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A counterexample concerning the relation between decoupling constants and -constants
Authors:Stefan Geiss
Affiliation:Mathematisches Institut der Friedrich--Schiller--Universität, Postfach, D--O7740 Jena, Germany
Abstract:For Banach spaces $X$ and $Y$ and a bounded linear operator
$T:X rightarrow Y$ we let $rho(T):=inf c$ such that

begin{displaymath}left( AV_{theta _l = pm 1} left|sumlimits _{l=1}^infty theta _l left( sumlimits _{k=tau _{l-1}+1}^{tau _l} h_k T x_k right)right|_{L_2^Y}^2 right)^{frac{1}{2}} le c left| sumlimits _{k=1}^infty h_k x_k right| _{L_2^X} end{displaymath}

for all finitely supported $(x_k)_{k=1}^infty subset X$ and all $0 = tau _0 < tau _1 < cdots$, where $(h_k)_{k=1}^infty subset L_1[0,1)$ is the sequence of Haar functions. We construct an operator $T:X rightarrow X$, where $X$ is superreflexive and of type 2, with $rho(T)<infty$ such that there is no constant $c>0$ with

begin{displaymath}sup _{theta _k = pm 1} left| sumlimits _{k=1}^infty theta _k h_k T x_k right| _{L_2^X} le c left| sumlimits _{k=1}^infty h_k x_k right| _{L_2^X}. end{displaymath}

In particular it turns out that the decoupling constants $rho(I_X)$, where $I_X$ is the identity of a Banach space $X$, fail to be equivalent up to absolute multiplicative constants to the usual $operatorname{UMD}$-constants. As a by-product we extend the characterization of the non-superreflexive Banach spaces by the finite tree property using lower 2-estimates of sums of martingale differences.

Keywords:Vector valued martingales   unconditional constants   superreflexive Banach spaces   interpolation of Banach spaces
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