We present an explicit construction of examples showing that the estimate $$\Vert T^\epsilon \Vert _{L^1(w)\rightarrow L^{1,\infty }(w)}$$ ‖ T ϵ ‖ L 1 ( w ) → L 1 , ∞ ( w ) $$\lesssim [w]_{A_1}\log (1+[w]_{A_1})$$ ≲ [ w ] A 1 log ( 1 + [ w ] A 1 ) for Haar multipliers is sharp in terms of the characteristic $$[w]_{A_1}$$ [ w ] A 1 .

中文翻译：

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弱 Muckenhoupt-Wheeden 猜想的显式反例

我们提出了一个明确的例子构造，表明估计 $$\Vert T^\epsilon \Vert _{L^1(w)\rightarrow L^{1,\infty }(w)}$$ ‖ T ϵ ‖ L 1 ( w ) → L 1 , ∞ ( w ) $$\lesssim [w]_{A_1}\log (1+[w]_{A_1})$$ ≲ [ w ] A 1 log ( 1 + [ w ) ] A 1) 对于 Haar 乘子，就特征 $$[w]_{A_1}$$ [ w ] A 1 而言是尖锐的。