The Muckenhoupt 
conjecture characterizes the weights for which the Hilbert
transform is controlled by the Hardy-Littlewood
maximal function in a weighted 
-norm. For 
, a nonnegative locally integrable weight 
on 
belongs to 
when there are constants 
and 
such that every bounded interval 
satisfies
 |
where 
is the uncentered Hardy-Littlewood
maximal function and 
is the characteristic
function of 
(Muckenhoupt 1981). The conjecture asserts that this condition
suffices for
 |
for bounded compactly supported 
, where 
is the Hilbert transform.
The condition is also necessary.
Lerner and Ombrosi (2026) proved the conjecture using weighted estimates for sparse operators. Lerner and Ombrosi (2026) report AI assistance with the mathematical proof.
Lerner and Ombrosi (2026) also obtained the weighted Fefferman-Stein inequality for 
weights, bounding the weighted 
-norm of 
by that of its sharp
maximal function.
Conjecture of Muckenhoupt." 25 Oct 2026.