  Among those transversally elliptic operators initiated by Atiyah and Singer,
Kohn's $\Box_b$ operator on CR manifolds with $S^1$ action is a natural one of
geometric significance for complex analysts. Our first main result establishes
an asymptotic expansion for the heat kernel of such an operator with values in
its Fourier components, which involves an unprecedented contribution in terms
of a distance function from lower dimensional strata of the $S^1$-action. Our
second main result computes a local index density, in terms of
\emph{tangential} characteristic forms, on such manifolds including
\emph{Sasakian manifolds} of interest in String Theory, by showing that certain
non-trivial contributions from strata in the heat kernel expansion will
eventually cancel out by applying Getzler's rescaling technique to off-diagonal
estimates. This leads to a local result which can be thought of as a type of
local index theorem on these CR manifolds. As applications of our CR index
theorem we can prove a CR version of Grauert-Riemenschneider criterion, and
produce many CR functions on a weakly pseudoconvex CR manifold with transversal
$S^1$ action and many CR sections on some class of CR manifolds, answering (on
this class of manifolds) some long-standing questions in several complex
variables and CR geometry. We give examples of these CR manifolds, some of
which arise from Brieskorn manifolds. Moreover in some cases, without use of
equivariant cohomology method nor keeping contributions arising from lower
dimensional strata as done in previous works, we can reinterpret Kawasaki's
Hirzebruch-Riemann-Roch formula for a complex orbifold with an orbifold
holomorphic line bundle, as an index theorem obtained by a single integral over
a smooth CR manifold which is essentially the circle bundle of this line
bundle.
