Operator-valued local Hardy spaces associated with anisotropic dilations and their dual spaces

Let script upper M M ℳ be a von Neumann algebra equipped with a normal semifinite faithful trace tau τ τ , and let upper A A A be an expansive dilation on double struck upper R d Rᵈ ℝ d . In this paper, the authors first investigate a class of local operator-valued anisotropic Hardy spaces, denoted by bold h pA left parenthesis double struck upper R d comma script upper M right parenthesis hₚA(Rᵈ,M) 𝐡 p A ( ℝ d , ℳ ) , and identify their dual spaces. This further develops the seminal work of T. Mei [MR2327840: Mem. Amer. Math. Soc., (2007)]. In addition, we also show that bold h 1 upper A left parenthesis double struck upper R d comma script upper M right parenthesis h₁A(Rᵈ,M) 𝐡 1 A ( ℝ d , ℳ ) and bmoA left parenthesis double struck upper R d comma script upper M right parenthesis bmoA(Rᵈ,M) b m o