We study sharp boundedness of special Fourier multiplier operators. We first investigate sharp $L^p-L^q$ estimates for the linear multiplier operators associated with non-elliptic quadratic surfaces. Using this, we completely characterize the admissible exponents $p,q$ for which the uniform Sobolev inequality $|u|_qle C|P(D)u|_p$ holds for a second order non-elliptic differential operator $P(D)$ with constants coefficients. As an application, we extend the class of functions of which the unique continuation property for $|P(D)u|le |Vu|$ holds. We are also concerned with boundedness of the bilinear Bochner-Riesz operator and its associated maximal operator. We make use of a decomposition which enable us to deduce bounds for the bilinear Bochner-Riesz operator (or its maximal operator) from square function estimates related to the classical Bochner-Riesz operator (or its weighted estimate). As consequences we can improve previously known bounds for the bilinear Bochner-Riesz operator and its maximal operator.