  Let $X$ be a space of homogeneous type and let $\mathfrak{L}$ be a
nonnegative self-adjoint operator on $L^2(X)$ enjoying Gaussian estimates. The
main aim of this paper is twofold. Firstly, we prove the (local) nontangential
and radial maximal function charaterization for the local Hardy spaces
associated to $\mathfrak{L}$. This deduces the maximal function charaterization
for local Hardy spaces in the sense of Coifman and Weiss provided that
$\mathfrak{L}$ satisfies certain extra conditions. Secondly, we introduce the
local Hardy space associated to the critical function $\rho$ which is motivated
by the theory of Hardy spaces related to Schr\"odinger operators and includes
the local Hardy spaces of Coifman and Weiss as a special case. Then we prove
that these local Hardy spaces can be characterized by the the local
nontangential and radial maximal function charaterization related to
$\mathfrak{L}$ and $\rho$ and the global maximal function charaterizations
associated to `perturbations' of $\mathfrak{L}$. As applications, we apply our
theory to obtain a number of new results on maximal characterizations for the
local Hardy type spaces in various settings ranging from Shr\"odinger operators
on manifolds to Shr\"odinger operators on connected and simply connected
nilpotent Lie groups.

Functional Analysis;Analysis of PDEs