%Revised version submitted 26 June 1995 % Template file generated by TeXmenu v2.2, June 1992, patchlevel 431. % \input amstex \documentstyle{amsppt} \topmatter \title A Note on UMD Spaces and Transference in Vector-valued Function Spaces\endtitle \author Nakhl\'e H. Asmar, Brian P. Kelly, and Stephen Montgomery-Smith \endauthor \thanks The work of the first and third authors was partially funded by NSF grants. The second and third authors' work was partially funded by the University of Missouri Research Board\endthanks \subjclass Primary: 43A17, Secondary: 42A50, 60G46\endsubjclass \abstract A Banach space $X$ is called an HT space if the Hilbert transform is bounded from $L^p(X)$ into $L^p(X)$, where $10\}\cup\{0 \}$ where as previously, if $J=(j_n)$, then $j_{n(J)}$ is the last non-zero coordinate of $J$. Thus, $\sgn_\od(\chi_J)=\sgn(j_{n(J)})$. Observe that if $\epsilon=(\epsilon_n)\in\{-1,1\}^\N$, then the set $\od(\epsilon)= \{J=(j_n)\in\dualz:\epsilon_{n(J)}j_{n(J)}>0\}\cup\{0 \}$ is also an order on $\dualz$. In this case, $\sgn_{\od(\epsilon)}(\chi_J)=\epsilon_{n(J)}\sgn(j_{n(J)})$. We now state a simple identity that links the unconditionality of martingale difference sequences to harmonic conjugation with respect to orders: for every $n\geq 1$, and all $J=(j_n)\in\Z^n\setminus\Z^{n- 1}$,\ we have $$\epsilon_n =\sgn_{\od(\epsilon)}(\chi_J)\, \sgn_{\od}(\chi_J).\qquad \tag{3.2}$$ To verify (3.2), simply note that for each $n\in\N$, if $J\in\Z^n\setminus\Z^{n-1}$, then $n(J)=n$. From (3.2), one immediately obtains that $$T_P \circ T_{P(\epsilon)} \bigg( \sum_{k=1}^n \sum_{J\in K_k} a_J\,\chi_J \bigg) = \sum_{k=1}^n\epsilon_k\bigg(\sum_{J\in K_k}a_J\,\chi_J \bigg), \tag{3.3}$$ which expresses the martingale transform on the right side as a composition of two conjugate function operators. Applying Theorem~2.1 twice yields (3.1) and implies our next and last result. \proclaim{(3.1) Theorem} Suppose $X$ is a Banach space, and let \$1