Integral operators with two-sided cusp singularities

We consider the Fourier integral operators associated to singular canonical relations, with the cusp singularities on both sides. We prove that such operators lose $\frac{1}{4}+\varepsilon$ of a derivative in smoothing properties, compared to nonsingular Fourier integral operators. We also state the results on regularity properties in L^p spaces. Our approach is based on almost orthogonality decompositions of singular oscillatory integral operators.