A sharp Lp estimate for the total variation process

For a given p≥2, let X be an Lp bounded martingale and let Y be a martingale of bounded mean oscillation. The paper contains the proof of the estimate ‖∫0∞|d〈X,Y〉t|‖p≤p‖X‖p‖Y‖bmo. The inequality is sharp for each p and the range p≥2 cannot be expanded without additional assumptions on X and Y. The proof rests on the existence of a certain special function, enjoying appropriate size and concavity requirements.