Espaces BMO, inégalités de Paley et multiplicateurs idempotents

Generalizing the classical BMO spaces defined on the unit circle with vector or scalar values, we define the spaces $\text{BMO}_{ψ_{q}}(\mathbb{T})$ and $\text{BMO}_{ψ_{q}}(\mathbb{T},B)$, where $ψ_{q}(x) = e^{x^q} -1$ for x ≥ 0 and q ∈ [1,∞[, and where B is a Banach space. Note that $\text{BMO}_{ψ_{1}}(\mathbb{T}) = \text{BMO}(\mathbb{T})$ and $\text{BMO}_{ψ_{1}}(\mathbb{T},B) = \text{BMO}(\mathbb{T},B)$ by the John-Nirenberg theorem. Firstly, we study a generalization of the classical Paley inequality and improve the Blasco-Pełczyński theorem in the vector case. Secondly, we compute the idempotent multipliers of $\text{BMO}_{ψ_{q}}(\mathbb{T})$. Pisier conjectured that the supports of idempotent multipliers of $L^{ψ_{q}}(\mathbb{T})$ form a Boolean algebra generated by the periodic parts and the finite parts for 2 < q < ∞. We confirm this conjecture with $L^{ψ_{q}}(\mathbb{T})$ replaced by $\text{BMO}_{ψ_{q}}(\mathbb{T})$