This is an announcement for the paper “Factorization in mixed norm Hardy and BMO spaces” by Richard Lechner<https://arxiv.org/find/math/1/au:+Lechner_R/0/1/0/all/0/1>.
Abstract: Let $1\leq p, q<\infty$ and $1\leq r\leq\infty$. We show that the direct sum of mixed norm Hardy spaces $(\sum_n H_n^p(H_n^q))_r$ and the sum of their dual spaces $(\sum_n H_n^p(H_n^q)^*)_r$ are both primary. We do so by using Bourgain's localization method and solving the finite dimensional factorization problem. In particular, we obtain that the spaces $(\sum_n H_n^1(H_n^s))_r, (\sum_n H_n^1(H_n^s))_r)$, as well as $(\sum_n BMO_n(H_n^s))_r$ and $(\sum_n H_n^s(BMO_n))_r$, $1<s<\infty, 1\leq r\leq\infty$ are all primary..
The paper may be downloaded from the archive by web browser from URL
https://arxiv.org/abs/1610.01506
This is an announcement for the paper “Dimension dependence of factorization problems: Hardy spaces and $SL_n^{\infty}$” by Richard Lechner<https://arxiv.org/find/math/1/au:+Lechner_R/0/1/0/all/0/1>.
Abstract: Given $1\leq p<\infty$, let $W_n$ denote the finite-dimensional dyadic Hardy space $H_n^p$, its dual or $SL_n^{\infty}$”. We prove the following quantitative result: The identity operator on $W_n$ factors through any operator $T: W_N\rightarrow W_N$ which has large diagonal with respect to the Haar system, where $N$ depends \emph{linearly} on $n$.
The paper may be downloaded from the archive by web browser from URL
https://arxiv.org/abs/1802.02857
