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

Let

be a von Neumann algebra equipped with a normal semifinite faithful trace

tau $\tau$ τ

, and let

upper A $A$ A

be an expansive dilation on

double struck upper R d $\mathbb{R}^d$ 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 $\mathbf{h}_p^A(\mathbb{R}^d,\mathcal{M})$ 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 $\mathbf{h}_1^A(\mathbb{R}^d,\mathcal{M})$ 𝐡 1 A ( ℝ d , ℳ )

and

bmoA left parenthesis double struck upper R d comma script upper M right parenthesis $\mathrm{bmo}_A(\mathbb{R}^d,\mathcal{M})$ b m o A ( ℝ d , ℳ )

serve as natural endpoint spaces for the noncommutative

Lp $L_p$ L p

-spaces

Lp left parenthesis upper L infinity left parenthesis double struck upper R d right parenthesis circled times quotation dash script upper M right parenthesis $L_p(L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M})$ L p ( L ∞ ( ℝ d ) ⊗ ― ℳ )

, and these endpoint spaces are then used to establish the

Lp $L_p$ L p

-estimates for certain operators via interpolation. Finally, as applications, an atomic decomposition for local Hardy spaces and Hardy–Lebesgue estimates for a local class of noncommutative Calderón–Zygmund operators are also established. To the best of our knowledge, these results are new in the operator-valued setting even for diagonal anisotropic dilations (when

upper A colon equals diag left parenthesis lamda 1 comma lamda 2 comma ellipsis comma lamda d right parenthesis $A:=\mathrm{diag}(\lambda_1, \lambda_2, \ldots, \lambda_d)$ A : = d i a g ( λ 1 , λ 2 , … , λ d )

with

vertical bar lamda j vertical bar greater than 1 $|\lambda_j| \gt 1$ | λ j | > 1

,

j equals 1 comma 2 comma ellipsis comma d $j=1,2,\ldots,d$ j = 1 , 2 , … , d

).