  We define and study dense Frechet subalgebras of compact quantum groups
consisting of elements rapidly decreasing with respect to an unbounded sequence
of real numbers. Further, this sequence can be viewed as the eigenvalues of a
Dirac-like operator and we characterize the boundedness of its commutators in
terms of the eigenvalues. Grotendieck's theory of topological tensor products
immediately yields a Schwartz kernel theorem for linear operators on compact
quantum groups and allows us to introduce a natural class of
pseudo-differential operators on compact quantum groups. As a by-product, we
develop elements of the distribution theory and corresponding Fourier analysis.
We give applications of our construction to obtain sufficient conditions for
$L^p-L^q$ boundedness of coinvariant linear operators. We provide necessary and
sufficient conditions for algebraic differential calculi on Hopf subalgebras of
compact quantum groups to extend to the proposed smooth structure. We check
explicitly that these conditions hold true on the quantum $SU(2)$ for both its
3-dimensional and 4-dimensional calculi.
