On Fourier integral operators with Hölder-continuous phase

In the study of the Boltzmann equation, one is led to consider integral operators TΦ,β with kernels of the form K(x,y)=∫RdΦ(u)exp(−2iπ(β(|u|)u⋅y−u⋅x))du,∀x,y∈Rd, and prove their boundedness on (possibly weighted) L1 spaces. For example, when the function β is constant (inelastic interactions with constant restitution coefficient), one has that TΦ,β is bounded on L1 if and only if the Fourier transform of Φ is in L1. In the more realistic case of a varying β, however, the problem is much more difficult. In the paper under review, the authors assume that Φ is in the Wiener amalgam space W(F(L1),L1) (covering a fairly typical case when Φ(u)=|u|(1+|u|2)m for m>d+12). Working with W(F(L1),L1)⊂L1 instead of all of L1 has the advantage that one has the following algebra property: ∥K(⋅,y)∥1=∥Φexp(−2iπϕ(y,⋅))ˆ∥1≤∥Φ∥W(F(L1),L1)∥exp(−2iπϕ(y,⋅))∥W(F(L1),L∞), where ϕ(y,u)=β(|u|)u⋅y. Therefore, Schur estimates for the kernel of TΦ,β can be obtained from W(F(L1),L∞) estimates involving only β. The authors do so in their main theorem, Theorem 3.1, finding appropriate conditions on β that guarantee the estimate ∥K(⋅,y)∥1≤C(1+|y|)α,∀y∈Rd, for some α>d2 depending on the function β (and hence a weighted L1 estimate for TΦ,β). An estimate for α