Möbius invariant Q_pspaces associated with the Green’s function on the unit ball of Cⁿ

Abstract

In this paper, function spaces Q p (B) and Q p,0 (B), associated with the Green's function, are defined and studied for the unit ball B of C n . We prove that Q p (B) and Q p,0 (B) are Möbius invariant Banach spaces and that Q p (B) = Bloch(B), Q p,0 (B) = B 0 (B) (the little Bloch space) when 1 < p < n/(n − 1), Q 1 = BMOA(∂B) and Q 1,0 (B) = VMOA(∂B). This fact makes it possible for us to deal with BMOA and Bloch space in the same way. And we give necessary and sufficient conditions on boundedness (and compactness) of the Hankel operator with antiholomorphic symbols relative to Q p (B) (and Q p,0 (B)). Moreover, other properties about the above spaces and |ϕ z (w)|, ϕ z (w) ∈ Aut(B), are obtained.

Riemann-Stieltjes operators from F(p,q,s) spaces to α-Bloch spaces on the unit ball

“…We call F(p, q,s) general function space because we can get many function spaces, such as BMOA space, Q p space (see [9]), Bergman space, Hardy space, Bloch space, if we take special parameters of p, q,s in the unit disk setting, see [20]. If q + s 1, then F(p, q,s) is the space of constant functions.…”

Composition Operators on -Type Spaces and a New Compactness Criterion for Composition Operators on Spaces

“…Furthermore the spaces Q s are the same for all 1 < s < ∞, and each equals the classical Bloch space B. For a summary of recent research for Q s spaces we refer to [11,20,21]. Bounded and compact composition operators on Q s spaces were characterized by Lou in [8], Wirths and Xiao in [16], and Li in [6].…”

An integral operator preserving s-Carleson measure on the unit ball

“…The holomorphic function spaces Q s associated with the Green's function is introduced in [4]. For s > 0, Q s is defined by ns+2 dτ (z) is a vanishing s-Carleson measure.…”