Interpolation between noncommutative martingale Hardy and BMO spaces: the case

Let $\mathcal {M}$ be a semifinite von Nemann algebra equipped with an increasing filtration $(\mathcal {M}_n)_{n\geq 1}$ of (semifinite) von Neumann subalgebras of $\mathcal {M}$ . For $0<p <\infty $ , let $\mathsf {h}_p^c(\mathcal {M})$ denote the noncommutative column conditioned martingale Hardy space and $\mathsf {bmo}^c(\mathcal {M})$ denote the column “little” martingale BMO space associated with the filtration $(\mathcal {M}_n)_{n\geq 1}$ . We prove the following real interpolation identity: if $0<p <\infty $ and $0<\theta <1$ , then for $1/r=(1-\theta )/p$ , $$ \begin{align*} \big(\mathsf{h}_p^c(\mathcal{M}), \mathsf{bmo}^c(\mathcal{M})\big)_{\theta, r}=\mathsf{h}_{r}^c(\mathcal{M}), \end{align*} $$ with equivalent quasi norms. For the case of complex interpolation, we obtain that if $0<p<q<\infty $ and $0<\theta <1$ , then for $1/r =(1-\theta )/p +\theta /q$ , $$ \begin{align*} \big[\mathsf{h}_p^c(\mathcal{M}), \mathsf{h}_q^c(\mathcal{M})\big]_{\theta}=\mathsf{h}_{r}^c(\mathcal{M}) \end{align*} $$ with equivalent quasi norms. These extend previously known results from $p\geq 1$ to the full range $0<p<\infty $ . Other related spaces such as spaces of adapted sequences and Junge’s noncommutative conditioned $L_p$ -spaces are also shown to form interpolation scale for the full range $0<p<\infty $ when either the real method or the complex method is used. Our method of proof is based on a new algebraic atomic decomposition for Orlicz space version of Junge’s noncommutative conditioned $L_p$ -spaces. We apply these results to derive various inequalities for martingales in noncommutative symmetric quasi-Banach spaces.

An equivalent quasinorm for the Lipschitz space of noncommutative martingales

In this paper, an equivalent quasinorm for the Lipschitz space of noncommutative martingales is presented. As an application, we obtain the duality theorem between the noncommutative martingale Hardy space {h}_{p}^{c}( {\mathcal M} ) (resp. {h}_{p}^{r}( {\mathcal M} ) ) and the Lipschitz space {\lambda }_{\beta }^{c}( {\mathcal M} ) (resp. {\lambda }_{\beta }^{r}( {\mathcal M} ) ) for 0\lt p\lt 1 , \beta =\tfrac{1}{p}-1 . We also prove some equivalent quasinorms for {h}_{p}^{c}( {\mathcal M} ) and {h}_{p}^{r}( {\mathcal M} ) for p=1 or 2\lt p\lt \infty .

Hoeffding decomposition in $$H^1$$ spaces

The well known result of Bourgain and Kwapień states that the projection $$P_{\le m}$$ P ≤ m onto the subspace of the Hilbert space $$L^2\left( \Omega ^\infty \right) $$ L 2 Ω ∞ spanned by functions dependent on at most m variables is bounded in $$L^p$$ L p with norm $$\le c_p^m$$ ≤ c p m for $$1<p<\infty $$ 1 < p < ∞ . We will be concerned with two kinds of endpoint estimates. We prove that $$P_{\le m}$$ P ≤ m is bounded on the space $$H^1\left( {\mathbb {D}}^\infty \right) $$ H 1 D ∞ of functions in $$L^1\left( {\mathbb {T}}^\infty \right) $$ L 1 T ∞ analytic in each variable. We also prove that $$P_{\le 2}$$ P ≤ 2 is bounded on the martingale Hardy space associated with a natural double-indexed filtration and, more generally, we exhibit a multiple indexed martingale Hardy space which contains $$H^1\left( {\mathbb {D}}^\infty \right) $$ H 1 D ∞ as a subspace and $$P_{\le m}$$ P ≤ m is bounded on it.

On the duality of some martingale spaces

Fefferman has proved that the dual space of the martingale Hardy space H1 is the BMO1-space. Garsia went further and proved that the dual of Hp is the so-called martingale Kp-space, where p and q are two conjugate numbers and 1 ≤ p < 2.The martingale Hardy spaces HΦ with general Young function Φ, were investigated by Bassily and Mogyoródi. In this paper we show that the dual of the martingale Hardy space HΦ is the martingale Hardy space HΦ where (Φ, Ψ) is a pair of conjugate Young functions such that both Φ and Ψ have finite power. Moreover, two other remarkable dualities are presented.