Abstract: We study the modified Benjamin-Ono equation for small, localized initial data where no solitons arise. Unlike the standard Benjamin-Ono equation, the modified version is not known to be completely integrable, so we cannot rely on integrable systems tools. First, to manage the nonlocal Hilbert transform in the linear part of the equation, we split it using positive and negative frequency projections. We then combine wave packet testing of Ifrim and Tataru, the Klainerman–Sobolev vector field method, a normal form transformation, and a bootstrap argument, to establish a global-in-time dispersive decay estimate.

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