\Vert _\mathbb {S}$. In general, $\boldsymbol {\delta }_m[\mathbb {S}_1,\mathbb {S}_2]=\boldsymbol {\tilde {\delta }}_m[\mathbb {S}_1,\mathbb {S}_2]$ whenever $\mathbb {S}_1$ and $\mathbb {S}_2$ are subsymmetric sequence spaces. As $\ell _p$-spaces are concerned, we have

(2.4)\begin{equation} \boldsymbol{\delta}_m[\ell_q,\ell_r]=\boldsymbol{\tilde{\delta}}_m[\ell_q,\ell_r]=m^{1/q-1/r},\quad m\in\mathbb{N},\; 0< q \le r\le \infty. \end{equation}

Note that $\ell _p$ is not locally convex space in the case when $0< p<1$. However, the parameters $\boldsymbol {\delta }$ and $\boldsymbol {\tilde {\delta }}$ still make sense in the nonlocally convex setting. Other subsymmetric sequence spaces of interest for us are Lorentz sequence spaces. Notice that

(2.5)\begin{equation} \left\|\sum_{n=1}^{m} \frac{1}{n^{1/p}} \text{e}_n\right\|_{\ell_{p,q}}=H_m^{1/q}, \quad m\in\mathbb{N},\; 0< p<\infty,\; 0< q \le \infty, \end{equation}

where $H_m=\sum _{n=1}^{m} 1/n$ is the $m$th harmonic number. Combining this identity with Hölder's inequality we obtain

(2.6)\begin{equation} \boldsymbol{\delta}_m[\ell_{p,q},\ell_{p,r}]=H_m^{1/q-1/r}, \quad m\in\mathbb{N},\; 0< p<\infty,\; 0< q \le r \le \infty. \end{equation}

We also record the weighted version of (2.6). Given $0< q\le \infty$ and a weight ${{\boldsymbol {w}}}=(w_n)_{n=1}^{\infty }$ whose primitive weight $(s_n)_{n=1}^{\infty }$ is doubling, the Lorentz sequence space $d_{1,q}({{\boldsymbol {w}}})$ is the quasi-Banach space consisting of all sequences $f\in c_0$ whose non-increasing rearrangement $(a_n)_{n=1}^{\infty }$ satisfies

\[ \Vert{f}\Vert_{d_{1,q}({{\boldsymbol{w}}})}=\left(\sum_{n=1}^{\infty} (s_n a_n)^{q} \frac{w_n}{s_n}\right)^{1/q}<\infty, \]

with the usual modification in $q=\infty$. If ${{\boldsymbol {w}}}=(n^{1/p-1})_{n=1}^{\infty }$, then $d_{1,q}({{\boldsymbol {w}}})=\ell _{p,q}$ up to an equivalent norm. We have

(2.7)\begin{equation} \boldsymbol{\delta}_m[d_{1,q}({{\boldsymbol{w}}}),d_{1,r}({{\boldsymbol{w}}})]=(H_m[{{\boldsymbol{w}}}])^{1/q-1/r},\quad m\in\mathbb{N},\ 0< q \le r \le \infty, \end{equation}

where $H_m[{{\boldsymbol {w}}}]=\sum _{n=1}^{m} w_n/s_n$. If

\[ s_{rm}\ge 2 s_m, \quad m\in\mathbb{N}, \]

for some integer $r$, in which case we say that $(s_m)_{m=1}^{\infty }$ has the lower regularity property (LRP for short), the growth of the sequence $(H_m[{{\boldsymbol {w}}}])_{m=1}^{\infty }$ can be computed.

Lemma 2.7 Let ${{\boldsymbol {w}}}$ be a weight whose primitive sequence $(s_m)_{m=1}^{\infty }$ has the LRP, and let $0< q\le \infty$. Then,

\[ \Vert{f}\Vert_{d_{1,q}({{\boldsymbol{w}}})}\approx \left(\sum_{n=1}^{\infty} (s_n a_n)^{q} \frac{1}{n}\right)^{1/q} \]

for every $f\in c_0$ with non-increasing rearrangement $(a_n)_{n=1}^{\infty }$. In particular, $H_m[{{\boldsymbol {w}}}]\approx H_m$ for $m\in \mathbb {N}$.

Proof. Let ${{\boldsymbol {w}}}'=(w_n')_{n=1}^{\infty }$ be the weight defined by $w_n'=s_n/n$, and let $(s_n')_{n=1}^{\infty }$ be its primitive sequence. Since $(s_n)_{n=1}^{\infty }$ has the LRP, $s_n\approx s_n'$ for $n\in \mathbb {N}$. Consequently, $\Vert {f}\Vert _{d_{1,q}({{\boldsymbol {w}}})}\approx \Vert {f}\Vert _{d_{1,q}({{\boldsymbol {w}}}')}$ for $f\in c_0$ (see, e.g., [Reference Albiac, Ansorena, Berná and Wojtaszczyk3, §9.2]). Since $w_n'/s_n'\approx 1/n$ for $n\in \mathbb {N}$, the desired equivalence of quasi-norms holds. To obtain the equivalence for $H_m[{{\boldsymbol {w}}}]$, we apply the equivalence between quasi-norms with $q=1$ and

\[ a_n=\begin{cases} 1/s_n & \text{ if } n\le m, \\ 0 & \text{ if } n>m,\end{cases} \]

where $m$ runs over $\mathbb {N}$.

We conclude this preliminary section with another equivalence for the quasi-norms of weighted Lorentz sequence spaces.

Lemma 2.8 (see [Reference Albiac, Ansorena, Berná and Wojtaszczyk3, equation (8.3)])

Let ${{\boldsymbol {w}}}$ be a weight whose primitive sequence $(s_n)_{n=1}^{\infty }$ is doubling, and let $0< q<\infty$. Then

\[ \Vert{f}\Vert_{d_{1,q}({{\boldsymbol{w}}})}\approx \left(\sum_{n=1}^{\infty} a_n^{q} (s_n^{q}-s_{n-1}^{q})\right)^{1/q} \]

for every $f\in c_0$ with non-increasing rearrangement $(a_n)_{n=1}^{\infty }$.

3. Fourier coefficients of functions in $L_2(\mathbb {T}, \vert {t}\vert ^{\lambda }\,{\rm d}t)$, $\vert {\lambda }\vert <1$

With the aid of proposition 2.1, we make our first move toward the proof of theorem 1.3.

Lemma 3.1 Let $0\le \alpha <1$. Then,

\[ J_B[\mathcal{T},\boldsymbol{H}_{-\alpha}]=[2,\infty]\quad \text{and}\quad J_H[\mathcal{T},\boldsymbol{H}_\alpha]=[1,2]. \]

The authors of [Reference Garrigós and Wojtaszczyk16] computed the norm in $\boldsymbol {H}_{\lambda }$, $-1<\lambda <1$, of the Dirichlet kernel $(D_m)_{m=0}^{\infty }$ defined by

\[ D_m=\sum_{n={-}m}^{m} \tau_n=\sum_{n=0}^{2m} \phi_m=\sum_{n=0}^{2m} \phi_m^\mathbb{R}, \quad m\in\mathbb{N}. \]

For the reader's ease, we record this result of Garrigós and Wojtaszczyk that we will use a couple of times.

Lemma 3.2 (see [Reference Garrigós and Wojtaszczyk16, lemma 3.7])

Let $-1<\lambda <1$. Then

\[ \Vert{D_m}\Vert_{\boldsymbol{H}_\lambda}\approx m^{(1-\lambda)/2},\quad m\in\mathbb{N}. \]

As a matter of fact, lemma 3.2 provides valuable information on the Besselian and Hilbertian intervals of the trigonometric system in $\boldsymbol {H}_\lambda$. Recall that, given $0\le \alpha <1$, $q_\alpha ={2}/{(1+\alpha )}$ and $r_\alpha ={2}/{(1-\alpha )}$.

Lemma 3.3 Let $0\le \alpha <1$. Then,

\[ J_B[\mathcal{T},\boldsymbol{H}_\alpha]\subseteq[r_\alpha,\infty]\quad \text{and}\quad J_H[\mathcal{T},\boldsymbol{H}_{-\alpha}]\subseteq[1,q_\alpha]. \]

Proof. Pick $s\in [1,\infty ]$ and suppose that $\mathcal {T}$ is a $s$-Besselian basis of $\boldsymbol {H}_\alpha$ (resp., a $s$-Hilbertian basis of $\boldsymbol {H}_{-\alpha }$). Then,

\[ \sup_m \frac{\Vert{ \sum_{n={-}m}^{m} \text{e}_n}\Vert_s}{\Vert{D_m}\Vert_{\boldsymbol{H}_\alpha}}<\infty\; \left(\text{resp.,}\; \sup_m \frac{\Vert{D_m}\Vert_{\boldsymbol{H}_{-\alpha}}}{\Vert{\sum_{n={-}m}^{m} \text{e}_n}\Vert_s}<\infty\right). \]

Since $\Vert {\sum _{n=-m}^{m} \text{e} _n}\Vert _s\approx m^{1/s}$ for $m\in \mathbb {N}$, we infer from lemma 3.2 that $1/s\le (1-\alpha )/2$ (resp., $(1+\alpha )/2\le 1/s$). Hence, $r_\alpha \le s$ (resp., $s\le q_\alpha$).

To help the reader to grasp the issue of the optimality of lemma 3.3, we note that combining the embedding

\[ \boldsymbol{H}_{\alpha} \subseteq L_q([{-}1/2,1/2]),\quad 0<\alpha<1,\ 1\le q< q_\alpha, \]

which follows from Hölder's inequality, with Hausdorff–Young inequality, and taking into account that $r_\alpha$ and $q_\alpha$ are conjugate exponents, yields $(r_\alpha,\infty ]\subseteq J_B[\mathcal {T},\boldsymbol {H}_\alpha ]$. So, only whether the trigonometric system is a $r_\alpha$-Besselian basis of $\boldsymbol {H}_{\alpha }$ is in doubt. To answer this question, we need to introduce some terminology. Let $\mathbb {E}=\{ a\in \mathbb {F} \colon \vert {a}\vert =1\}$. The fundamental function of a basis $\mathcal {X}=(\boldsymbol {x}_n)_{n=1}^{\infty }$ of a Banach space $\mathbb {X}$ is defined as

\[ \boldsymbol{\varphi}_m[\mathcal{X},\mathbb{X}]= \sup\left\{\Vert{\mathbf{1}_{\varepsilon,A}[\mathcal{X}]}\Vert \colon \varepsilon\in\mathbb{E}^{A},\; \vert{A}\vert\le m\right\}, \]

where, for $A\subseteq \mathbb {N}$ finite and scalars $\varepsilon =(\varepsilon _n)_{n\in A}\in \mathbb {E}^{A}$,

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

Theorem 3.4 below is the last step on our route toward proving theorem 1.3.

Proof. Let $-1<\lambda <1$. We have $\boldsymbol {H}_\lambda ^{*}=\boldsymbol {H}_{-\lambda }$ via the dual pairing

\[ (f,g)\mapsto \int_{{-}1/2}^{1/2} f(t)g(t)\,{\rm d}t, \quad f\in \boldsymbol{H}_\lambda, \; g\in \boldsymbol{H}_{-\lambda}. \]

Moreover, $\mathcal {T}$, regarded as a basis of $\boldsymbol {H}_{-\lambda }$, is the dual basis of the own system $\mathcal {T}$ regarded as a basis of $\boldsymbol {H}_\lambda$. Consequently, by duality, it suffices to prove (ii). To that end, we pick $0<\alpha <\beta <1$. Given $A\subseteq \mathbb {Z}$ we set

\[ J_n[A]=\{1+\vert{n-k}\vert\colon k\in A\},\quad n\in A. \]

Since, for some constant $C_\beta$,

(3.1)\begin{equation} \vert{\widehat{{{\boldsymbol{w}}}_{-\beta}}(n)}\vert\le \frac{C_\beta}{(1+\vert{n}\vert)^{1-\beta}}, \quad n\in\mathbb{Z}, \end{equation}

(see [Reference Garrigós and Wojtaszczyk16, lemma A.2]), for every $m\in \mathbb {N}$, every $A\subseteq \mathbb {Z}$ with $\vert {A}\vert \le m$, and every $\varepsilon =(\varepsilon _n)_{n\in A}\in \mathbb {T}^{A}$ we have

\begin{align*} \Vert{\mathbf{1}_{\varepsilon,A}[\mathcal{T},\boldsymbol{H}_{-\beta}]}\Vert^{2}& =\sum_{(n,k)\in A^{2}}\varepsilon_n \overline{\varepsilon_k} \widehat{{{\boldsymbol{w}}}_{-\beta}}(k-n)\\ & \le C_\beta \sum_{n\in A} \sum_{k\in A} \frac{1}{(1+\vert{n-k}\vert)^{1-\beta}}\\ & \le 2 C_\beta \sum_{n\in A} \sum_{j\in J_n[A]} \frac{1}{j^{1-\beta}}\\ & \le 2 C_\beta m \sum_{j=1}^{m} \frac{1}{j^{1-\beta}}\\ & \le \frac{2C_\beta}{\beta} m^{1+\beta}. \end{align*}

We have obtained

\[ \boldsymbol{\varphi}_m[\mathcal{T}, \boldsymbol{H}_{-\beta}]\le \sqrt{\frac{2 C_\beta} {\beta}} m^{1/q_\beta}, \quad m\in\mathbb{N}. \]

Therefore, by [Reference Berná, Blasco, Garrigós, Hernández and Oikhberg11, theorem 1.2], the Fourier series transform is a bounded operator from $\ell _{q_\beta,1}$ into $\boldsymbol {H}_{-\beta }$.

By orthogonality, the Fourier series transform is a bounded operator from $\ell _2$ into $\boldsymbol {H}_0$. Pick $0<\theta <1$. By interpolation (by means of the real method) the Fourier series transform is a bounded operator from $\ell _{q,2}$ into $\boldsymbol {H}_{-\gamma }$, where

\[ \frac{1}{q}= \frac{\theta}{q_\beta} + \frac{ 1-\theta}{2}, \quad \gamma=\theta\beta. \]

(see [Reference Bergh and Löfström10, theorems 5.3.1 and 5.4.1]). If we choose $\theta =\alpha /\beta$, then $\gamma =\alpha$ and

\[ \frac{1}{q}= \frac{\alpha(1+\beta)}{2\beta} + \frac{\beta-\alpha}{2\beta}=\frac{1+\alpha}{2}, \]

that is, $q=q_\alpha$.

Theorem 1.3 says, in particular, that $q_\alpha$ is the optimal index $q$ such that the trigonometric system in $\boldsymbol {H}_{-\alpha }$ if $q$-Hilbertian, and $r_\alpha$ is the optimal index $r$ such that the trigonometric system in $\boldsymbol {H}_\alpha$ if $r$-Besselian. What remains of this section is devoted to proving that the estimates obtained in theorem 3.4 are also optimal in the ‘secondary’ index. To that end, we need to compute, up to equivalence, the Fourier coefficients of the power weight ${{\boldsymbol {w}}}_\lambda$, $-1<\lambda <0$.

Lemma 3.5 (cf. equation (3.1))

Let $0<\alpha <1$. Then,

\[ \widehat{{{\boldsymbol{w}}}_{-\alpha}}(n)\approx \frac{1}{(1+\vert{n}\vert)^{1-\alpha}}, \quad n\in\mathbb{Z}. \]

Proof. Pick $n\in \mathbb {N}$. We have $\widehat {{{\boldsymbol {w}}}_{-\alpha }}(-n)=\widehat {{{\boldsymbol {w}}}_{-\alpha }}(n)$ and

\[ \widehat{{{\boldsymbol{w}}}_{-\alpha}}(n)=2\int_0^{1/2} \frac{\cos(2\pi n t)}{t^{\alpha}}\,{\rm d}t=\frac{2^{\alpha}}{n^{1-\alpha}} A_n, \]

where $A_n=\int _0^{n} \cos (\pi x) x^{-\alpha }\,{\rm d}x$. Set $G(k,x)=(k+x-1)^{-\alpha }-(k-x)^{-\alpha }$ for $k\in \mathbb {N}$ and $0< x<1/2$. We have

\begin{align*} A_n& = \sum_{k=1}^{n} \int_{k-1}^{k} \frac{\cos(\pi x)}{ x^{\alpha}}\,{\rm d}x\\ & =\sum_{k=1}^{n} ({-}1)^{k-1} \int_{0}^{1} \frac{\cos(\pi x)}{ (k-1+x)^{\alpha}}\,{\rm d}x\\ & =\sum_{k=1}^{n} ({-}1)^{k-1} \int_0^{1/2} G(k,x) \cos(\pi x)\,{\rm d}x. \end{align*}

Since $(G(k,x))_{k=1}^{\infty }$ is nonnegative and nonincreasing for each $x\in (0,1/2)$, an application of Leibniz test for alternating series gives that there exists $\lim _n A_n\in (0,\infty )$, and that $A_n>0$ for all $n\in \mathbb {N}$.

Remark 3.6 The precise value of $A(\alpha ):=\lim _n A_n$, where $(A_n)_{n=1}^{\infty }$ is as in the proof of lemma 3.5, is known. Namely,

\[ A(\alpha)=\pi^{\alpha-1} \cos \left(\frac{(1-\alpha)\pi}{2} \right)\Gamma(1-\alpha), \quad 0<\alpha<1, \]

(see [Reference Zygmund29, §2, equation (3.10)]).

Proof. Since $\ell _{p_1,s_1}\subseteq \ell _{p_2,s_2}$ if $p_1< p_2$, or $p_1=p_2$ and $s_1\le s_2$, in light of theorem 3.4 it suffices to prove that

• • if the coefficient transform is bounded from $\boldsymbol {H}_\alpha$ into $\ell _{r_\alpha,s}$, then $s\ge 2$; and that

• • if Fourier series transform is a bounded operator from $\ell _{q_\alpha,s}$ into $\boldsymbol {H}_{-\alpha }$, then $s\le 2$.

Since the former assertion can be deduced from the latter by duality, it suffices to prove the latter one.

Let $(f_m)_{m=1}^{\infty }$ be the sequence of trigonometric polynomials defined by

\[ f_m(t)=\sum_{n=1}^{m} \frac{1}{n^{1/q_\alpha}}\,{\rm e}^{2\pi i n t}, \quad \frac{-1}{2} \le t \le \frac{1}{2}, \quad m\in\mathbb{N}. \]

By lemma 3.5, for $m\in \mathbb {N}$ we have

\begin{align*} \Vert{f_m}\Vert^{2}_{\boldsymbol{H}_{-\alpha}} & =\sum_{1\le n,k\le m} \frac{\widehat{{{\boldsymbol{w}}}_{-\alpha}}(k-n)}{(nk)^{(1+\alpha)/2}} \approx \sum_{n=1}^{m}\sum_{k=1}^{n} \frac{(1+n-k)^{\alpha-1}}{ (nk)^{(1+\alpha)/2}}\\ & =\sum_{n=1}^{m} \frac{D_n}{n^{(1+\alpha)/2}}, \end{align*}

where

\[ D_n=\sum_{k=1}^{n} \frac{1}{k^{(1+\alpha)/2}} \frac{1}{(1+n-k)^{1-\alpha}}. \]

Set

\begin{align*} B_n& =\int_0^{n} \frac{{\rm d}x}{x^{(1+\alpha)/2} (n-x)^{1-\alpha}},\quad n\in\mathbb{N}, \ \text{and}\\ R_n& ={-}\frac{1}{n^{1-\alpha}}-\frac{1}{n^{(1+\alpha)/2}} +\frac{2/(1-\alpha)}{(n-1)^{1-\alpha}}+\frac{1/\alpha}{(n-1)^{(1+\alpha)/2}},\quad n\in\mathbb{N},\ n\ge 2. \end{align*}

We have

\begin{align*} B_n& =\frac{\beta((1-\alpha)/2,\alpha)}{n^{(1-\alpha)/2}},\ n\in\mathbb{N},\\ D_n& \le B_n\le D_n+R_n,\ n\in\mathbb{N},\ n\ge 2,\ \text{and}\\ 0& =\lim_n n^{(1-\alpha)/2} R_n. \end{align*}

We infer that $D_n\approx n^{-(1-\alpha )/2}$ for $n\in \mathbb {N}$. Consequently,

(3.2)\begin{equation} \Vert{f_m}\Vert_{\boldsymbol{H}_{-\alpha}}\approx H_m^{1/2},\quad m\in\mathbb{N}. \end{equation}

Suppose that the Fourier series transform is a bounded operator from $\ell _{q_\alpha,s}$ into $\boldsymbol {H}_{-\alpha }$. Combining (3.2) with (2.5) gives $s\le 2$.

4. Conditionality parameters of Schauder bases

The trigonometric system $\mathcal {T}$, regarded as a sequence in $\boldsymbol {H}_\lambda$, $0<\vert {\lambda }\vert <1$, is the first example of a conditional Schauder basis of a Hilbert space arisen in the literature (see [Reference Gelbaum17]). In hindsight, that $\mathcal {T}$ is not unconditional can be deduced from combining the papers [Reference Köthe and Toeplitz22], where it is proved that every semi-normalized unconditional basis of $\ell _2$ is equivalent to its unit vector system, and [Reference Al'tman1], where it is proved that $\mathcal {T}$, regarded as a sequence in $\boldsymbol {H}_\lambda$, is not equivalent to the unit vector system of $\ell _2$. Notice that the last-mentioned result can be deduced from theorem 1.3, and also from lemma 3.2. So, it shouldn't be surprising that theorem 1.3 enables us to move forward with the theory of conditional bases.

Theorem 4.1 For each $1< q\le 2 \le r<\infty$ there is a $q$-Hilbertian $r$-Besselian Schauder basis $\mathcal {X}$ of $\ell _2$ with

\[ \boldsymbol{k}_m[\mathcal{X},\ell_2]\approx \boldsymbol{\tilde{k}}_m[\mathcal{X},\ell_2] \approx m^{1/q-1/r}, \quad m\in\mathbb{N}. \]

Moreover, $\mathcal {X}$ is not $\ell _{q_1}$-Hilbertian for any $q_1>q$ nor $\ell _{r_1}$-Besselian for any $r_1< r$.

Proof. Pick $0<\alpha <1$ with $r_\alpha =r$, and $0<\beta <1$ with $q_\beta =q$. By theorem 1.3, lemma 2.3 and lemma 2.4, the rotated system $\mathcal {X}:=\mathcal {T}^{\mathbb {R}} \diamond \mathcal {T}^{\mathbb {R}}$ is a $q$-Hilbertian and $r$-Besselian Schauder basis of the Hilbert space $\boldsymbol {H}:=\boldsymbol {H}_{-\beta }\oplus \boldsymbol {H}_\alpha$ with

\[ 2 \boldsymbol{\tilde{k}}_{2m} [\mathcal{X},\boldsymbol{H}] \ge d_m:=\boldsymbol{\tilde{\delta}}_m[ \mathcal{T}^{\mathbb{R}},\boldsymbol{H}_{-\beta},\mathcal{T}^{\mathbb{R}},\boldsymbol{H}_\alpha], \quad m\in\mathbb{N}. \]

In turn, by lemma 3.2,

\[ d_{2m+1}\ge \frac{\Vert{D_m}\Vert_{\boldsymbol{H}_{-\beta}}} {\Vert{D_m}\Vert_{\boldsymbol{H}_\alpha}}\approx m^{(1+\beta)/2-(1-\alpha)/2}= m^{1/q-1/r}, \quad m\in\mathbb{N}. \]

We infer that $\boldsymbol {\tilde {k}}_{m}[\mathcal {X}]\gtrsim m^{1/q-1/r}$ for $m\in \mathbb {N}$. In light of (2.4), an application of corollary 2.6 puts an end to the proof.

We will derive theorem 1.4 from the first of the two consequences of theorem 4.1 that we record below. The second one will be used in § 5.

Before proceeding with the proof of these results, it will be convenient to introduce some notation. Given a sequence $\mathcal {X}=(\boldsymbol {x}_n)_{n=1}^{\infty }$ in a Banach space $\mathbb {X}$ we set

\[ \mathcal{X}^{(m)}=(\boldsymbol{x}_n)_{n=1}^{m}, \quad m\in\mathbb{N}. \]

If $(\mathbb {X}_j)_{j=1}^{\infty }$ are Banach spaces, we denote by $L_j$ the canonical embedding of $\mathbb {X}_j$ into $\prod _{j=1}^{\infty }\mathbb {X}_j$, $j\in \mathbb {N}$. For each $j\in \mathbb {N}$, let $\mathcal {X}_j=(\boldsymbol {x}_{j,n})_{n=1}^{m_j}$ be a finite family in the Banach space $\mathbb {X}_j$. We denote by $\bigoplus _{j=1}^{\infty } \mathcal {X}_j=(\boldsymbol {y}_k)_{k=1}^{\infty }$ the natural arrangement of the family

\[ \overline{\boldsymbol{x}}_{j,n}:=L_j(\boldsymbol{x}_{j,n}),\quad j\in\mathbb{N},\ 1\le n \le m_j, \]

that is, $\boldsymbol {y}_k=\overline {\boldsymbol {x}}_{j,n}$ if $k=\sum _{i=1}^{j-1}m_i +n$. Given a Schauder basis $\mathcal {X}$ and a sequence ${{\boldsymbol {m}}}=(m_j)_{j=1}^{\infty }$ in $\mathbb {N}$ we set

\[ \mathcal{X}^{({{\boldsymbol{m}}})}= \bigoplus_{j=1}^{\infty} \mathcal{X}^{(m_j)}. \]

Lemma 4.4 Let $\mathcal {X}=(\boldsymbol {x}_j)_{j=1}^{\infty }$ be a Schauder basis of a Banach space $\mathbb {X}$, ${{\boldsymbol {m}}}=(m_j)_{j=1}^{\infty }$ be a sequence in $\mathbb {N}$, and $\mathbb {U}$ be a sequence space. Set

\[ \mathbb{Y}:=\left(\bigoplus_{j=1}^{\infty} [\boldsymbol{x}_n \colon 1\le n \le m_j]\right)_\mathbb{U}. \]

1. (i) $\mathcal {X}^{({{\boldsymbol {m}}})}$ is a Schauder basis of $\mathbb {Y}$.

2. (ii) If ${{\boldsymbol {m}}}$ is unbounded, then $\boldsymbol {k}_m[\mathcal {X}^{({{\boldsymbol {m}}})},\mathbb {Y}]=\boldsymbol {k}_m[\mathcal {X},\mathbb {X}]$.

3. (iii) Let $1\le q\le \infty$. Suppose that both $\mathcal {X}$ and the unit vector system of $\mathbb {U}$ are $q$-Hilbertian (resp., $q$-Besselian). Then, $\mathcal {X}^{({{\boldsymbol {m}}})}$ is $q$-Hilbertian (resp., $q$-Besselian) regarded as a basis of $\mathbb {Y}$.

4. (iv) Suppose that $\sum _{i=1}^{j-1} m_i \lesssim m_j$ for $j\in \mathbb {N}$. Let $\delta \colon (0,\infty )\to (0,\infty )$ be a doubling function such that $\boldsymbol {\tilde {k}}_m[\mathcal {X},\mathbb {X}] \gtrsim \delta (m)$ for $m\in \mathbb {N}$. Then, $\boldsymbol {\tilde {k}}_m[\mathcal {X}^{({{\boldsymbol {m}}})},\mathbb {Y}] \gtrsim \delta (m)$ for $m\in \mathbb {N}$.

Proof of corollaries 4.2 and 4.3. To prove corollary 4.2, we choose $1< q\le 2\le r<\infty$ such that $\alpha =1/q-1/r$. Then, both to prove corollary 4.2 and corollary 4.3, we consider the Schauder basis $\mathcal {X}_{q,r}$ of $\ell _2$ provided by theorem 4.1. For each $m\in \mathbb {N}$, let $\boldsymbol {H}_m$ be the subspace of $\ell _2$ spanned by $\mathcal {X}_{q,r}^{(m)}$. Set ${{\boldsymbol {m}}}=(2^{j})_{j=1}^{\infty }$. By (2.3), and (2.4), lemma 4.4 and corollary 2.6, the sequence $\mathcal {B}\oplus \mathcal {X}^{({{\boldsymbol {m}}})}$ is a Schauder basis of $\mathbb {Y}:=\mathbb {X}\oplus \left (\bigoplus _{n=1}^{\infty } \boldsymbol {H}_{2^{n}} \right )_p$ satisfying the desired properties. We have

\[ \left( \bigoplus_{n=1}^{\infty} \boldsymbol{H}_{2^{n}} \right)_p\equiv \left( \bigoplus_{n=1}^{\infty} \ell_2^{2^{n}} \right)_p\simeq \ell_p \]

(see [Reference Pełczyński26, proof of theorem 7]). In turn, since $\ell _p$ is isomorphic to its square, $\mathbb {X}\oplus \ell _p\simeq \mathbb {X}$. We infer that $\mathbb {Y}\simeq \mathbb {X}$.

5. Conditionality parameters of almost greedy bases

Let us draw reader's attention to the existence of an almost greedy counterpart of theorem 1.1. Namely, the authors of [Reference Ansorena, Bello and Wojtaszczyk8] proved the following result.

In light of theorem 5.1, lemma 2.5 and equation (2.7), looking for embeddings involving Lorentz sequence spaces is a reasonable way of obtaining upper estimates for the conditionality parameters of almost greedy bases. Here, we focus on almost greedy bases arising from the so-called DKK-method invented in [Reference Dilworth, Kalton and Kutzarova12].

Let $(\mathbb {X},\Vert {\cdot }\Vert _\mathbb {X})$ be a Banach space with a semi-normalized Schauder basis $\mathcal {X}=(\boldsymbol {x}_n)_{n=1}^{\infty }$, and let $(\mathbb {S},\Vert {\cdot }\Vert _\mathbb {S})$ be a symmetric or, more generally, subsymmetric sequence space. Let $(\Lambda _m)_{m=1}^{\infty }$ be the fundamental function of the unit vector system of $\mathbb {S}$, that is,

\[ \Lambda_m=\Vert{\sum_{j=1}^{m} \text{e}_j}\Vert_\mathbb{S}, \quad m\in\mathbb{N}. \]

Let $\sigma =(\sigma _n)_{n=1}^{\infty }$ be an ordered partition of $\mathbb {N}$, i.e., a partition into integer intervals with

\[ \max(\sigma_n)<\min(\sigma_{n+1}),\quad n\in\mathbb{N}. \]

The averaging projection $P_\sigma \colon \mathbb {F}^{\mathbb{N}}\to \mathbb {F}^{\mathbb{N}}$ associated with the ordered partition $\sigma$ can be expressed as

\[ P_\sigma(f)=\sum_{n=1}^{\infty} \langle \boldsymbol{v}_n^{*},f\rangle\, \boldsymbol{v}_n, \]

where

(5.1)\begin{equation} \boldsymbol{v}_n= \frac{1}{\Lambda_{\vert{\sigma_n}\vert}} \sum_{j\in\sigma_n} \text{e}_j,\quad \boldsymbol{v}_n^{*}= \frac{\Lambda_{\vert{\sigma_n}\vert}}{\vert{\sigma_n}\vert}\sum_{j\in\sigma_n} \text{e}_j, \quad n\in\mathbb{N}, \end{equation}

and $\langle \cdot,\cdot \rangle$ is the natural dual pairing defined by

\[ \langle f, g\rangle = \sum_{n=1}^{\infty} a_n b_n, \quad f=(a_n)_{n=1}^{\infty}\in\mathbb{F}^\mathbb{N},\; g=(b_n)_{n=1}^{\infty}\in\mathbb{F}^\mathbb{N},\ fg\in\ell_1. \]

Let $Q_\sigma ={{\mathrm {Id}}}_{\mathbb {F}^{\mathbb {N}}}-P_\sigma$ be the complementary projection. We define $\Vert {\cdot }\Vert _{\mathcal {X},\mathbb {S},\sigma }$ on $c_{00}$ by

\begin{align*} \Vert{f}\Vert_{\mathcal{X},\sigma,\mathbb{S}}=\Vert{Q_\sigma(f)}\Vert_\mathbb{S}+\left\|\sum_{n=1}^{\infty} \boldsymbol{v}_n^{*}(f)\, \boldsymbol{x}_n\right\|\Vert_\mathbb{X}, \quad f\in c_{00}. \end{align*}

The completion of the normed space $(c_{00},\Vert {\cdot }\Vert _{\mathcal {X},\mathbb {S},\sigma })$ will be denoted by $\mathbb {Y}[\mathcal {X},\mathbb {S},\sigma ]$. This method for building Banach spaces was invented in [Reference Dilworth, Kalton and Kutzarova12]. The authors of [Reference Albiac, Ansorena, Dilworth and Kutzarova4] delved into its study and named it the DKK-method. For the purposes of this paper, it will be necessary to achieve some new properties of the unit vector system of $\mathbb {Y}[\mathcal {X},\mathbb {S},\sigma ]$. To properly enunciate them, we introduce some terminology. Let us regard the functionals on $\mathbb {S}$ as sequences acting on $\mathbb {S}$ through the natural dual pairing. With this convention, we have $c_{00} \subseteq \mathbb {S}^{*}\subseteq \ell _{\infty }$, and the closed subspace of $\mathbb {S}^{*}$ spanned by $c_{00}$ is a subsymmetric sequence space that we denote by $\mathbb {S}_0^{*}$. Let $(\Gamma _m)_{m=1}^{\infty }$ be the fundamental function of the unit vector system of $\mathbb {S}_0^{*}$, that is,

\begin{align*} \Gamma_m=\left\|\sum_{j=1}^{m} \text{e}_j\right\|_{\mathbb{S}^{*}}, \quad m\in\mathbb{N}. \end{align*}

By [Reference Lindenstrauss and Tzafriri24, proposition 3.a.6],

(5.2)\begin{equation} \frac{m}{\Lambda_n} = c_m \Gamma_m,\quad 1\le c_m\le 2,\quad m\in\mathbb{N}. \end{equation}

Recall that the dual basis $\mathcal {Y}^{*}$ of a Schauder basis $\mathcal {Y}$ of a Banach space $\mathbb {Y}$ is a Schauder basis of the Banach space it spans in $\mathbb {Y}^{*}$, and $\mathcal {Y}^{**}$ is equivalent to $\mathcal {Y}$ (see [Reference Albiac and Kalton7, corollary 3.2.4]). In particular, we have $(\mathbb {S}_0^{*})^{*}_0=\mathbb {S}$ up to an equivalent norm.

Note also that the operator $P_\sigma$ is self-adjoint, i.e.,

\[ \langle f,P_\sigma(g) \rangle =\langle P_\sigma(f), g \rangle, \quad f,g\in\mathbb{F}^\mathbb{N},\ fg\in\ell_1. \]

Consequently, $Q_\sigma$ also is self-adjoint.

Proof. By [Reference Albiac, Ansorena, Dilworth and Kutzarova4, theorem 3.6], the mapping

\[ (g,x)\mapsto g+ \sum_{n=1}^{\infty} \boldsymbol{x}_n^{*}(x)\, \boldsymbol{v}_n, \quad g\in c_{00}\cap Q_\sigma(\mathbb{S}),\; x\in\mathbb{X},\;\vert{\mathop{\mathrm{supp}}\nolimits(x)}\vert<\infty. \]

defines an isomorphism from $Q_\sigma (\mathbb {S})\oplus \mathbb {X}$ onto $\mathbb {Y}[\mathcal {X},\mathbb {S},\sigma ]$. In turn, since $P_\sigma$ and $Q_\sigma$ are complementary linear bounded projections on $\mathbb {S}$ (see [Reference Lindenstrauss and Tzafriri24, proposition 3.a.4]), the map

\[ u^{*}\mapsto (u^{*}(Q_\sigma(\text{e}_n))_{n=1}^{\infty} \]

defines an isomorphism from $(Q_\sigma (\mathbb {S}))^{*}$ onto

\[ \mathbb{U}:=\{ f \in\mathbb{S}^{*} \colon P_\sigma(f)=0\}. \]

We infer that, if we regard the functionals in $(\mathbb {Y}[\mathcal {X},\mathbb {S},\sigma ])^{*}$ as sequences acting on $\mathbb {Y}[\mathcal {X},\mathbb {S},\sigma ]$ by means of the natural dual pairing $\langle \cdot,\cdot \rangle$, the mapping

(5.3)\begin{equation} f\mapsto T(f):=\left( (\langle f, Q_\sigma(\text{e}_n )\rangle)_{n=1}^{\infty}, \sum_{n=1}^{\infty} \langle f,\boldsymbol{v}_n\rangle \, \boldsymbol{x}_n^{*}\right) \end{equation}

defines an isomorphism from $(\mathbb {Y}[\mathcal {X},\mathbb {S},\sigma ])^{*}$ onto $\mathbb {U}\times \mathbb {X}^{*}$. In (5.3), we use the convention that $x^{*}=\sum _{n=1}^{\infty } a_n\, \boldsymbol {x}_n^{*}$ means that $x^{*}\in \mathbb {X}^{*}$ and $(a_n)_{n=1}^{\infty }\in \mathbb {F}^{\mathbb{N}}$ satisfy $\boldsymbol {x}^{*}(x)=\sum _{n=1}^{\infty } a_n\, \boldsymbol {x}_n^{*}(x)$ for every $x\in \mathbb {X}$ finitely supported.

Since $Q_\sigma$ is selft-adjoint,

\[ (\langle f, Q_\sigma(\text{e}_n )\rangle)_{n=1}^{\infty}= (\langle Q_\sigma(f), \text{e}_n\rangle)_{n=1}^{\infty}=Q_\sigma(f),\quad f\in\mathbb{F}^\mathbb{N}. \]

If $({{\boldsymbol {u}}}_n^{*})_{n=1}^{\infty }$ are the vectors defined as in (5.1) corresponding to the subsymmetric space $\mathbb {S}_0^{*}$, and $(c_n)_{n=1}^{\infty }$ is as in (5.2), then $\boldsymbol {v}_n =c_n {{\boldsymbol {u}}}_n^{*}$ for all $n\in \mathbb {N}$. Summing up, we have

\[ T(f)=\left( Q_\sigma(f), \sum_{n=1}^{\infty} \langle {{\boldsymbol{u}}}_n^{*},f\rangle \, c_n\, \boldsymbol{x}_n^{*}\right), \quad f\in (\mathbb{Y}[\mathcal{X},\mathbb{S},\sigma])^{*}. \]

Let $\mathbb {X}^{*}_0$ denote the subspace of $\mathbb {X}^{*}$ spanned by $\mathcal {X}^{*}$. Applying [Reference Albiac, Ansorena, Dilworth and Kutzarova4, theorem 3.6] with the Schauder basis $\mathcal {B}=(c_n\, \boldsymbol {x}_n^{*})_{n=1}^{\infty }$ of $\mathbb {X}^{*}_0$, the subsymmetric sequence space $\mathbb {S}_0^{*}$, and the partition $\sigma$ gives that the mapping

\[ f \mapsto \left( Q_\sigma(f),\sum_{n=1}^{\infty} \langle {{\boldsymbol{u}}}_n^{*}, f\rangle \, c_n \, \boldsymbol{x}_n^{*}\right), \quad f\in c_{00}, \]

extend to an isomorphism from $\mathbb {Y}[\mathcal {B}^{*},\mathbb {S}_0^{*},\sigma ]$ onto $Q_\sigma (\mathbb {S}^{*}_0)\oplus \mathbb {X}^{*}_0$. We infer that $\mathbb {Y}[\mathcal {B}^{*},\mathbb {S}_0^{*},\sigma ]$ is, up to an equivalent norm, the closed subspace of $(\mathbb {Y}[\mathcal {X},\mathbb {S},\sigma ])^{*}$ spanned by $c_{00}$.

Proof. We have

\[ \boldsymbol{\varphi}_{km} \le C k^{1/q} \boldsymbol{\varphi}_m, \quad k,\,m\in\mathbb{N}. \]

where $C$ is upper $q$-estimate constant of $\mathbb {U}$. Given $n$, $m\in \mathbb {N}$ with $n>m$, we pick $k\in \mathbb {N}$ with $km< n\le (k+1)m$. Then,

\[ \frac{\boldsymbol{\varphi}_{n}^{q}}{n} \le \frac{\boldsymbol{\varphi}_{(k+1)m}^{q}}{km}\le C^{q} \frac{k+1}{k} \frac{\boldsymbol{\varphi}_{m}^{q}}{m} \le 2 C^{q} \frac{\boldsymbol{\varphi}_{m}^{q}}{m}. \]

Besides the LRP, we will use the upper regularity property (URP for short) of sequences of positive scalars. We say that a sequence $(s_m)_{m=1}^{\infty }$ has the URP for short if there is $r\in \mathbb {N}$ such that

\[ s_{rm}\le \frac{1}{2} r s_m, \quad m\in\mathbb{N}. \]

Proof. Pick $f=(a_n)_{n=1}^{\infty }\in c_{00}$. Put $t=\max _n\vert {a_n}\vert$, and for each $k\in \mathbb {N}$ consider the set

\[ E_k=\{n\in\mathbb{N} \colon t 2^{{-}k}<\vert{a_n}\vert \le t 2^{{-}k+1}\}. \]

Notice that $(E_k)_{k=1}^{\infty }$ is a partition of $\{n\in \mathbb {N} \colon a_n\not =0\}$. Set $n_k=\vert {E_k}\vert$ ($n_0=0$) and $m_k=\sum _{j=1}^{k} n_j$, so that, if $(b_n)_{n=1}^{\infty }$ is the non-increasing rearrangement of $f$,

\[ \{n\in\mathbb{N} \colon t 2^{{-}k}< b_n \le t 2^{{-}k+1}\}=\{n\in\mathbb{N} \colon 1+m_{k-1}\le n \le m_k\}. \]

For $n\in \mathbb {N}$, let $s_n=\boldsymbol {\varphi }_n[\mathcal {E},\mathbb {U}]$ and $w_n'=s_n^{r}-s_{n-1}^{r}$. Let $C$ be the upper $C$-estimate constant of $\mathbb {U}$. Using Abel's summation formula gives

\begin{align*} \Vert{f}\Vert^{q} & =\Vert{\sum_{k=1}^{\infty} \sum_{n\in E_k} a_n \, \text{e}_n}\Vert^{q}\\ & \le C^{q} \sum_{k=1}^{\infty} \Vert{\sum_{n\in E_k} a_n \,\text{e}_n}\Vert^{q} \\ & \le C^{q} \sum_{k=1}^{\infty} (t 2^{{-}k+1} s_{m_k})^{q} \\ & = (2tC)^{q} \sum_{k=1}^{\infty} 2^{{-}kq} \sum_{j=1}^{k} ( s_{m_j}^{q}-s_{m_{j-1}}^{q})\\ & = \frac{(2tC)^{q}}{1-2^{{-}q}} \sum_{j=1}^{\infty} 2^{{-}jq} ( s_{m_j}^{q}-s_{m_{j-1}}^{q})\\ & = \frac{(4C)^{q}}{2^{q}-1} \sum_{j=1}^{\infty} \sum_{n=1+m_{j-1}}^{m_j} ( t2^{{-}j})^{q} ( s_{n}^{q}-s_{n-1}^{q})\\ & \le \frac{(4C)^{q}}{2^{q}-1} \sum_{n=1}^{\infty} b_n^{q} ( s_{n}^{q}-s_{n-1}^{q}). \end{align*}

Applying lemma 2.8 puts an end to the proof.

Proposition 5.6 Let $\mathcal {X}$ be a Schauder basis of a Banach space $\mathbb {X}$, $\mathbb {S}$ be a subsymmetric sequence space, and $\sigma =(\sigma _n)_{n=1}^{\infty }$ be an ordered partition of $\mathbb {N}$ with

\[ \sum_{n=1}^{m-1} \vert{\sigma_n}\vert\le D \vert{\sigma_m}\vert, \quad m\in \mathbb{N}, \]

for some $D\in \mathbb {N}$. Let $1< q<\infty$. Suppose that $\mathbb {S}$ satisfies an upper $q$-estimate, that $\mathcal {X}$ $q$-Hilbertian, and that $(\Lambda _m)_{m=1}^{\infty }$ has the LRP. Then, the unit vector system of $\mathbb {Y}[\mathcal {X},\mathbb {S},\sigma ]$ is $d_{1,q}({{\boldsymbol {w}}})$-Hilbertian, that is,

\[ d_{1,q}({{\boldsymbol{w}}})\subseteq \mathbb{Y}[\mathcal{X},\mathbb{S},\sigma] \]

continuously, where ${{\boldsymbol {w}}}=(\Lambda _n-\Lambda _{n-1})_{n=1}^{\infty }$.

Proof. Taking into account that $Q_\sigma$ is bounded on $\mathbb {S}$, we infer from lemma 5.5 that there is a constant $C_0$ such that

\[ \Vert{Q_{\sigma}(f)}\Vert_\mathbb{S}\le C_0 \Vert{f}\Vert_{d_{1,q}({{\boldsymbol{w}}})}, \quad f\in c_{00}. \]

By lemma 5.3, there is a constant $C_1$ such that

\[ \frac{1}{C_1} \frac{\Lambda_n^{q}}{n}\le T_m:=\inf_{j\le m} \frac{\Lambda_j^{q}}{j} ,\quad n\ge m. \]

Therefore, if $n\in \mathbb {N}$ and $j\in \sigma _n$,

\[ \frac{\Lambda^{q}_{\vert{\sigma_n}\vert}}{\vert{\sigma_n}\vert}\le (D+1) \frac{\Lambda^{q}_{(D+1)\vert{\sigma_n}\vert}}{(D+1)\vert{\sigma_n}\vert} \le (D+1)C_1 T_j. \]

By lemma 2.7, there is a constant $C_2$ such that

\[ \sum_{j=1}^{\infty} \frac{\Lambda_j^{q} b_j^{q}}{j} \le C_2^{q} \Vert{f}\Vert_{d_{1,q}({{\boldsymbol{w}}})}^{q},\; (b_j)_{j=1}^{\infty}\; \text{non-increasing rearrangement of}\ f. \]

Let $C_3$ be the norm of the series transform with respect to $\mathcal {X}$, regarded as an operator from $\ell _q$ into $\mathbb {X}$. Pick $f=(a_j)_{j=1}^{\infty }\in c_{00}$, and let $(b_j)_{j=1}^{\infty }$ be its non-increasing rearrangement. Using Hölder's inequality and the rearrangement inequality we obtain

\begin{align*} \Vert{\sum_{n=1}^{\infty} \langle \boldsymbol{v}_n^{*},f\rangle\, \boldsymbol{x}_n}\Vert^{q}& \le C_3^{q} \sum_{n=1}^{\infty} \vert{\langle \boldsymbol{v}_n^{*},f\rangle}\vert^{q} \\ & \le C_3^{q} \sum_{n=1}^{\infty} \frac{\Lambda^{q}_{\vert{\sigma_n}\vert}}{\vert{\sigma_n}\vert} \sum_{j\in\sigma_n} \vert{a_j}\vert^{q} \\ & \le (D+1) C_1^{q}C_3^{q}\sum_{j=1}^{\infty} T_j \vert{a_j}\vert^{q} \\ & \le (D+1) C_1^{q}C_3^{q} \sum_{j=1}^{\infty} T_j b_j^{q} \\ & \le (D+1) C_1^{q}C_3^{q} \sum_{j=1}^{\infty} \frac{\Lambda_j^{q}}{j} b_j^{q} \\ & \le (D+1) C_1^{q}C_2^{q}C_3^{q} \Vert{f}\Vert_{d_{1,q}({{\boldsymbol{w}}})}^{q}. \end{align*}

Summing up, we obtain $\Vert {f}\Vert _{\mathbb {Y}[\mathcal {X},\mathbb {S},\sigma ]}\le C \Vert {f}\Vert _{d_{1,q}({{\boldsymbol {w}}})}$, where

\[ C=C_0+(1+D)^{1/q} C_1 C_2 C_3. \]

Proposition 5.7 Let $\mathcal {X}$ be a Schauder basis of a Banach space $\mathbb {X}$, $\mathbb {S}$ be a subsymmetric sequence space, and $\sigma =(\sigma _n)_{n=1}^{\infty }$ be an ordered partition of $\mathbb {N}$ with

(5.4)\begin{equation} \sum_{n=1}^{m-1} \vert{\sigma_n}\vert\lesssim \vert{\sigma_m}\vert, \quad m\in \mathbb{N}. \end{equation}

Let $1< r<\infty$. Suppose that $\mathbb {S}$ satisfies a lower $r$-estimate, that $\mathcal {X}$ is $r$-Besselian, and that $(\Lambda _m)_{m=1}^{\infty }$ has the URP. Then, the unit vector system of $\mathbb {Y}[\mathcal {X},\mathbb {S},\sigma ]$ is $d_{1,r}({{\boldsymbol {w}}})$-Besselian, that is,

\[ \mathbb{Y}[\mathcal{X},\mathbb{S},\sigma] \subseteq d_{1,r}({{\boldsymbol{w}}}) \]

continuously, where ${{\boldsymbol {w}}}=(\Lambda _n-\Lambda _{n-1})_{n=1}^{\infty }$.

Proof. Let ${{\boldsymbol {w}}}^{*}$ be the weight whose primitive sequence is $(\Gamma _m)_{m=1}^{\infty }$. By [Reference Lindenstrauss and Tzafriri25, proposition 1.f.r], $\mathbb {S}_0^{*}$ satisfies an upper $r'$-estimate. In turn, by duality, $\mathcal {X}^{*}$ is an $r'$-Hilbertian. Hence, any perturbation $\mathcal {Y}$ of $\mathcal {X}^{*}$ is $r'$-Hilbertian. An application of proposition 5.6 gives

(5.5)\begin{equation} d_{1,r'}({{\boldsymbol{w}}}^{*})\subseteq \mathbb{Y}[\mathcal{Y},\mathbb{S}_0^{*},\sigma]. \end{equation}

By lemma 5.4, $(\Lambda _n)_{n=1}^{\infty }$ has the LRP. Consequently, by [Reference Ansorena, Bello and Wojtaszczyk8, theorem 3.10] and equation (5.2), $d_{1,r'}({{\boldsymbol {w}}}^{*})$ is, via the natural dual pairing, and up to an equivalent norm, the dual space of $d_{1,r}({{\boldsymbol {w}}})$. In turn, by proposition 5.2, we can choose $\mathcal {Y}$ so that $\mathbb {Y}[\mathcal {Y},\mathbb {S}_0^{*},\sigma ]$ is, via the natural dual pairing, and up to an equivalent norm, the dual space of $\mathbb {Y}[\mathcal {X},\mathbb {S},\sigma ]$. Therefore, dualizing (5.5) yields the desired embedding.

Our next result must be regarded as a paradigm to prove the existence of almost greedy bases whose conditionality parameters grow as a given sequence.

Proof. The unconditional basis $\mathcal {V}=(\boldsymbol {v}_n)_{n=1}^{\infty }$ of $P_\sigma (\mathbb {S})$ is $q$-Hilbertian and $r$-Besselian. Therefore, by lemmas 2.3, 5.4, propositions 5.6 and 5.7,

\[ d_{1,q}({{\boldsymbol{w}}}) \subseteq \mathbb{Y}:=\mathbb{Y}[\mathcal{B}\oplus\mathcal{V}, \mathbb{S},\sigma] \subseteq d_{1,r}({{\boldsymbol{w}}}), \]

provided that the ordered partition $\sigma =(\sigma _n)_{n=1}^{\infty }$ satisfies (5.4). Choose $\sigma$ so that

\[ \log\left(\sum_{n=1}^{m} \vert{\sigma_n}\vert\right) \lesssim m, \quad m\in\mathbb{N}, \]

also holds. Then, by (2.2) and [Reference Albiac, Ansorena, Dilworth and Kutzarova4, proposition 3.8 and theorem 3.17], the unit vector system is an almost greedy basis of $\mathbb {Y}$ with fundamental function equivalent to $(\Lambda _m)_{m=1}^{\infty }$, and we have

\[ \boldsymbol{\tilde{k}}_m[\mathcal{E},\mathbb{Y}] \gtrsim (\log m)^{1/q-1/r}, \quad m\ge 2. \]

Applying [Reference Albiac, Ansorena, Dilworth and Kutzarova4, theorem 3.6], and using that $\mathbb {S}$ is isomorphic to its square, gives

\[ \mathbb{Y}\simeq Q_\sigma(\mathbb{S}) \oplus \mathbb{X} \oplus P_\sigma(\mathbb{S}) \simeq \mathbb{X}\oplus\mathbb{S}\simeq\mathbb{X}. \]

Combining equation (2.7), lemma 2.7 and corollary 2.6 puts an end to the proof.

Theorem 5.9, of which theorem 1.5 is a simple consequence, lies within the line of research initiated by Konyagin and Telmyakov [Reference Konyagin and Temlyakov23] of finding conditional quasi-greedy bases in general Banach spaces. This topic has evolved towards the more specific quest of finding quasi-greedy bases with suitable conditionality parameters. The reader will find a detailed account of this process in the papers [Reference Albiac, Ansorena and Wojtaszczyk4–Reference Albiac and Kalton6, Reference Dilworth, Kalton and Kutzarova12, Reference Dilworth, Kutzarova and Wojtaszczyk14, Reference Garrigós, Hernández and Oikhberg15, Reference Gogyan18, Reference Wojtaszczyk28].