  In this paper, the authors propose a new framework under which a theory of
generalized Besov-type and Triebel-Lizorkin-type function spaces is developed.
Many function spaces appearing in harmonic analysis fall under the scope of
this new framework. Among others, the boundedness of the Hardy-Littlewood
maximal operator or the related vector-valued maximal function on any of these
function spaces is not required to construct these generalized scales of
smoothness spaces. Instead of this, a key idea used in this framework is an
application of the Peetre maximal function. This idea originates from recent
findings in the abstract coorbit space theory obtained by Holger Rauhut and
Tino Ullrich. Under this new setting, the authors establish the boundedness of
pseudo-differential operators based on atomic and molecular characterizations
and also the boundedness of the Fourier multipliers. The characterizations of
these function spaces by means of differences and oscillations are also
established. As further applications of this new framework, the authors
reexamine and polish some existing results for many different scales of
function spaces.
