A Bellman function counterexample to the $A_1$ conjecture: the blow-up of the weak norm estimates of weighted singular operators

Abstract

We consider several weak type estimates for singular operators using the Bellman function approach. We disprove the $A_1$ conjecture, thus strengthening the counterexamples built by Reguera--Thiele. We show a certain logarithmic blow-up for the end-point estimates of important weighted singular operators.

arXiv comment23 pages

References in corpus (1)

• Extremizers and Bellman function for martingale weak type inequality

Cited by in corpus (6)

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• On the sharp upper bound related to the weak Muckenhoupt-Wheeden conjecture
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• $A_1$ theory of weights for rough homogeneous singular integrals and commutators
• Strong weighted and restricted weak weighted estimates of the square function