  The theory of generalized Besov-Morrey spaces and generalized
Triebel-Lizorkin-Morrey spaces is developed. Generalized Morrey spaces, which
T. Mizuhara and E. Nakai proposed, are equipped with a parameter and a
function. The trace property is one of the main focuses of the present paper,
which will clarify the role of the parameter of generalized Morrey spaces. The
quarkonial decomposition is obtained as an application of atomic decomposition.
In the end, the relation between the function spaces dealt in the present paper
and the foregoing researches is discussed.
