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Register a new userDimension dependence of factorization problems: bi-parameter Hardy spaces
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Authors
Richard Lechner
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Given $1 leq p,q < infty$ and $ninmathbb{N}_0$, let $H_n^p(H_n^q)$ denote the canonical finite-dimensional bi-parameter dyadic Hardy space. Let $(V_n : ninmathbb{N}_0)$ denote either $bigl(H_n^p(H_n^q) : ninmathbb{N}_0bigr)$ or $bigl( (H_n^p(H_n^q))^* : ninmathbb{N}_0bigr)$. We show that the identity operator on $V_n$ factors through any operator $T : V_Nto V_N$ which has large diagonal with respect to the Haar system, where $N$ depends emph{linearly} on $n$.
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Let $1leq p,q < infty$ and $1leq r leq infty$. We show that the direct sum of mixed norm Hardy spaces $big(sum_n H^p_n(H^q_n)big)_r$ and the sum of their dual spaces $big(sum_n H^p_n(H^q_n)^*big)_r$ are both primary. We do so by using Bourgains localization method and solving the finite dimensional factorization problem. In particular, we obtain that the spaces $big(sum_{nin mathbb N} H_n^1(H_n^s)big)_r$, $big(sum_{nin mathbb N} H_n^s(H_n^1)big)_r$, as well as $big(sum_{nin mathbb N} BMO_n(H_n^s)big)_r$ and $big(sum_{nin mathbb N} H^s_n(BMO_n)big)_r$, $1 < s < infty$, $1leq r leq infty$, are all primary.
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