Factorization of Hardy spaces on local fields (Q1095348)

R. Coifman, R. Rochberg, and G. Weiss provided the extension to \(H^ 1\) of several real variables of well-known factorization theorems on the unit disk involving conjugate functions. A. Uchiyama generalized this work to \(H^ p\) on spaces of homogeneous type for a restricted range of p. In this note, these results are extended to \(H^ p\) on a local field for all p and provide an application to commutator operators. The primary result is that if \(f\in H^ p\) and \(1/p=1/q+1/r\), \(0<p,q,r<\infty\), then there exist sequences \((g_ k)\subset H^ q\), \((h_ k)\subset H^ r\) such that \(\| f\| _ p\approx \inf (\sum \| g_ i\| _ q\| h_ i\| _ r)\) where \(f=\sum (h_ iT_{\omega}g_ i-g_ iT'_{\omega}h_ i)\) and where \(T_{\omega}\) is a Calderon-Zygmund singular integral with locally constant kernel \(\omega\).