Sharp bounds for the distribution of martingale transforms of bounded functions

The paper suggests a characterization of a positive function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis psi Subscript normal infinity Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi> ψ </mml:mi> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f(\psi _\infty )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is integrable for any limit value <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="psi Subscript normal infinity"> <mml:semantics> <mml:msub> <mml:mi> ψ </mml:mi> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\psi _\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the martingale transform for an indicator function. The characterizing condition is that a version of the Lipschitz majorant of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is integrable with respect to an exponential weight. The arguments are based on the calculation of particular minimal biconcave functions on a strip.