Upper Estimates for Banach Spaces

We study the relationship of dominance for sequences and trees in Banach spaces. In the context of sequences, we prove that domination of weakly null sequences is a uniform property. More precisely, if $(vi)$ is a normalized basic sequence and $X$ is a Banach space such that every normalized weakly null sequence in $X$ has a subsequence that is dominated by $(vi)$, then there exists a uniform constant $C\ge 1$ such that every normalized weakly null sequence in $X$ has a subsequence that is $C$-dominated by $(vi)$. We prove as well that if $V=(vi)i=1\infty$ satisfies some general conditions, then a Banach space $X$ with separable dual has subsequential $V$ upper tree estimates if and only if it embeds into a Banach space with a shrinking FDD which satisfies subsequential $V$ upper block estimates. We apply this theorem to Tsirelson spaces to prove that for all countable ordinals $\alpha$ there exists a Banach space $X$ with Szlenk index at most $\omega \alpha \omega +1$ which is universal for all Banach spaces with Szlenk index at most $\omega \alpha \omega$.