Parseval p-frames and the Feichtinger conjecture

In this paper, we give a precise characterization of Parseval p  -frames by the known Clarkson's inequality for ℓpℓp. As a direct application, we show that every tight p  -frame {gj}j=1∞ for ℓpℓp, with frame bound B>0B>0 and infj⁡‖gj‖≥C>0infj⁡‖gj‖≥C>0, can be decomposed into ⌊B/Cp⌋⌊B/Cp⌋ standard q  -Riesz basic sequences, and we show that the estimate ⌊B/Cp⌋⌊B/Cp⌋ is optimal. Moreover, we prove the existence of a p-frame which is not equivalent to any Parsevel p  -frame for ℓpℓp, and a Parseval p-frame which is not a Schauder frame sequence for the space or its dual space, while we obtain that every p  -frame can become a pseudo-framing with ℓqℓq coefficients for the dual space.