A weighted weak-type bound for Haar multipliers
Published online by Cambridge University Press: 22 April 2018
Abstract
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.
- Type
- Research Article
- Information
- Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 148 , Issue 3 , June 2018 , pp. 643 - 658
- Copyright
- Copyright © Royal Society of Edinburgh 2018
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