Operator-valued anisotropic Hardy and BMO spaces

Abstract

Let M be a von Neumann algebra equipped with a normal semifinite faithful trace τ, and A be an expansive dilation on Rn. In this article, the author introduces the operator-valued anisotropic BMO space BMOA(Rn,M) and obtains a predual space of BMOA(Rn,M), which is the operator-valued anisotropic Hardy space HA1(Rn,M). Moreover, as the Lq-spaces analogues of the BMO spaces, the author introduces the operator-valued anisotropic BMO-type space LqMOA(Rn,M)(2<q<∞), and establishes the duality between LqMOA(Rn,M) and the operator-valued anisotropic Hardy space HAp(Rn,M) with p=(q−1)/q. In addition, the author also obtains some interpolation results and the equivalence of HAp(Rn,M) and Lp(Rn,M) with p∈(1,∞). As an application, the boundedness of anisotropic Calderón-Zygmund operators on BMOA(Rn,M) is established. When A:=2In×n, part of results coincides with those of T. Mei [33], where In×n denotes the n×n unit matrix.