The Szlenk index of $L_p(X)$ and $A_p$

TL;DR

This paper establishes optimal relationships between the Szlenk index of a Banach space and its associated $L_p$ spaces, extending previous results to uncountable ordinals and providing new estimates for operators.

Contribution

It introduces optimal bounds relating the Szlenk index of $X$ and $L_p(X)$, extending prior work to uncountable ordinals and general operators.

Findings

Provides optimal estimates of Szlenk index for $L_p(X)$ in terms of $X$.

Extends results to uncountable ordinals.

Estimates Szlenk index of operators $A_p$ in terms of $A$.

Abstract

Given a Banach space $X$, a $w^*$-compact subset of $X^*$, and $1<p<\infty$, we provide an optimal relationship between the Szlenk index of $K$ and the Szlenk index of an associated subset of $L_p(X)^*$. As an application, given a Banach space $X$, we prove an optimal estimate of the Szlenk index of $L_p(X)$ in terms of the Szlenk index of $X$. This extends a result of H\'ajek and Schlumprecht to uncountable ordinals. More generally, given an operator $A:X\to Y$, we provide an estimate of the Szlenk index of the "pointwise $A$" operator $A_p:L_p(X)\to L_p(Y)$ in terms of the Szlenk index of $A$.

Equations

$$T.X.\mathcal{C}=\{({\zeta }_i,Z_i,C_i{)}_{i=1}^k:({\zeta }_i{)}_{i=1}^k,Z_i\in \text{codim}(X),C_i\in \mathcal{C}\}.$$

$${\mathbb{P}}_{\xi ,n}(({\zeta }_i,Z_i{)}_{i=1}^k)={\mathbb{P}}_{\xi ,n}(({\zeta }_i{)}_{i=1}^k)$$

$${\mathbb{P}}_{\xi ,n}(({\zeta }_i,Z_i,C_i{)}_{i=1}^k)={\mathbb{P}}_{\xi ,n}(({\zeta }_i{)}_{i=1}^k).$$

$$\lambda <\underset{t\in MAX({\Gamma }_{\xi .n}.X)}{\inf }\sum _{s⪯t}{\mathbb{P}}_{\xi ,n}(s)f(s,t),$$

$$s_{\epsilon }^0(L)=L,$$

$$s_{\epsilon }^{\xi +1}(L)=s_{\epsilon }(s_{\epsilon }^{\xi }(L)),$$

$$s_{\epsilon }^{\xi }(L)=\bigcap _{\zeta <\xi }{s}_{\epsilon }^{\zeta }(L).$$

$${\text{p}}_{\xi }(K)=\underset{\epsilon \to 0^{+}}{\mathrm{limsup}}\frac{\log S{z}_{\xi }(K,\epsilon )}{|\log (\epsilon )|}.$$

$$f\mapsto \int ⟨g,f⟩,$$

$$\inf\bigl{\{}\|\varrho(f)\|_{L_{p}}:t\in T.L_{p}(X),f\in\text{\emph{co}}(f_{s}:\varnothing\prec s\preceq t)\bigr{\}}=0,$$

$$\inf\bigl{\{}\|\varrho(f)\|_{L_{p}}:t\in T.L_{p}(X),f\in\text{\emph{co}}(f_{s}:\varnothing\prec s\preceq t)\bigr{\}}\leqslant Cn^{-1/\alpha},$$

$$\varnothing \ne \bigcap _{Z\in \text{codim}(X)}\overline{\{(x_1^Y,\dots ,x_{|s|}^Y,x_Y^{\ast }):Z⩾Y\in \text{codim}(X)\}}\subset \mathfrak{s}(s)\times K.$$

$$\text{diam}(s_{\delta }^{\gamma }(K)\cap V)⩾\Vert x_Z^{\ast }-x^{\ast }\Vert ⩾\text{Re }(x_Z^{\ast }-x^{\ast })(x_Z)=\text{Re }{x}_Z^{\ast }(x_Z)⩾\epsilon >\delta .$$

$$\bigcap _{\beta <\gamma }\overline{\{(x_1^{\mu },\dots ,x_{|s|}^{\mu },x_{\mu }^{\ast }):\mu ⩾\beta \}}\ne \varnothing .$$

$$\{(({\zeta }_i,Z_i,C_i{)}_{i=1}^{n-1},({\zeta }_n,Z_n)):$$

$$({\zeta }_i{)}_{i=1}^n\in T\}$$

$$\omega \gamma (s_{k-1})⩾\omega (\gamma (s_k)+1)=\omega \gamma (s_k)+\omega ,$$

$$\Vert \sum _{i=1}^n{f}_i{e}_i{\Vert }_{L_p(E)}⩽C\Vert \sum _{i=1}^n\Vert f_i{\Vert }_{L_p}{e}_i{\Vert }_E.$$

$$\bigl{(}\sum_{i=1}^{m}\|x_{i}\|^{\beta}\bigr{)}^{1/\beta}\leqslant\|\sum_{i=1}^{m}x_{i}\|.$$

$$\displaystyle N_{\theta,\phi,T}=\{0\}\cup\Bigl{\{}\theta\sum_{i=1}^{k}e^{*}_{t|_{j_{i}}}:$$

$$\displaystyle\hskip 8.53581pt\prod_{i=1}^{k}Z_{j_{i}}\cap C_{j_{i}}\in\mathcal{H}^{K}_{\phi}\Bigr{\}}\subset\text{span}(e^{*}_{t}:t\in T.X.\mathcal{C}).$$

$$\displaystyle M_{\theta,\phi,\alpha,T}=\Bigl{\{}\sum_{i=1}^{k}a_{i}g_{i}:$$

$$\Vert x{\Vert }_{\theta ,\varphi ,\alpha ,T}=\sup \{f(|x|):f\in M_{\theta ,\varphi ,\alpha ,T}\}.$$

$$\|\sum_{i=1}^{n}x_{i}\|_{\theta,\phi,\alpha,T}\geqslant g\bigl{(}\bigl{|}\sum_{i=1}^{n}x_{i}\bigr{|}\bigr{)}=\sum_{i=1}^{n}a_{i}g_{i}(|x_{i}|)=\bigl{(}\sum_{i=1}^{n}\|x_{i}\|_{\theta,\phi,\alpha,T}^{\beta})^{1/\beta}.$$

$$\varrho (\sum _{i=1}^k{a}_i{x}_i)⩽\frac{1}{\theta -\varphi }\Vert \sum _{i=1}^k{a}_i{e}_{t{|}_i}{\Vert }_{\theta ,\varphi ,\alpha ,T}.$$

$${\varphi }^{j-1}\sum _{i\in B_j}{a}_i={\varphi }^{-1}(\varphi /\theta {)}^j({\theta }^j\sum _{i\in B_j}{e}_{t{|}_i}^{\ast })(\sum _{i=1}^k{a}_i{e}_{t{|}_i})⩽{\varphi }^{-1}(\varphi /\theta {)}^j\Vert \sum _{i=1}^k{a}_i{e}_{t{|}_i}{\Vert }_{\theta ,\varphi ,\alpha ,T}.$$

$$\varrho (\sum _{i=1}^k{a}_i{x}_i)$$

$$=\frac{1}{\theta -\varphi }\Vert \sum _{i=1}^k{a}_i{e}_{t{|}_i}{\Vert }_{\theta ,\varphi ,\alpha ,T}.$$

$$\Bigl{\{}t\in MAX(\Gamma_{\xi,n}.X.\mathcal{C}):\|\sum_{s\preceq t}\mathbb{P}_{\xi,n}(s)e_{s}\|_{\theta,\phi,\alpha,\Gamma_{\xi,n}}>\varepsilon\Bigr{\}}.$$

$$\inf\bigl{\{}\|\varrho(f)\|_{L_{p}}:t\in T.L_{p}(X),f\in\text{\emph{co}}(f_{s}:\varnothing\prec s\preceq t)\bigr{\}}\leqslant\frac{C^{\prime}\varepsilon}{n(\theta-\phi)}.$$

$$\varrho (\sum _{i=1}^k{a}_i{h}_i(\varpi ))=\varrho (\sum _{i=1}^k{a}_i\Vert h_i(\varpi )\Vert {\overline{h}}_i(\varpi ))⩽\frac{1}{\theta -\varphi }\Vert \sum _{i=1}^k{a}_i\Vert h_i(\varpi )\Vert e_{s{|}_i}{\Vert }_{\theta ,\varphi ,\alpha ,{\Gamma }_{\xi ,n}}.$$

$$\Vert \varrho (\sum _{i=1}^k{n}^{-1}{\mathbb{P}}_{\xi ,n}(s{|}_i)h_i){\Vert }_{L_p}$$

Full text

1. Introduction

Throughout this work, $X$ will be a fixed Banach space and $K\subset X^{*}$ will be a $w^{*}$-compact, non-empty subset. For $1<p<\infty$, we let $K_{p}$ denote the $w^{*}$-closure in $L_{p}(X)^{*}$ of all functions of the form $gh\in L_{q}(X^{*})\subset L_{p}(X)^{*}$, where $g:[0,1]\to K$ is simple and Lebesgue measurable, and $h\in B_{L_{q}}$. Recall that these functions act on $L_{p}(X)$ by $\langle gh,f\rangle=\int_{0}^{1}\langle g(\varpi),f(\varpi)\rangle h(\varpi)d\varpi$ for $f\in L_{p}(X)$. Note that if $R\geqslant 0$ is such that $K\subset RB_{X^{*}}$, $K_{p}\subset RB_{L_{p}(X)^{*}}$, so that $K_{p}$ is also $w^{*}$-compact. If $K=B_{X^{*}}$, $K_{p}=B_{L_{p}(X)^{*}}$ by the Hahn-Banach theorem. If $A:X\to Y$ is an operator, then there exists a “pointwise $A$” operator $A_{p}:L_{p}(X)\to L_{p}(Y)$ given by $(A_{p}f)(\varpi)=A(f(\varpi))$ for all $\varpi\in[0,1]$. Then if $K=A^{*}B_{Y^{*}}$, $K_{p}=(A_{p})^{*}B_{L_{p}(Y)^{*}}$, which follows from the Hahn-Banach theorem. Thus it is natural to examine what relationship exists between $K$ and $K_{p}$. In particular, one may ask what relationship exists between the Szlenk indices of these sets. To that end, we obtain the optimal relationship. In what follows, $\omega$ denotes the first infinite ordinal.

Theorem 1**.**

Fix $1<p<\infty$. Suppose that $\xi$ is an ordinal such that $Sz(K)\leqslant\omega^{\xi}$. Then $Sz(K_{p})\leqslant\omega^{1+\xi}$. If $K$ is convex, $Sz(K_{p})\leqslant\omega Sz(K)$. If $K$ is convex and $Sz(K)\geqslant\omega^{\omega}$, $Sz(K)=Sz(K_{p})$.

Using the facts stated in the introduction that $K_{p}=(A_{p})^{*}B_{L_{p}(Y)^{*}}$ if $K=A^{*}B_{Y^{*}}$, we immediately deduce the following from Theorem 1.

Corollary 2**.**

Fix $1<p<\infty$. If $A:X\to Y$ is an operator and $K=A^{*}B_{Y^{*}}$, then $Sz(A_{p})\leqslant\omega Sz(A)$, and if $Sz(A)\geqslant\omega^{\omega}$, $Sz(A_{p})=Sz(A)$. In particular, $A_{p}$ is Asplund if and only if $A$ is.

Applying Corollary 2 to the identity of a Banach space, we extend the result of Hájek and Schlumprecht from [9] to uncountable ordinals.

We recall that $K$ is said to be $w^{*}$-fragmentable if for any non-empty subset $L$ of $K$ and any $\varepsilon>0$, there exists a $w^{*}$-open subset $U$ of $X^{*}$ such that $L\cap U\neq\varnothing$ and $\text{diam}(L\cap U)<\varepsilon$. We recall that $K$ is $w^{*}$-dentable if for any non-empty subset $L$ of $K$ and any $\varepsilon>0$, there exists a $w^{*}$-open slice $S$ of $X^{*}$ such that $L\cap U\neq\varnothing$ and $\text{diam}(L\cap S)<\varepsilon$. We recall that a $w^{*}$-open slice is a subset of $X^{*}$ of the form $\{x^{*}\in X^{*}:\text{Re\ }x^{*}(x)>a\}$ for some $x\in X$ and $a\in\mathbb{R}$. As mentioned in [6], a consequence of Corollary 2 is that if $Sz(K)\leqslant\omega^{\xi}$, then $Sz(K)\leqslant Dz(K)\leqslant\omega^{1+\xi}$, where $Dz(K)$ denotes the $w^{*}$-dentability index of $K$. Thus Corollary 2 implies that $K$ is $w^{*}$-dentable if and only if it is $w^{*}$-fragmentable.

In addition to considering the Szlenk index of a set, one may consider the $\xi$-Szlenk power type $\textbf{p}_{\xi}(L)$ of the set $L$, which is important in $\xi$-asymptotically uniformly smooth renormings of Banach spaces and operators. The concept of a $\xi$-asymptotically uniformly smooth operator was introduced in [7], and further sharp renorming results regarding the $\xi$-Szlenk power type of an operator were established in [5]. To that end, we have the following.

Theorem 3**.**

For any ordinal $\xi$ and any $1<p<\infty$, if $1/p+1/q=1$, $\textbf{\emph{p}}_{1+\xi}(K_{p})\leqslant\max\{q,\textbf{\emph{p}}_{\xi}(K)\}$.

In the case that $\xi\geqslant\omega$ and $\textbf{p}_{\xi}(K)\leqslant p$, $\textbf{p}_{\xi}(K)=\textbf{p}_{\xi}(K_{p})$, in showing that Theorem 3 is sharp in some cases.

The author wishes to thank P.A.H. Brooker, P. Hájek, N. Holt, and Th. Schlumprecht for helpful remarks during the preparation of this work.

2. $L_{p}(X)$, Trees, Szlenk index, games

2.1. Trees, $\Gamma_{\xi,n}$, $\mathbb{P}_{\xi,n}$, and stablization results

Given a set $\Lambda$, we let $\Lambda^{<\mathbb{N}}$ denote the finite, non-empty sequences in $\Lambda$. Given two members $s,t$ of $\Lambda^{<\mathbb{N}}$, we let $s^{\smallfrown}t$ denote the concatenation of $s$ and $t$, $|s|$ denotes the length of $s$, $s\preceq t$ means $s$ is an initial segment of $t$, and $s|_{i}$ denotes the initial segment of $s$ having length $i$. Given $t\in\Lambda^{<\mathbb{N}}$, we let $[\preceq t]=\{s\in\Lambda^{<\mathbb{N}}:s\preceq t\}$.

Any subset $T$ of $\Lambda^{<\mathbb{N}}$ which contains all non-empty initial segments of its members will be called a $B$-tree. We define by transfinite induction the derived$B$* trees* of $T$. We let $MAX(T^{\prime})$ denote the $\preceq$-maximal members of $T$ and $T^{\prime}=T\setminus MAX(T)$. We then define $T^{0}=T$, $T^{\xi+1}=(T^{\xi})^{\prime}$, and if $\xi$ is a limit ordinal, $T^{\xi}=\cap_{\gamma<\xi}T^{\gamma}$. We let $o(T)$ denote the smallest ordinal $\xi$ such that $T^{\xi}=\varnothing$, provided such an ordinal exists. If no such ordinal exists, we write $o(T)=\infty$. We say $T$ is well-founded if $o(T)$ is an ordinal, and $T$ is ill-founded if $o(T)=\infty$. For convenience, we agree to the convention that if $\xi$ is an ordinal $\xi<\infty$, and that $\omega\infty=\infty$.

Given a $B$-tree $T$ and a Banach space $Y$, we let $T.Y=\{(\zeta_{i},Z_{i})_{i=1}^{k}:(\zeta_{i})_{i=1}^{k}\in T,Z_{i}\in\text{codim}(Y)\}$, where $\text{codim}(Y)$ denotes the closed subspaces of $Y$ having finite codimension in $Y$. We let $\mathcal{C}$ denote the norm compact subsets of $B_{X}$ and

[TABLE]

We note that $T.Y$ and $T.X.\mathcal{C}$ are $B$-trees. Furthermore, for any ordinal $\gamma$, $(T.Y)^{\gamma}=T^{\gamma}.Y$ and $(T.X.\mathcal{C})^{\gamma}=T^{\gamma}.X.\mathcal{C}$. In particular, $T.Y$ and $T.X.\mathcal{C}$ have the same order as $T$.

Given a $B$-tree $T$, a Banach space $Y$, and a collection $(x_{t})_{t\in T.Y}\subset Y$, we say $(x_{t})_{t\in T.Y}$ is normally weakly null provided that for any $t=(\zeta_{i},Z_{i})_{i=1}^{k}\in T.Y$, $x_{t}\in Z_{k}$. Given another $B$-tree $S$ and a function $\sigma:S.Y\to T.Y$, we say $\sigma$ is a pruning provided that for every $s,s_{1}\in S.Y$ with $s\prec s_{1}$, $\sigma(s)\prec\sigma(s_{1})$, and if $s_{1}=s^{\smallfrown}(\zeta,Z)$ and $\sigma(s_{1})=t^{\smallfrown}(\mu,W)$ for some $t\in T.Y$, $W\leqslant Z$. If $\sigma:S.Y\to T.Y$ is a pruning and $\tau:MAX(S.Y)\to MAX(T.Y)$ is such that for every $s\in MAX(S.Y)$, $\sigma(s)\preceq\tau(s)$, we say the pair $(\sigma,\tau)$ is an extended pruning, and denote this by $(\sigma,\tau):S.Y\to T.Y$.

For every $\xi\in\mathbb{N}$ and $n\in\mathbb{N}$, a $B$-tree $\Gamma_{\xi,n}$ was defined in [4] so that $o(\Gamma_{\xi,n})=\omega^{\xi}n$. Furthermore, a function $\mathbb{P}_{\xi}:\Gamma_{\xi}\to[0,1]$ was defined so that for every $t\in MAX(\Gamma_{\xi})$, $\sum_{s\preceq t}\mathbb{P}_{\xi}(s)=1$. Furthermore, $\Gamma_{\xi+1}$ is the disjoint union of $\Gamma_{\xi,n}$, $n\in\mathbb{N}$. For convenience, we define $\mathbb{P}_{\xi,n}:\Gamma_{\xi,n}\to[0,n]$ by $\mathbb{P}_{\xi,n}(s)=n\mathbb{P}_{\xi+1}(s)$. It follows from the definitions that $\Gamma_{\xi,1}=\Gamma_{\xi}$ and $\mathbb{P}_{\xi,1}=\mathbb{P}_{\xi}$. For every $\xi$ and every $n\in\mathbb{N}$, there exist disjoint subsets $\Lambda_{\xi,n,1}$, $\ldots$, $\Lambda_{\xi,n,n}$ of $\Gamma_{\xi,n}$ such that $\Gamma_{\xi,n}=\cup_{i=1}^{n}\Lambda_{\xi,n,i}$. It follows from the facts regarding $\mathbb{P}_{\xi+1}$ discussed in [4] that, with these definitions, for every ordinal $\xi$, every $n\in\mathbb{N}$, every $1\leqslant i\leqslant n$, and every $t\in MAX(\Gamma_{\xi,n})$, $\sum_{\Lambda_{\xi,n,i}\ni s\preceq t}\mathbb{P}_{\xi,n}(s)=1$. For any Banach space $Y$, we may define $\mathbb{P}_{\xi,n}$ on $\Gamma_{\xi,n}.Y$ and $\Gamma_{\xi,n}.X.\mathcal{C}$ by letting

[TABLE]

and

[TABLE]

We say an extended pruning $(\sigma,\tau):\Gamma_{\xi,n}.X\to\Gamma_{\xi,n}.X$ is level preserving provided that for every $1\leqslant i\leqslant n$, $\sigma(\Lambda_{\xi,n,i})\subset\Lambda_{\xi,n,i}$.

The following theorem collects results from Theorem $3.3$, Propositions $3.2$, $3.3$, and Lemma $3.4$ of [5].

Theorem 4**.**

Suppose $\xi$ is an ordinal and $n$ is a natural number.

1. (i)

If $f:\Pi(\Gamma_{\xi}.n.X)\to\mathbb{R}$ is bounded and $\lambda\in\mathbb{R}$ is such that

[TABLE]

then there exist a level preserving extended pruning $(\sigma,\tau):\Gamma_{\xi,n}.X\to\Gamma_{\xi,n}.X$ and real numbers $b_{1},\ldots,b_{n}$ such that $\lambda<\sum_{i=1}^{n}b_{i}$ and for every $1\leqslant i\leqslant n$ and every $\Lambda_{\xi,n,i}\ni s\preceq t\in MAX(\Gamma_{\xi,n}.X)$, $b_{i}\leqslant f(\sigma(s),\tau(t))$. 2. (ii)

If $(M,d)$ is a compact metric space and $f:\Pi(\Gamma_{\xi,n}.X)\to M$ is any function, then for any $\delta>0$, there exist $x_{1},\ldots,x_{n}\in M$ and a level preserving extended pruning $(\sigma,\tau):\Gamma_{\xi,n}.X\to\Gamma_{\xi,n}.X$ such that for every $1\leqslant i\leqslant n$ and every $\Lambda_{\xi,n,i}\ni s\preceq t\in MAX(\Gamma_{\xi,n}.X)$, $d(x_{i},f(\sigma(s),\tau(t)))<\delta$. 3. (iii)

If $F$ is a finite set and $f:MAX(\Gamma_{\xi,n}.X)\to F$ is any function, there exists a level preserving extended pruning $(\sigma,\tau):\Gamma_{\xi,n}.X\to\Gamma_{\xi,n}.X$ such that $f\circ\tau|_{MAX(\Gamma_{\xi,n}.X)}$ is constant. 4. (iv)

For any natural numbers $k_{1}<\ldots<k_{r}\leqslant n$, there exists an extended pruning $(\sigma,\tau):\Gamma_{\xi,r}.X\to\Gamma_{\xi,n}.X$ such that for every $1\leqslant i\leqslant r$, $\sigma(\Lambda_{\xi,n,i})\subset\Lambda_{\xi,n,k_{i}}$.

2.2. The Szlenk index, Szlenk power type

Given a $w^{*}$-compact subset $L$ of $X^{*}$ and $\varepsilon>0$, we let $s_{\varepsilon}(K)$ denote the set consisting of those $x^{*}\in L$ such that for every $w^{*}$-neighborhood $V$ of $x^{*}$, $\text{diam}(L\cap V)>\varepsilon$. We define the transfinite derivations

[TABLE]

[TABLE]

and if $\xi$ is a limit ordinal,

[TABLE]

If there exists an ordinal $\xi$ such that $s_{\varepsilon}^{\xi}(L)=\varnothing$, we let $Sz(L,\varepsilon)$ be the minimum such ordinal. Otherwise we write $Sz(L,\varepsilon)=\infty$. Since $s_{\varepsilon}^{\xi}(L)$ is $w^{*}$-compact, we deduce that $Sz(L,\varepsilon)$ cannot be a limit ordinal. We agree to the conventions that $\omega\infty=\infty$ and $\xi<\infty$ for any ordinal $\xi$. We let $Sz(L)=\sup_{\varepsilon>0}Sz(L,\varepsilon)$. If $B:Z\to W$ is an operator, we let $Sz(B,\varepsilon)=Sz(B^{*}B_{W^{*}},\varepsilon)$, $Sz(B)=Sz(B^{*}B_{W^{*}})$. If $Z$ is a Banach space, $Sz(Z,\varepsilon)=Sz(I_{Z},\varepsilon)$ and $Sz(Z)=Sz(I_{Z})$.

We recall that a set $L\subset X^{*}$ is called $w^{*}$-fragmentable if for any $\varepsilon>0$ and any $w^{*}$-compact, non-empty subset $M$ of $L$, $s_{\varepsilon}(M)\subsetneq M$. This is equivalent to $Sz(L)<\infty$. We say an operator $B:Z\to W$ is Asplund if $B^{*}B_{W^{*}}$ is $w^{*}$-fragmentable, which happens if and only if $Sz(B)<\infty$. We say a Banach space $Z$ is Asplund if $I_{Z}$ is Asplund. These are not the original definitions of Asplund spaces and operators, but they are equivalent to the original definitions (see [2]).

If $Sz(K)\leqslant\omega^{\xi+1}$, then for any $\varepsilon>0$, $Sz(K,\varepsilon)\leqslant\omega^{\xi}n$ for some $n\in\mathbb{N}$. We let $Sz_{\xi}(K,\varepsilon)$ be the smallest $n\in\mathbb{N}$ such that $Sz(K,\varepsilon)\leqslant\omega^{\xi}n$. We define the $\xi$-Szlenk power type $\textbf{p}_{\xi}(K)$ of $K$ by

[TABLE]

This value need not be finite. By convention, we let $\textbf{p}_{\xi}(K)=\infty$ if $Sz(K)>\omega^{\xi+1}$. We let $\textbf{p}_{\xi}(A)=\textbf{p}_{\xi}(A^{*}B_{Y^{*}})$ and $\textbf{p}_{\xi}(X)=\textbf{p}_{\xi}(B_{X^{*}})$. The quantities $\textbf{p}_{\xi}(X)$, $\textbf{p}_{\xi}(A)$ are important for the renorming theorem of $\xi$-asymptotically uniformly smooth norms with power type modulus.

Given a $w^{*}$-compact subset $L$ of $X^{*}$ and $\varepsilon>0$, we let $\mathcal{H}_{\varepsilon}^{L}$ denote the set of Cartesian products $\prod_{i=1}^{n}C_{i}$ such that $C_{i}\in\mathcal{C}$ for each $1\leqslant i\leqslant n$ and such that there exist $(x_{i})_{i=1}^{n}\in\prod_{i=1}^{n}C_{i}$ and $x^{*}\in K$ such that for each $1\leqslant i\leqslant n$, $\text{Re\ }x^{*}(x_{i})\geqslant\varepsilon$.

2.3. The Szlenk index of $K_{p}$

Recall that for $1<p<\infty$, $L_{p}(X)$ denotes the space of equivalence classes of Bochner integrable functions $f:[0,1]\to X$ such that $\int\|f\|^{p}<\infty$, where $[0,1]$ is endowed with its Lebesgue measure. Recall also that if $1<q<\infty$, $L_{q}(X^{*})$ is isometrically included in $L_{p}(X)^{*}$ by the action

[TABLE]

for $g\in L_{q}(X^{*})$. We also recall that if $\varrho:X\to\mathbb{R}$ is any Lipschitz function, then for any $f\in L_{p}(X)$, $\varrho\circ f\in L_{p}$.

We note that the Szlenk index and the $\xi$ Szlenk power type of $K$ are unchanged by scaling $K$ by a positive scalar or by replacing $K$ with its balanced hull. Moreover, for a positive scalar $c$, $(cK)_{p}=cK_{p}$, which has the same Szlenk index and $\xi$-Szlenk power type as $K_{p}$. If $\mathbb{T}K$ is the balanced hull of $K$, $K_{p}\subset(\mathbb{T}K)_{p}$ and $Sz(K)=Sz(\mathbb{T}K)$ ([4, Lemma $2.2$]) so that Theorem 1, Corollary 2, and Theorem 3 hold in general if they hold under the assumption that $K\subset B_{X^{*}}$ is balanced. Therefore we can and do assume throughout that $K\subset B_{X^{*}}$ and $K$ is balanced.

Let $\varrho:X\to\mathbb{R}$ be given by $\varrho(x)=\max_{x^{*}\in K}\text{Re\ }x^{*}(x)$. Since we have assumed $K$ is balanced, $\varrho(x)=\max_{x^{*}\in K}|x^{*}(x)|$. It is easy to see that for any $1<p<\infty$ and any $f\in L_{p}(X)$, $\|\varrho(f)\|_{L_{p}}=\max_{f^{*}\in K_{p}}\text{Re\ }f^{*}(f)$. Combining this fact with [5, Corollary $2.4$] and the proof of that corollary, we obtain the following.

Theorem 5**.**

Fix $1<p,\alpha<\infty$.

1. (i)

If for every $B$-tree $T$ with $o(T)=\omega^{1+\xi}$ and every normally weakly null $(f_{t})_{t\in T.L_{p}(X)}\subset B_{L_{p}(X)}$,

[TABLE]

then $Sz(K_{p})\leqslant\omega^{1+\xi}$. 2. (ii)

If there exists a constant $C$ such that for every $n\in\mathbb{N}$, every $B$-tree $T$ with $o(T)=\omega^{1+\xi}n$, and every normally weakly null collection $(f_{t})_{t\in T.L_{p}(X)}\subset B_{L_{p}(X)}$,

[TABLE]

then $\textbf{\emph{p}}_{1+\xi}(K_{p})\leqslant\alpha$.

Proposition 6**.**

Suppose $T$ is a non-empty $B$-tree. Suppose also that $(C_{s})_{s\in T.X}\subset\mathcal{C}$ is fixed and for $s=(\zeta_{i},Z_{i})_{i=1}^{k}\in T.X$, let $\lambda(s)=Z_{k}\cap C_{s}$. Suppose that $S$ is a non-empty, well-founded $B$-tree and $\theta:S.X\to T.X$ is a pruning. For $s\in S.X$, let $\mathfrak{s}(s)=\prod_{i=1}^{|s|}\lambda(\theta(s|_{i}))$. If $\varepsilon>0$ is such that for every $t\in S.X$, $\mathfrak{s}(s)\in\mathcal{H}^{K}_{\varepsilon}\neq\varnothing$, then for any $0<\delta<\varepsilon$, any $0\leqslant\gamma<o(S)$, and any $s\in S^{\gamma}.X$, $\mathfrak{s}(s)\in\mathcal{H}^{s_{\delta}^{\gamma}(K)}_{\varepsilon}\neq\varnothing$. Moreover, for any $0<\delta<\varepsilon$, $Sz(K,\delta)>o(S)$.

Proof.

We induct on $\gamma$. The base case is the hypothesis. Assume $\gamma+1<o(S)$ and the result holds for $\gamma$. Assume $s\in S^{\gamma+1}.X$, which means there exists $\zeta$ such that $s^{\smallfrown}(\zeta,Z)\in S^{\gamma}.X$ for all $Z\in\text{codim}(X)$. Then for every $Z\in\text{codim}(X)$, there exists $Z\geqslant W_{Z}\in\text{codim}(X)$ such that $\mathfrak{s}(s^{\smallfrown}(\zeta,Z))\subset\mathfrak{s}(s)\times B_{W_{Z}}$. From this and the inductive hypothesis, for every $Z\in\text{codim}(X)$, we fix $x_{Z}\in B_{W_{Z}}$, $(x^{Z}_{i})_{i=1}^{|s|}\in\mathfrak{s}(s)$, and $x^{*}_{Z}\in s_{\delta}^{\gamma}(K)$ such that $\text{Re\ }x^{*}_{Z}(x_{Z})\geqslant\varepsilon$ and $\text{Re\ }x^{*}_{Z}(x^{Z}_{i})\geqslant\varepsilon$ for each $1\leqslant i\leqslant|s|$. By compactness of $\mathfrak{s}(s)\times K$ with the product topology, where $\lambda(\theta(s|_{i}))$ has its norm topology and $K$ has its $w^{*}$-topology,

[TABLE]

Fix $(x_{1},\ldots,x_{|s|},x^{*})$ lying in this intersection. Obviously $x^{*}\in s_{\delta}^{\gamma}(K)$. Moreover, for any $w^{*}$-neighborhood $V$ of $x^{*}$, there exists $Z\in\text{codim}(X)$ such that $\ker(x^{*})\subset Z$ and $x^{*}_{Z}\in V$, whence

[TABLE]

This implies $x^{*}\in s^{\gamma+1}_{\delta}(K)$. It is obvious that $\text{Re\ }x^{*}(x_{i})\geqslant\varepsilon$ for all $1\leqslant i\leqslant|s|$. This shows that $\mathfrak{s}(s)\in\mathcal{H}_{\varepsilon}^{s^{\gamma+1}_{\delta}(K)}$ and completes the successor case.

Finally, assume $\gamma<o(S)$ is a limit ordinal and the result holds for all ordinals less than $\gamma$. Fix $s\in S^{\gamma}.X$ and let $\mathfrak{s}(s)\times K$ be topologized as in the successor case. By the inductive hypothesis, for all $\beta<\gamma$, there exists $(x^{\beta}_{1},\ldots,x^{\beta}_{|s|},x^{*}_{\beta})\in\mathfrak{s}(s)\times K$ such that $x^{*}_{\beta}\in s^{\beta}_{\varepsilon}(K)$ and for all $1\leqslant i\leqslant|s|$, $\text{Re\ }x^{*}_{\beta}(x_{i}^{\beta})\geqslant\varepsilon$. By compactness of \bigl{(}\prod_{i=1}^{|s|}\lambda(\theta(s|_{i}))\bigr{)}\times K,

[TABLE]

Clearly any $(x_{1},\ldots,x_{|s|},x^{*})$ lying in this intersection is such that $x^{*}\in s^{\gamma}_{\delta}(K)$ and for any $1\leqslant i\leqslant|s|$, $\text{Re\ }x^{*}(x_{i})\geqslant\varepsilon$. This shows that $\mathfrak{s}(s)\in\mathcal{H}_{\varepsilon}^{s^{\gamma}_{\delta}(K)}$ and completes the induction.

We have shown that for any $0<\delta<\varepsilon$, $Sz(K,\delta)\geqslant o(S)$. If $o(S)$ is a limit ordinal, we deduce that $Sz(K,\delta)>o(S)$ since $Sz(K,\delta)$ cannot be a limit ordinal. If $o(S)$ is a successor, say $o(S)=\xi+1$, then there exists a length $1$ sequence $(\zeta)\in S^{\xi}$. For every $Z\in\text{codim}(X)$, $\mathfrak{s}((\zeta,Z))=W_{Z}\cap C_{\theta((\zeta,Z))}$ for some $W\subset Z$. The first part of the proof yields that for each $Z\in\text{codim}(X)$, there exists $x_{Z}\in W_{Z}\cap C_{\theta((\zeta,Z))}\subset W_{Z}\cap B_{X}$ and some $x^{*}_{Z}\in s_{\zeta}^{\xi}(K)$ such that $\text{Re\ }x^{*}_{Z}(x_{Z})\geqslant\varepsilon$. Arguing as in the successor case, we deduce that any $w^{*}$-limit of a subnet of $(x^{*}_{Z})_{Z\in\text{codim}(X)}$ lies in $s^{\xi+1}_{\delta}(K)$, whence $Sz(K,\delta)>\xi+1=o(S)$.

∎

2.4. Games

Suppose $T\subset\Lambda^{<\mathbb{N}}$ is a well-founded, non-empty $B$-tree and $\mathcal{E}\subset MAX(T.X.\mathcal{C})$ is some subset. We define the game on $T.X.\mathcal{C}$ with target set $\mathcal{E}$. Player I first chooses $(\zeta_{1},Z_{1})\in\Lambda\times\text{codim}(X)$ such that $(\zeta)\in T$ and Player II then chooses $C_{1}\in\mathcal{C}$. Assuming $(\zeta_{i},Z_{i})_{i=1}^{n}\in T.X$ and $C_{1},\ldots,C_{n}\in\mathcal{C}$ have been chosen, the game terminates if $(\zeta_{i},Z_{i})_{i=1}^{n}\in MAX(T.X)$. Otherwise Player I chooses $(\zeta_{n+1},Z_{n+1})\in\Lambda\times\text{codim}(X)$ such that $(\zeta_{i})_{i=1}^{n+1}\in T$ and Player II chooses $C_{n+1}\in\mathcal{C}$. Since $T$ is well-founded, this game must terminate after finitely many steps. Suppose that the resulting choices are $(\zeta_{i},Z_{i})_{i=1}^{n}$ and $C_{1},\ldots,C_{n}\in\mathcal{C}$. We say that Player II wins if $(\zeta_{i},Z_{i},C_{i})_{i=1}^{n}\in\mathcal{E}$, and Player I wins otherwise.

A strategy for Player I for the game on $T.X.\mathcal{C}$ with target set $\mathcal{E}$ is a function $\psi:T^{\prime}.X.\mathcal{C}\cup\{\varnothing\}\to\Lambda\times\text{codim}(X)$ such that if $\psi((\zeta_{i},Z_{i},C_{i})_{i=1}^{n-1})=(\zeta_{n},Z_{n})$, $(\zeta_{i})_{i=1}^{n}\in T$. We say $\psi$ is a winning strategy for Player I provided that for any sequence $(\zeta_{i},Z_{i},C_{i})_{i=1}^{n}\in MAX(T.X.\mathcal{C})$ such that $(\zeta_{i},Z_{i})=\psi((\zeta_{j},Z_{j},C_{j})_{j=1}^{i-1})$ for every $1\leqslant i\leqslant n$, $(\zeta_{i},Z_{i},C_{i})_{i=1}^{n}\notin\mathcal{E}$.

A strategy for Player II for the game on $T.X.\mathcal{C}$ with target set $\mathcal{E}$ is a function $\psi$ defined on the set

[TABLE]

and taking values in $\mathcal{C}$. We say $\psi$ is a winning strategy for Player II provided that for any sequence $(\zeta_{i},Z_{i},C_{i})_{i=1}^{n}\in MAX(T.X.\mathcal{C})$ such that $C_{i}=\psi((\zeta_{j},Z_{j},C_{j})_{j=1}^{i-1},(\zeta_{i},Z_{i}))$ for all $1\leqslant i\leqslant n$, $(\zeta_{i},Z_{i},C_{i})_{i=1}^{n}\in\mathcal{E}$.

Proposition 7**.**

[6, Proposition $3.1$]** For any non-empty, well-founded $B$-tree $T$ and any $\mathcal{E}\subset T.X.\mathcal{C}$, either Player I or Player II has a winning strategy for the game on $T.X.\mathcal{C}$ with target set $\mathcal{E}$.

Proposition 8**.**

Suppose that Player II has a winning strategy for a game on $T.X.\mathcal{C}$ with target set $\mathcal{E}$. Then there exists $(C_{s})_{s\in T.X}\subset\mathcal{C}$ such that for every $t=(\zeta_{i},Z_{i})_{i=1}^{k}\in MAX(T.X)$, $(\zeta_{i},Z_{i},C_{t|_{i}})_{i=1}^{k}\in\mathcal{E}$.

Proof.

Fix a winning strategy $\psi$ for Player II in the game. We define $C_{s}$ by induction on $|s|$. We let $C_{(\zeta,Z)}=\psi(\varnothing,(\zeta,Z))$. If $|s|=k+1$, $C_{s|_{i}}$ has been defined for every $1\leqslant i\leqslant k$, and $s=s|_{k}^{\smallfrown}(\zeta,Z)$, we let $C_{s}=\psi(s|_{k},(\zeta,Z))$.

∎

For the next proposition, if $h\in L_{p}(X)$ is a simple function, we let $\overline{h}$ be the function in $L_{p}(X)$ such that $\overline{h}(\varpi)=0$ if $h(\varpi)=0$ and $\overline{h}(\varpi)=h(\varpi)/\|h(\varpi)\|$ otherwise.

Proposition 9**.**

Let $\xi$ be an ordinal, $n$ a natural number, and let $T$ be a $B$-tree with $o(T)\geqslant\omega^{1+\xi}n$. If $\psi$ is a strategy for Player I for some game on $\Gamma_{\xi,n}.X.\mathcal{C}$, then for any $1<p<\infty$, any $\delta>0$, and any normally weakly null $(f_{t})_{t\in T.L_{p}(X)}\subset B_{L_{p}(X)}$, there exist $s=(\zeta_{i},Z_{i})_{i=1}^{k}\in MAX(\Gamma_{\xi,n}.X)$, $\varnothing=t_{0}\prec t_{1}\prec\ldots\prec t_{k}\in T.L_{p}(X)$, $g_{i}\in\text{\emph{co}}(f_{u}:t_{i-1}\prec u\preceq t_{i})$, $h_{i}\in B_{L_{p}(X)}$, and $C_{i}\in\mathcal{C}$ such that for every $1\leqslant i\leqslant k$,

1. (i)

$h_{i}$* is simple,* 2. (ii)

$\text{\emph{range}}(\overline{h}_{i})=C_{i}\subset B_{Z_{i}}$, 3. (iii)

$\|g_{i}-h_{i}\|_{L_{p}(X)}<\delta$, 4. (iv)

$(\zeta_{i},Z_{i})=\psi((\zeta_{j},Z_{j},C_{j})_{j=1}^{i-1})$.

Remark 10**.**

For a $B$-tree $S$ on $\Lambda$ and $s\in S$, we let $S(s)$ denote those non-empty sequences $u\in\Lambda^{<\mathbb{N}}$ such that $s^{\smallfrown}u\in S$. An easy induction argument yields that for any ordinals $\xi,\zeta$, $S^{\xi}(s)=(S(s))^{\xi}$ for any ordinal $\xi$. From this it follows that $s\in S^{\xi}$ if and only if $o(S(s))\geqslant\xi$. Furthermore, another easy induction yields that if $(S^{\xi})^{\zeta}=S^{\xi+\zeta}$, from which it follows that if $o(S)\geqslant\xi+\zeta$, $o(S^{\xi})\geqslant\zeta$. Therefore if $s\in S^{\xi+\omega}$, $o(S^{\xi}(s))\geqslant\omega$.

Proof of Proposition 9.

We first note that if $Z\in\text{codim}(X)$, $L_{p}(X)/L_{p}(Z)$ is either the zero vector space or isomorphic to $L_{p}$, and therefore has Szlenk index not exceeding $\omega$. As explained in [6], this means that for any $B$-tree $T$ with $o(T)\geqslant\omega$, any $\delta>0$, and any normally weakly null $(f_{t})_{t\in T.L_{p}(X)}\subset B_{L_{p}(X)}$, there exist $t\in T.L_{p}(X)$, $g\in\text{co}(f_{s}:\varnothing\prec s\preceq t)$, and $h\in B_{L_{p}(Z)}$ such that $\|g-h\|_{L_{p}(X)}<\delta$. Moreover, by the density of simple functions, we may assume this $h$ is simple.

Let $\psi$ be a strategy for Player I for a game on $\Gamma_{\xi,n}.X.\mathcal{C}$. Let $T$ be a $B$-tree with $o(T)=\omega^{1+\xi}n$ and define $\gamma:\Gamma_{\xi,n}.X\cup\{\varnothing\}\to[0,\omega^{\xi}n]$ by letting $\gamma(t)=\max\{\mu\leqslant\omega^{\xi}n:t\in(\Gamma_{\xi,n}.X)^{\mu}\}$ for $t\in\Gamma_{\xi,n}.X$ and $\gamma(\varnothing)=\omega^{\xi}n$. Let $s_{0}=t_{0}=\varnothing$. Now assume that for some $k\in\mathbb{N}$ and all $1\leqslant i<k$, $s_{i}\in\Gamma_{\xi,n}.X$, $\zeta_{i}\in[0,\omega^{\xi}n]$, $Z_{i}\in\text{codim}(X)$, $t_{i}\in T.L_{p}(X)$, $g_{i},h_{i}\in B_{L_{p}(X)}$, and $C_{i}\in\mathcal{C}$ have been chosen such that for all $1\leqslant i<k$,

1. (i)

$h_{i}$ is simple, 2. (ii)

$s_{i}=(\zeta_{j},Z_{j})_{j=1}^{i}$, 3. (iii)

$t_{0}\prec t_{1}\prec\ldots\prec t_{k-1}$, 4. (iv)

$t_{i}\in(T.L_{p}(X))^{\omega\gamma(s_{i})}$, 5. (v)

$(\zeta_{i},Z_{i})=\psi((\zeta_{j},Z_{j},C_{j})_{j=1}^{i-1})$, 6. (vi)

$g_{i}\in\text{co}(f_{u}:t_{i-1}\prec u\preceq t_{i})$, 7. (vii)

$\|g_{i}-h_{i}\|_{L_{p}(X)}<\delta$, 8. (viii)

$\text{range}(\overline{h}_{i})=C_{i}\subset B_{Z_{i}}$.

If $s_{k-1}$ is maximal in $\Gamma_{\xi,n}.X$, we let $s=s_{k-1}$, and one easily checks that the conclusions are satisfied. Otherwise let $(\zeta_{k},Z_{k})=\psi((\zeta_{j},Z_{j},C_{j})_{j=1}^{k-1})$ and $s_{k}=s_{k-1}^{\smallfrown}(\zeta_{k},Z_{k})$. Let $u_{k-1}$ be the sequence of first members of the pairs of $t_{k-1}$ and let $U$ denote the proper extensions of $u_{k-1}$ in $T^{\omega\gamma(s_{k})}$. Then $(f_{t_{k-1}^{\smallfrown}u})_{u\in U.L_{p}(X)}\subset B_{L_{p}(X)}$ is normally weakly null and $o(U)\geqslant\omega$ by the remark preceding the proof, so that the previous paragraph yields the existence of some $u^{\prime}\in U.L_{p}(X)$, $g_{k}\in\text{co}(f_{u}:t_{k-1}\prec u\preceq t_{k-1}^{\smallfrown}u^{\prime})$, and some simple function $h_{k}\in L_{p}(Z_{k})$ such that $\|g_{k}-h_{k}\|_{L_{p}(X)}<\delta$. Let $t_{k}=t_{k-1}^{\smallfrown}u^{\prime}$. In order to apply the remark before the proof, we note that since $s_{k-1}\prec s_{k}$, $\gamma(s_{k-1})\geqslant\gamma(s_{k})+1$. Since

[TABLE]

the remark preceding the proof applies. Note that $C_{k}:=\text{range}(\overline{h}_{k})\subset B_{Z_{k}}$. This completes the recursive construction. Since $\Gamma_{\xi,n}.X$ is well-founded, eventually this process terminates. The resulting $s=(\zeta_{i},Z_{i})_{i=1}^{k}\in MAX(\Gamma_{\xi,n}.X)$ clearly satisfies the conclusions.

∎

3. Definition of an associated space and two games

3.1. The associated space and its properties

If $E$ is a vector space with seminorm $\|\cdot\|$, we say a sequence $(e_{i})_{i=1}^{n}$ in $E$ is $1$-unconditional provided that for any scalars $(a_{i})_{i=1}^{n}$ and any $(\varepsilon_{i})_{i=1}^{n}\in\{\pm 1\}^{n}$, $\|\sum_{i=1}^{n}\varepsilon_{i}a_{i}e_{i}\|=\|\sum_{i=1}^{n}a_{i}e_{i}\|$. Recall that for $1<p<\infty$, a vector space $E$ with seminorm $\|\cdot\|$ which is spanned by the $1$-unconditional basis $(e_{i})_{i=1}^{n}$ is called $p$-concave provided there exists a constant $C$ such that for any $(f_{i})_{i=1}^{n}\subset L_{p}$,

[TABLE]

The smallest such constant $C$ is denoted by $M_{(p)}(E)$.

Given $x\in\text{span}(e_{i}:1\leqslant i\leqslant n)$, where $(e_{i})_{i=1}^{n}$ is a Hamel basis for the seminormed space $E$, we write $x=\sum_{i=1}^{n}a_{i}e_{i}$ and $\mathrm{supp}(x)=\{i\leqslant n:a_{i}\neq 0\}$. We say the vectors $x_{1},\ldots,x_{n}\in\text{span}(e_{i}:1\leqslant i\leqslant n)$ are disjointly supported if the sets $\mathrm{supp}(x_{1})$, $\ldots$, $\mathrm{supp}(x_{n})$ are pairwise disjoint.

For $1<\beta<\infty$, we say that an unconditional Hamel basis $(e_{i})_{i=1}^{n}$ for a seminormed space $E$ satisfies an $1$-lower $\ell_{\beta}$ estimate provided that for any $m\in\mathbb{N}$ and any disjointly supported elements $(x_{i})_{i=1}^{m}\subset E$,

[TABLE]

Theorem 11**.**

[8, Theorem $1.$f$.7$]** Fix $1<\beta<p<\infty$. There exists a constant $C^{\prime}=C^{\prime}(\beta,p)$ such that if $(e_{i})_{i=1}^{n}$ is a $1$-unconditional basis for the seminormed space $E$ which satisfies a $1$-lower $\ell_{\beta}$ estimate, then $E$ is $p$-concave and $M_{(p)}(E)\leqslant C^{\prime}$.

For the remainder of this section, $T$ is a fixed, non-empty $B$-tree.

For a non-empty set $J$, we let $c_{00}(J)$ be the span of the canonical Hamel basis $(e_{j})_{j\in J}$ in the space of scalar-valued functions on $J$, where $e_{j}$ is the indicator of the singleton $\{j\}$. We let $e^{*}_{j}$ denote the coordinate functional to $e_{j}$. Given $x\in c_{00}(J)$, we may write $x=\sum_{j\in J}a_{j}e_{j}$. Then we define $|x|$ to be $\sum_{j\in J}|a_{j}|e_{j}$. A suppression projection is an operator $P$ from $\text{span}(e_{j}^{*}:j\in J)$ into itself such that there exists a subset $F$ of $J$ such that $P\sum_{j\in J}a_{j}e_{j}^{*}=\sum_{j\in F}a_{j}e_{j}^{*}$.

For $0<\phi<\theta<1$, let

[TABLE]

For $0<\phi<\theta<1$ and $1<\alpha<\infty$, let

[TABLE]

Note that the set $M_{\theta,\phi,\alpha,T}$ is closed under suppression projections.

We define the seminorm $\|\cdot\|_{\theta,\phi,\alpha,T}$ on $c_{00}(T.X.\mathcal{C})$ by

[TABLE]

Claim 12**.**

Fix $1<\alpha<\infty$ and $0<\phi<\theta<1$. For any $t\in T.X.\mathcal{C}$, $(e_{t|_{i}})_{i=1}^{|t|}$ is $1$-unconditional and satisfies a $1$-lower $\ell_{\beta}$ estimate in its span, where $1/\alpha+1/\beta=1$.

Proof.

Note that $1$-unconditionality is obvious. Fix $x_{1},\ldots,x_{n}\in\text{span}(e_{t|_{i}}:1\leqslant i\leqslant|t|)$ with disjoint supports. That is, there exist pairwise disjoint subsets $S_{1},\ldots,S_{n}$ of $\{1,\ldots,|t|\}$ such that $x_{i}\in\text{span}(e_{t|_{j}}:j\in S_{i})$. Then there exist $g_{1},\ldots,g_{n}\in M_{\theta,\phi,\alpha,T}$ such that for each $1\leqslant i\leqslant n$, $g_{i}(|x_{i}|)=\|x_{i}\|_{\theta,\phi,\alpha,T}$. Since $M_{\theta,\phi,\alpha,T}$ is closed under suppression projections, we may assume that $\mathrm{supp}(g_{i})\subset S_{i}$ for each $1\leqslant i\leqslant n$. Then if $(a_{i})_{i=1}^{n}$ are such that $\sum_{i=1}^{n}a_{i}^{\alpha}=1$, $a_{i}\geqslant 0$, and $\sum_{i=1}^{n}a_{i}\|x_{i}\|_{\theta,\phi,\alpha,T}=(\sum_{i=1}^{n}\|x_{i}\|_{\theta,\phi,\alpha,T}^{\beta})^{1/\beta}$, $g:=\sum_{i=1}^{n}a_{i}g_{i}\in M_{\theta,\phi,\alpha,T}$ and

[TABLE]

∎

Claim 13**.**

Fix $1<\alpha<\infty$ and $0<\phi<\theta<1$. For any $t=(\zeta_{i},Z_{i},C_{i})_{i=1}^{k}\in T.X.\mathcal{C}$, any sequence $(x_{i})_{i=1}^{k}\in\prod_{i=1}^{k}Z_{i}\cap C_{i}$, and any sequence $(a_{i})_{i=1}^{k}$ of non-negative scalars,

[TABLE]

Proof.

We recall that if $C\in\mathcal{C}$, $C\subset B_{X}$ by the definition of $\mathcal{C}$. With $t$, $(x_{i})_{i=1}^{k}\in\prod_{i=1}^{k}Z_{i}\cap C_{i}$, and $(a_{i})_{i=1}^{k}$ as in the statement, fix $x^{*}\in K$ such that $\text{Re\ }x^{*}(\sum_{i=1}^{k}a_{i}x_{i})=\varrho(\sum_{i=1}^{k}a_{i}x_{i})$. For all $j\in\mathbb{N}$, let $B_{j}=\{i\leqslant k:\text{Re\ }x^{*}(x_{i})\in(\phi^{j},\phi^{j-1}]\}$. Note that for every $j\in\mathbb{N}$, $\theta^{j}\sum_{i\in B_{j}}e^{*}_{t|_{i}}\in N_{\theta^{j},\phi^{j},T}$, so

[TABLE]

Then

[TABLE]

∎

Corollary 14**.**

Fix $1<p,\alpha,\beta<\infty$ with $1/\alpha+1/\beta=1$ and $\beta<p$. Let $C^{\prime}=C^{\prime}(\beta,p)$ be the constant from Theorem 11. Suppose that $\xi$ is an ordinal, $n$ is a natural number, $\varepsilon>0$, and $0<\phi<\theta<1$ are such that Player I has a winning strategy in the game with target set

[TABLE]

Then for any $B$-tree $T$ with $o(T)\geqslant\omega^{1+\xi}n$ and any normally weakly null $(f_{t})_{t\in T.L_{p}(X)}\subset B_{L_{p}(X)}$,

[TABLE]

Proof.

Recall for the proof that for a simple function $h\in L_{p}(X)$, $\overline{h}$ is the function in $L_{p}(X)$ such that $\overline{h}(\varpi)=0$ if $h(\varpi)=0$ and $\overline{h}(\varpi)=h(\varpi)/\|h(\varpi)\|$ otherwise.

Fix a winning strategy $\psi$ for Player I in the game with the indicated target set. Fix $\delta>0$. By Proposition 9, there exist $s=(\zeta_{i},Z_{i})_{i=1}^{k}\in MAX(\Gamma_{\xi,n}.X)$, $\varnothing=t_{0}\prec\ldots\prec t_{k}$, $g_{i}\in\text{co}(f_{u}:t_{i-1}\prec u\preceq t_{i})$, simple functions $h_{i}\in B_{L_{p}(X)}$, and $C_{i}\in\mathcal{C}$ such that $\|g_{i}-h_{i}\|_{L_{p}(X)}<\delta$, $\text{range}(\overline{h}_{i})=C_{i}\subset B_{Z_{i}}$, and $(\zeta_{i},Z_{i})=\psi((\zeta_{j},Z_{j},C_{j})_{j=1}^{i-1})$. This means that for any $\varpi\in[0,1]$, $(\overline{h}_{i}(\varpi))_{i=1}^{k}\in\prod_{i=1}^{k}Z_{i}\cap C_{i}$, whence by Claim 13, for any non-negative scalars $(a_{i})_{i=1}^{k}$,

[TABLE]

Since by Claim 13 $(e_{u})_{u\preceq s}$ satisfies a lower $\ell_{\beta}$ estimate in its span, we deduce that

[TABLE]

Here we have used $1$-unconditionality, $\|h_{i}\|_{L_{p}(X)}\leqslant 1$ for each $1\leqslant i\leqslant k$, and the fact that since $\psi$ is a winning strategy for Player I,

[TABLE]

Let $g=n^{-1}\sum_{u\preceq s}\mathbb{P}_{\xi,n}(u)g_{i}\in\text{co}(f_{u}:u\preceq t_{k})$ and $h=n^{-1}\sum_{u\preceq s}\mathbb{P}_{\xi,n}(u)h_{i}$. Since $\varrho$ is $1$-Lipschitz, it follows that $\|\varrho(g)-\varrho(g)\|_{L_{p}}\leqslant\|g-h\|_{L_{p}(X)}<\delta$, so that

[TABLE]

Since $\delta>0$ was arbitrary, we are done.

∎

3.2. Particular games on $\Gamma_{\xi,n}.X.\mathcal{C}$

The statement of Proposition 6 is notationally cumbersome. We isolate the following result as a way of using Proposition 6.

Lemma 15**.**

Fix $0<\phi<\theta<1$. Suppose that $\xi$ is an ordinal, $m,n$ are natural numbers, $(C_{s})_{s\in\Gamma_{\xi,n}.X}\subset\mathcal{C}$, and $(\sigma,\tau):\Gamma_{\xi,m}.X\to\Gamma_{\xi,n}.X$ is an extended pruning. For $t=(\zeta_{i},Z_{i})_{i=1}^{k}\in\Gamma_{\xi,n}.X$, let $r(t)=(\zeta_{i},Z_{i},C_{t|_{i}})_{i=1}^{k}$. If $\nu\in\mathbb{N}$ is such that for every $s\in MAX(\Gamma_{\xi,m}.X)$, there exists a functional $h_{s}\in\cup_{l=1}^{\nu}N_{\theta^{i},\phi^{i},\Gamma_{\xi,n}}$ such that $\cup_{t\preceq s}r(\sigma(t))\subset\text{\emph{supp}}(h_{s})$, then $Sz(K,\phi^{\nu}/2)>\omega^{\xi}m$.

Proof.

For $s=(\zeta_{i},Z_{i})_{i=1}^{k}\in\Gamma_{\xi,n}.X$, let $\lambda(s)=Z_{k}\cap C_{s}$. For $s\in\Gamma_{\xi,m}.X$, let $\mathfrak{s}(s)=\prod_{i=1}^{|s|}\lambda(\sigma(s|_{i}))$.

Fix $s\in MAX(\Gamma_{\xi,m}.X)$ and let $h_{s}\in\cup_{l=1}^{\nu}N_{\theta^{l},\phi^{l},\Gamma_{\xi,n}}$ be as in the statement of the lemma and fix $1\leqslant l\leqslant\nu$ such that $h_{s}\in N_{\theta^{l},\phi^{l},\Gamma_{\xi,n}}$. We will prove that $\mathfrak{s}(s)\in\mathcal{H}^{K}_{\phi^{\nu}}$. Since for any $1\leqslant m\leqslant k$ and any $C_{1}^{\prime},\ldots,C_{k}^{\prime}\in\mathcal{C}$ such that $\prod_{i=1}^{k}C_{i}^{\prime}\in\mathcal{H}^{K}_{\phi^{\nu}}$, $\prod_{i=1}^{m}C_{i}^{\prime}\in\mathcal{H}^{K}_{\phi^{\nu}}$, this will show that for any non-empty initial segment $s_{1}$ of $s$, $\mathfrak{s}(s_{1})\in\mathcal{H}^{K}_{\phi^{\nu}}$. From here, an appeal to Proposition 6 will finish the proof.

Fix $u=(\mu_{i},W_{i},C_{i})_{i=1}^{|u|}\in\Gamma_{\xi,n}.X.\mathcal{C}$ and $1\leqslant j_{1}<\ldots<j_{\mu}\leqslant|u|$ such that $h_{s}=\theta^{l}\sum_{i=1}^{\mu}e^{*}_{u|_{j_{i}}}$ and $\prod_{i=1}^{\mu}W_{j_{i}}\cap C_{j_{i}}\in\mathcal{H}^{K}_{\phi^{l}}$. Let $\tau(s)=t=(\zeta_{i},Z_{i})_{i=1}^{\eta}$. For each $1\leqslant i\leqslant|s|$, let $l_{i}=|\sigma(s|_{i})|$. Note that for all $1\leqslant i\leqslant|s|$, $r(\sigma(s|_{i}))=(\zeta_{j},Z_{j},C_{t|_{j}})_{j=1}^{l_{i}}$ and $\mathfrak{s}(s)=\prod_{j=1}^{|s|}Z_{l_{j}}\cap C_{t|_{l_{j}}}.$ By hypothesis,

[TABLE]

From this it follows that there exist $m_{1}<\ldots<m_{|s|}$ such that for every $1\leqslant i\leqslant|s|$, $r(\sigma(s|_{i}))=u|_{j_{m_{i}}}$. Choose $(x_{i})_{i=1}^{\mu}\in\prod_{i=1}^{\mu}W_{j_{i}}\cap C_{j_{i}}$ such that there exists $x^{*}\in K$ so that $\text{Re\ }x^{*}(x_{i})\geqslant\phi^{l}$ for each $1\leqslant i\leqslant\mu$, which exists because $\prod_{i=1}^{\mu}W_{j_{i}}\cap C_{j_{i}}\in\mathcal{H}^{K}_{\phi^{l}}$. Since $Z_{l_{i}}=W_{j_{m_{i}}}$ and $C_{t|_{l_{i}}}=C_{j_{m_{i}}}$, $(x_{m_{i}})_{i=1}^{|s|}\in\prod_{i=1}^{|s|}Z_{l_{i}}\cap C_{t|_{l_{i}}},$ which shows that $\mathfrak{s}(s)\in\mathcal{H}^{K}_{\phi^{l}}.$ Since $l\leqslant\nu$, $\mathcal{H}^{K}_{\phi^{l}}\subset\mathcal{H}^{K}_{\phi^{\nu}}$, so that $\mathfrak{s}(s)\in\mathcal{H}^{K}_{\phi^{\nu}}$.

∎

Lemma 16**.**

Fix $1<\alpha<\infty$ and $0<\phi<\theta<1$. If $Sz(K)\leqslant\omega^{\xi}$, then for any $\varepsilon>0$, Player I has a winning strategy in the game with target set

[TABLE]

Proof.

Suppose not. Then by Proposition 8, there exist $\varepsilon>0$ and $(C_{s})_{s\in\Gamma_{\xi}.X}\subset\mathcal{C}$ such that

[TABLE]

For $s=(\zeta_{i},Z_{i})_{i=1}^{k}\in\Gamma_{\xi}.X$, let $r(s)=(\zeta_{i},Z_{i},C_{s|_{i}})_{i=1}^{k}$. For every $t\in MAX(\Gamma_{\xi}.X)$, fix $f_{t}\in M_{\theta,\phi,\alpha,\Gamma_{\xi}}$ such that $\mathrm{supp}(f_{t})\subset[\preceq r(t)]$ and $f_{t}(\sum_{s\preceq t}\mathbb{P}_{\xi}(s)e_{r(s)})=\|\sum_{s\preceq t}\mathbb{P}_{\xi}(s)e_{r(s)}\|_{\theta,\phi,\alpha,\Gamma_{\xi}}$. Define $F:\Pi(\Gamma_{\xi}.X)\to\mathbb{R}$ by letting $F(s,t)=f_{t}(e_{r(s)})$. By Theorem 4, there exists an extended pruning $(\sigma,\tau):\Gamma_{\xi}.X\to\Gamma_{\xi}.X$ such that

[TABLE]

Fix $\nu\in\mathbb{N}$ such that $\varepsilon>\theta^{\nu}$ and for each $t\in MAX(\Gamma_{\xi}.X)$, write $f_{\tau(t)}=\sum_{i=1}^{k_{t}}a_{i,t}g_{i,t}$ where $a_{i,t}\geqslant 0$, $\sum_{i=1}^{k_{t}}a_{i,t}^{\alpha}\leqslant 1$, and $g_{i,t}\in\cup_{n=1}^{\infty}N_{\theta^{n},\phi^{n},\Gamma_{\xi}}$ have pairwise disjoint supports. For each $t\in MAX(\Gamma_{\xi}.X)$, let

[TABLE]

Since $\sum_{i=1}^{k_{t}}a_{i,t}^{\alpha}\leqslant 1$, $|R_{t}|\leqslant\lfloor 1/\varepsilon^{\alpha}\rfloor=:k_{0}$. Note that since $\varepsilon<f_{\tau(t)}(e_{r(\sigma(s))})$ for any $\varnothing\prec s\preceq t$, $r(\sigma(s))\in\cup_{i\in R_{t}}\mathrm{supp}(g_{i,t})$. We write $\sum_{i\in R_{t}}a_{i,t}g_{i,t}=\sum_{i=1}^{l_{t}}b_{i,t}h_{i,t}$ where $l_{t}\leqslant k_{0}$, $(b_{i,t})_{i=1}^{l_{t}}$ is an enumeration of $(a_{i,t})_{i\in R_{t}}$, and $(h_{i,t})_{i=1}^{l_{t}}$ is the corresponding enumeration of $(g_{i,t})_{i\in R_{t}}$. Define $\kappa:\Pi(\Gamma_{\xi}.X)\to\{1,\ldots,k_{0}\}$ by letting $\kappa(\sigma,\tau)$ be the unique $i\leqslant l_{t}$ such that $r(\sigma(s))\in\text{supp}(h_{i,t})$. By Theorem 4$(ii)$, there exists an extended pruning $(\sigma^{\prime},\tau^{\prime}):\Gamma_{\xi}.X\to\Gamma_{\xi}.X$ and $1\leqslant l\leqslant k_{0}$ such that $\kappa(\sigma^{\prime}(s),\tau^{\prime}(t))=l$ for all $(s,t)\in\Pi(\Gamma_{\xi}.X)$. We now note that for any $s\in MAX(\Gamma_{\xi}.X)$, $h_{l,\tau^{\prime}(s)}\in\cup_{i=1}^{\nu}N_{\theta^{i},\phi^{i},\Gamma_{\xi}}$ is such that

[TABLE]

and an appeal to Lemma 15 yields that $Sz(K,\phi^{\nu}/2)>\omega^{\xi}$. This contradiction finishes the proof. To see that $h_{l,\tau^{\prime}(s)}\in\cup_{i=1}^{\nu}N_{\theta^{i},\phi^{i},\Gamma_{\xi}}$, we note that if $h_{l,\tau^{\prime}(s)}\in N_{\theta^{i},\phi^{i},\Gamma_{\xi}}$,

[TABLE]

This shows that $i\leqslant\nu$ by our choice of $\nu$.

∎

Lemma 17**.**

Fix $1<\alpha,\beta<\infty$ and $0<\phi<2^{-1/\alpha}$ and assume that $1/\alpha+1/\beta=1$. Assume that for some $C\geqslant 1$ and all $i\in\mathbb{N}$, $Sz_{\xi}(K,\phi^{i}/2)\leqslant C2^{i}$. Let $\theta=2^{-1/\alpha}$. Then for any $n\in\mathbb{N}$ and any $C_{1}>C$, Player I has a winning strategy in the game with target set

[TABLE]

Proof.

Suppose not. Then for some $n\in\mathbb{N}$, there exist $(C_{s})_{s\in\Gamma_{\xi,n}.X}\subset\mathcal{C}$ and

[TABLE]

such that

[TABLE]

We may assume as in Lemma 16 that $\mathrm{supp}(f_{t})\subset[\preceq r(t)]$ for each $t\in MAX(\Gamma_{\xi,n}.X)$. Then by Theorem 4$(i)$, there exist a level preserving extended pruning $(\sigma,\tau):\Gamma_{\xi,n}\to\Gamma_{\xi,n}$ and numbers, $b_{1},\ldots,b_{n}$ such that $Cn^{1/\beta}<\sum_{i=1}^{n}b_{i}$ and for all $1\leqslant i\leqslant n$ and all $\Lambda_{\xi,n,i}\ni s\preceq t\in MAX(\Gamma_{\xi,n})$, $f_{\tau(t)}(e_{r(\sigma(s))})\geqslant b_{i}$. Fix $\delta>0$ such that $Cn^{1/\beta}+n\delta<\sum_{i=1}^{n}b_{i}$. Let $R=\{i\leqslant n:b_{i}\geqslant\delta\}$.

Sublemma 18**.**

There exist a level preserving extended pruning $(\sigma_{0},\tau_{0}):\Gamma_{\xi,n}.X\to\Gamma_{\xi,n}.X$, $l,w\in\mathbb{N}$, $(a_{i})_{i=1}^{l}\in B_{\ell_{\alpha}^{l}}$, $(k_{i})_{i\in R}\subset\{1,\ldots,l\}$, $(w_{i})_{i=1}^{l}\subset\{1,\ldots,w\}$, and $(g_{t})_{t\in MAX(\Gamma_{\xi,n}.X)}\subset M_{\theta,\phi,\alpha,\Gamma_{\xi,n}}$ such that

1. (i)

for each $t\in MAX(\Gamma_{\xi,n}.X)$, $\|g_{t}-f_{\tau\circ\tau_{0}(t)}\|_{\infty}<\delta$, 2. (ii)

for any $t\in MAX(\Gamma_{\xi,n}.X)$, there exist disjointly supported functionals $h_{1,t},\ldots,h_{l,t}$ such that $h_{i,t}\in N_{\theta^{w_{i}},\phi^{w_{i}},\Gamma_{\xi,n}}$ and $g_{t}=\sum_{i=1}^{l}a_{i}h_{i,t}$, 3. (iii)

for $i\in R$ and $\Lambda_{\xi,n,i}\ni s\preceq t\in MAX(\Gamma_{\xi,n}.X)$, $r(\sigma\circ\sigma_{0}(s))\in\text{\emph{supp}}(h_{k_{i},t})$,

We first finish the proof of the lemma and then return to the proof of the sublemma. Note that item $(iii)$ of the sublemma implies that for $i\in R$ and $\Lambda_{\xi,n,i}\ni s\preceq t\in MAX(\Gamma_{\xi,n})$,

[TABLE]

From this and our choice of $\delta$ we deduce that

[TABLE]

Partition $R$ into sets $R_{1},\ldots,R_{l}$, where $R_{j}=\{i\in R:k_{i}=j\}$, so that

[TABLE]

We claim that for each $j$, $|R_{j}|\leqslant C2^{w_{j}}$. Indeed, suppose $|R_{j}|>C2^{w_{j}}$ for some $j$. By Theorem 4$(iv)$, if $R_{j}=\{r_{1},\ldots,r_{m}\}$, with $r_{1}<\ldots<r_{m}$, there exists extended pruning $(\sigma^{\prime},\tau^{\prime}):\Gamma_{\xi,m}.X\to\Gamma_{\xi,n}.X$ such that $\sigma^{\prime}(\Lambda_{\xi,m,i})\subset\Lambda_{\xi,n,r_{i}}$. We now use Lemma 15 to deduce that $Sz_{\xi}(K,\phi^{w_{j}}/2)>C2^{w_{j}}$, which is a contradiction. Thus we deduce that $|R_{j}|\leqslant C2^{w_{j}}$ for each $j$. This means that for each $1\leqslant j\leqslant l$,

[TABLE]

Then

[TABLE]

Thus we reach a contradiction.

We now return to the proof of the sublemma. First fix $w\in\mathbb{N}$ such that $\theta^{w}<\delta$. For each $t\in MAX(\Gamma_{\xi,n})$, write $f_{\tau(t)}=\sum_{i=1}^{k_{t}}a_{i,t}f_{i,t}$ for some disjointly supported $f_{i,t}\in\cup_{j=1}^{\infty}N_{\theta^{j},\phi^{j},\Gamma_{\xi,n}}$ and $a_{i,t}\geqslant 0$ such that $\sum_{i=1}^{k_{t}}a_{i,t}^{\alpha}\leqslant 1$. Let $S_{t}=\{i\leqslant k_{t}:\|a_{i,t}f_{i,t}\|_{\infty}\geqslant\delta\}$. Note that since $\sum_{i=1}^{k_{t}}a_{i,t}^{\alpha}\leqslant 1$, $|S_{t}|\leqslant\lfloor 1/\delta^{\alpha}\rfloor=:k_{0}$. As in the previous lemma, we write $\sum_{i\in S_{t}}a_{i,t}f_{i,t}=\sum_{i=1}^{l_{t}}a^{\prime}_{i,t}f_{i,t}^{\prime}$ for some $l_{t}\leqslant k_{0}$. Considering the function from $MAX(\Gamma_{\xi,n}.X)$ given by $t\mapsto l_{t}\in\{1,\ldots,k_{0}\}$, we use Theorem 4$(iii)$ to obtain $l\in\mathbb{N}$ and a level preserving extended pruning $(\sigma^{\prime},\tau^{\prime}):\Gamma_{\xi,n}.X\to\Gamma_{\xi,n}.X$ such that for all $t\in MAX(\Gamma_{\xi,n}.X)$, $l_{\tau^{\prime}(t)}=l$. Note that since $\|a^{\prime}_{i,\tau^{\prime}(t)}f^{\prime}_{i,\tau^{\prime}(t)}\|_{\infty}\geqslant\delta$ for every $1\leqslant i\leqslant l$ and $t\in MAX(\Gamma_{\xi,n})$, if $f^{\prime}_{i,\tau^{\prime}(t)}\in N_{\theta^{j},\phi^{j},\Gamma_{\xi,n}}$, $j\leqslant w$. Let $w_{i,\tau^{\prime}(t)}$ be the value $j\in\{1,\ldots,w\}$ such that $f^{\prime}_{i,\tau^{\prime}(t)}\in N_{\theta^{j},\phi^{j},\Gamma_{\xi,n}}$. By considering the map from $MAX(\Gamma_{\xi,n}.X)$ into $B_{\ell_{\alpha}^{l}}\times\{1,\ldots,w\}^{l}$ given by

[TABLE]

we use Theorem 4$(iii)$ again to find another level preserving extended pruning $(\sigma^{\prime\prime},\tau^{\prime\prime}):\Gamma_{\xi,n}.X\to\Gamma_{\xi,n}.X$, $(a_{i})_{i=1}^{l}\in B_{\ell_{\alpha}^{l}}$ and $(w_{i})_{i=1}^{l}\subset\{1,\ldots,w\}$ such that for all $t\in MAX(\Gamma_{\xi,n})$, $\|(a_{i,\tau^{\prime}\circ\tau^{\prime\prime}(t)})_{i=1}^{l}-(a_{i})_{i=1}^{l}\|_{\ell_{\alpha}^{l}}<\delta$ and for all $1\leqslant i\leqslant l$, $f_{i,\tau^{\prime}\circ\tau^{\prime\prime}(t)}^{\prime}\in N_{\theta^{w_{i}},\phi^{w_{i}},\Gamma_{\xi,n}}$. Note that for all $t\in MAX(\Gamma_{\xi,n}.X)$,

[TABLE]

This implies that for any $i\in R$ and any $\Lambda_{\xi,n,i}\ni s\preceq t$, since

[TABLE]

$r(\sigma\circ\sigma^{\prime}\circ\sigma^{\prime\prime}(s))\in\cup_{j=1}^{l}\text{supp}(f^{\prime}_{j,\tau^{\prime}\circ\tau^{\prime\prime}(t)})$. Thus we may let $\kappa(s,t)$ be the unique $j\in\{1,\ldots,l\}$ such that $r(\sigma\circ\sigma^{\prime}\circ\sigma^{\prime\prime}(s))\in\text{supp}(f_{j,\tau^{\prime}\circ\tau^{\prime\prime}(t)})$ if $s\in\cup_{i\in R}\Lambda_{\xi,n,i}$, and $\kappa(s,t)=0$ otherwise. Applying Theorem 4$(ii)$, we deduce the existence of $(k_{i})_{i\in R}\subset\{1,\ldots,l\}$ and a level preserving extended pruning $(\sigma^{\prime\prime\prime},\tau^{\prime\prime\prime}):\Gamma_{\xi,n}.X\to\Gamma_{\xi,n}.X$ such that setting $\sigma_{0}=\sigma^{\prime}\circ\sigma^{\prime\prime}\circ\sigma^{\prime\prime\prime}$, $\tau_{0}=\tau^{\prime}\circ\tau^{\prime\prime}\circ\tau^{\prime\prime\prime}$, $h_{i,t}=f^{\prime}_{i,\tau_{0}(t)}$, and $g_{t}=\sum_{i=1}^{l}a_{i}h_{i,t}$, finishes the proof.

∎

4. Proof of the main results

Proof of Theorem 1.

Let $\phi=1/3$ and $\theta=2/3$, so that $\frac{1}{\theta-\phi}=3$. Fix $1<p<\infty$. Fix any $1<\alpha,\beta<\infty$ such that $\beta<p$ and $1/\alpha+1/\beta=1$. Let $C^{\prime}=C^{\prime}(\beta,p)$ be the constant from Theorem 11. Fix $\varepsilon>0$. By Lemma 16, Player I has a winning strategy in the game with target set

[TABLE]

By Corollary 14, for any $B$-tree $T$ with $o(T)=\omega^{1+\xi}$ and any normally weakly null collection $(f_{t})_{t\in T.L_{p}(X)}\subset B_{L_{p}(X)}$,

[TABLE]

We deduce $Sz(K_{p})\leqslant\omega^{1+\xi}$ by Theorem 5$(i)$.

It is clear that $Sz(K)\leqslant Sz(K_{p})$ for any $1<p<\infty$. If $K$ is convex, then either $Sz(K)=\infty$, in which case $Sz(K_{p})=\infty=\omega\infty=Sz(K)$, or there exists an ordinal $\xi$ such that $Sz(K)=\omega^{\xi}$ [3, Proposition $4.2$]. We deduce that $Sz(K_{p})\leqslant\omega^{1+\xi}=\omega Sz(K)$ by the previous paragraph. In the case that $\xi\geqslant\omega$, $1+\xi=\xi$.

∎

Proof of Theorem 3.

If $\textbf{p}_{\xi}(K)=\infty$, there is nothing to show, so assume $\textbf{p}_{\xi}(K)<\infty$. Fix $1<p,q<\infty$ with $1/p+1/q=1$. Fix $1<\alpha,\beta,\gamma<\infty$ such that $\max\{\textbf{p}_{\xi}(K),q\}<\gamma<\alpha$ and $1/\alpha+1/\beta=1$. Let $C^{\prime}=C^{\prime}(\beta,p)$ be the constant from Theorem 11. Let $\phi=2^{-1/\gamma}$ and note that $\sup_{i\in\mathbb{N}}\varepsilon^{\gamma}Sz_{\xi}(K,\phi^{i}/2)/2^{i}<\infty$. By Lemma 17, with $\theta=2^{-1/\alpha}$, there exists a constant $C_{1}$ such that for every $n\in\mathbb{N}$, Player I has a winning strategy in the game with target set

[TABLE]

By Corollary 14, for every $n\in\mathbb{N}$, every $B$-tree $T$ with $o(T)=\omega^{1+\xi}n$, and every normally weakly null $(f_{t})_{t\in T.L_{p}(X)}\subset B_{L_{p}(X)}$,

[TABLE]

By Theorem 5$(ii)$, $\textbf{p}_{1+\xi}(K_{p})\leqslant\alpha$. Since $\alpha>\max\{\textbf{p}_{\xi}(K),q\}$ was arbitrary, we deduce that $\textbf{p}_{1+\xi}(K_{p})\leqslant\max\{\textbf{p}_{\xi}(K),q\}$.

∎