Assuming[{b >0 }, FourierTransform[  Sinc[b (ω1 - ω2)], {ω1, ω2}, {t1, t2}, !(TraditionalForm`FourierParameters -> {1, (-1)})]]

(I [Pi] DiracDelta[t1+t2] (-Log[I b-I t1]+Log[-I b+I t1]+Log[-I (b+t1)]-Log[I (b+t1)]))/b

Log[-I (b - t1)] + Log[-I (b + t1)] - Log[I (b - t1)] - Log[I (b + t1)]

    = Log[I (b - t1)*I (b + t1)] - Log[I (b - t1)*I (b + t1)]

    = Log[-b^2 + t1^2] - Log[-b^2 + t1^2]

    = 0

FourierTransform[ Sinc[ω1 - ω2], {ω1, ω2}, {t1,   t2}, !(TraditionalForm`FourierParameters -> {1, (-1)})]

FourierTransform[ Sinc[b ω], {ω}, {t}, !(TraditionalForm`FourierParameters -> {1, (-1)})]

Assuming[{a > b },  FourierTransform[Sinc[3 (ω1 - ω2)], {ω1, ω2}, {t1, t2}, !(TraditionalForm`FourierParameters -> {1, (-1)})]]

PiecewiseExpand[
 FourierTransform[
  Sinc[ω1 - ω2],
  {ω1, ω2}, {t1, t2},
  FourierParameters -> {1, -1}
  ],
 t1 ∈ Reals
 ]

Sqrt[2π]*InverseFourierTransform[1, x, s]
(* 2π*DiracDelta[s] *)

Sqrt[2π]*InverseFourierTransform[Sinc[Sqrt[2]*b*y], y, t] // Simplify

(* π*(Sign[Sqrt[2]*b-t] + Sign[Sqrt[2]*b+t])/(2*Sqrt[2]*b) *)

f = π/b I DiracDelta[t1 + t2]
     (-Log[I b - I t1] + Log[-I b + I t1] + Log[-I (b + t1)] - Log[I (b + t1)]);
Simplify[f, Assumptions -> 0 < b < t1]
Simplify[f, Assumptions -> -b < t1 < b]
Simplify[f, Assumptions -> t1 < -b < 0]

(*
    0
    (2 π^2 DiracDelta[t1 + t2])/b
    0
*)

FourierTransform[ Sinc[b (ω1 - ω2)], {ω1, ω2}, {t1, t2},
 FourierParameters -> {1, -1}, Assumptions -> -b < t1 < b]


(*   (4 π^2 DiracDelta[t1 + t2])/b   *)