We study a weighted maximal weak-type inequality for Haar multipliers that can be regarded as a dual problem of Muckenhoupt and Wheeden. More precisely, if Tε is the Haar multiplier associated with the sequence ε with values in [−1, 1], and is the r-maximal operator, then for any weight w and any function f on [0, 1) we have

for some constant Cr depending only on r. We also show that the analogous result does not hold if we replace by the dyadic maximal operator Md. The proof rests on the Bellman function method; using this technique we establish related weighted Lp estimates for p close to 1, and then deduce the main result by extrapolation arguments.

We prove that the extremal sequences for the Bellman function of the dyadic maximal operator behave approximately as eigenfunctions of this operator for a specific eigenvalue. We use this result to prove the analogous one with respect to the Hardy operator.

Let f:ℝ→ℝ be a locally integrable function of bounded lower oscillation. The paper contains the proofs of sharp strong-type, weak-type and exponential estimates for the mean oscillation of f. In particular, this yields the precise value of the norm of the embedding BLO⊂BMOp, 1≤p<∞. Higher-dimensional analogues for anisotropic BLO spaces are also established.