Composition of some singular Fourier integral operators and estimates for restricted X -ray transforms

We establish a composition calculus for Fourier integral operators associated with a class of smooth canonical relations C ⊂ ( T * X ∖ 0 ) × ( T * Y ∖ 0 ) . These canonical relations, which arise naturally in integral geometry, are such that π : C → T * Y is a Whitney fold and ρ : C → T * X is a blow-down mapping. If A ∈ I m ( C ) , B ∈ I m ′ ( C t ) , then B A ∈ I m + m ′ , 0 ( Δ , Λ ) a class of pseudodifferential operators with singular symbols. From this follows L 2 boundedness of A with a loss of 1/4 derivative.