PiecewiseExpand[
     FourierTransform[
      b Sinc[(\[Omega]1 - \[Omega]2)], {\[Omega]1, \[Omega]2}, {t1, t2}, 
      \!\(TraditionalForm\`FourierParameters -> {1, \(-1\)}\),
      Assumptions -> b > 0
      ],
     t1 \[Element] Reals
     ]

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/8Mmpe.png