  In this paper, we give a complete real-variable theory of local variable
Hardy spaces. First, we present various real-variable characterizations in
terms of several local maximal functions. Next, the new atomic and the finite
atomic decomposition for the local variable Hardy spaces are established. As an
application, we also introduce the local variable Campanato space which is
showed to be the dual space of the local variable Hardy spaces. Analogous to
the homogeneous case, some equivalent definitions of the dual of local variable
Hardy spaces are also considered. Finally, we show that the boundedness of
inhomogeneous Calder\'on-Zygmund singular integrals and local fractional
integrals on local variable Hardy spaces and their duals.

Classical Analysis and ODEs