  Braverman and Kazhdan proposed a conjecture, later refined by Ng\^o and
partially set in the framework of spherical varieties by Sakellaridis, that
asserts that affine spherical varieties admit Schwartz spaces, Fourier
transforms, and Poisson summation formulae. The first author in joint work with
B.~Liu and later the first two authors proved these conjectures for certain
spherical varieties $Y$ built out of triples of quadratic spaces. However, in
these works the Fourier transform was only proven to exist. In the present
paper we give, for the first time, an explicit formula for the Fourier
transform on $Y.$ We also prove that it is unitary when $F$ is nonarchimedean.
As preparation for this result, we give explicit formulae for Fourier
transforms on Braverman-Kazhdan spaces attached to maximal parabolic subgroups
of split, simple, simply connected groups. These Fourier transforms are of
independent interest (for example, from the point of view of analytic number
theory).
