  This paper defines local weighted Hardy spaces with variable exponent. Local
Hardy spaces permit atomic decomposition, which is one of the main themes in
this paper. A consequence is that the atomic decomposition is obtained for the
functions in the Lebesgue spaces with exponentially decaying exponent. As an
application, we obtain the boundedness of singular integral operators, the
Littlewood--Paley characterization and wavelet decomposition.

Functional Analysis