On those weights satisfying a weak-type inequality for the maximal operator and fractional maximal operator

Muckenhoupt and Wheeden [Indiana Univ. Math. J. 26 (1977), pp. 801–816] formulated a weighted weak <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis p comma p right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(p,p)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> inequality where the weight for the weak <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript p"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">L^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> space is treated as a multiplier rather than a measure. They proved such inequalities for the Hardy-Littlewood maximal operator and the Hilbert transform for weights in the class <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript p"> <mml:semantics> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">A_p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , while also deriving necessary conditions to characterize the weights for which these estimates hold. In this paper, we establish the sufficiency of these conditions for the maximal operator when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than 1"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p &gt; 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and present corresponding results for the fractional maximal operators. This completes the characterization and resolves the open problem posed by Muckenhoupt and Wheeden for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than 1"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p &gt; 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .