Pointwise multipliers and decomposition theorems in analytic Besov spaces (Q1586857)

The authors proved the corona problem for holomorphic Besov space \(\mathcal{B}_s^p\) in the unit ball \({B}\) of \(\mathbb{C}^n\). For \(g_1, \cdots, g_m \in {H} = {H(B)},\) the space of holomorphic functions, it is well-known that the operator \[ \begin{aligned} {M}_g : {H}^m & = {H} \times \cdots \times {H} \rightarrow {H} \\ & f = (f_1, \cdots, f_m) \rightarrow \sum_{j=1}^m g_jf_j \end{aligned} \] is surjective if and only if the functions \(g_j\) satisfy \(|g(z)|= \sum_{j=1}^m |g_j(z)|> 0\) for all \(z \in {B}\). The purpose of this paper is analogously to consider the corona problem for Besov space \(\mathcal{B}_s^p\) of a subspace of \({H(B)}\): Let \(0<p\leq\infty\) and \(s \in {R}\). Let \(g_1, g_2\) be pointwise multiplies of \(\mathcal{B}_s^p\) satisfying the condition \(\inf \{|g(z)|; z \in {B} \} > 0\). Then there exist linear bounded operators \(T_1, T_2 : \mathcal{B}_s^p \rightarrow \mathcal{B}_s^p,\) which satisfy \(g_1T_1(f) + g_2T_2(f) = f\) for all \(f \in \mathcal{B}_s^p\), in the following cases: 1) \(0<p\leq 1,\) all \(s\), 2) \( 1<p<\infty, s<0 \), 3) \(1<p<\infty, n-sp <1\), 4) \(1<p\leq 2, s=0\), 5) \(p=\infty,\) all \(s\). The special cases for Lipschitz space, Bloch space, Begman space, and so on, are already proved by the authors and others. The authors also mention that the decomposition problem remains open in the following cases for 3) and 4) : \(1<p<\infty, n-sp \geq 1 \) and \(2<p<\infty , s=0\).