Endpoint estimates about certain oscillatory integral operators on torus

On the [Formula: see text] torus [Formula: see text] we study two oscillatory singular integrals [Formula: see text] and [Formula: see text], which are raised from studying the Cauchy problem of fractional wave equation and are extensively studied by many authors. We prove that, when [Formula: see text] the inequality [Formula: see text] holds if and only if [Formula: see text]. Also we prove that [Formula: see text] if and only if [Formula: see text].