This paper is oriented to an elementary introduction to function spaces
with variable exponents and a survey of related function spaces.
After providing basic and elementary properties of generalized Lebesgue spaces
$L^{p(\cdot)}(\mathbb{R}^n)$ with variable exponents,
we give rearranged proofs of the theorems by Diening (2004),
Cruz-Uribe, Fiorenza and Neugebauer (2003, 2004),
Nekvinda (2004) and Lerner (2005).
They are maybe simpler than the originals.
Moreover, we deal with topics related to $L^{p(\cdot)}(\mathbb{R}^n)$.
For example,
we present an alternative proof
for Lerner's theorem on the modular inequality
and a detailed proof of the density
in Sobolev spaces with variable exponents.
Furthermore, we will describe
the recent results
of fractional integral operators
and Calder$\acute{o}$n-Zygmund operators on $L^{p(\cdot)}(\mathbb{R}^n)$.
Finally, we survey recent results (without proofs)
on several function spaces with variable exponents,
for example,
generalized Morrey and Campanato spaces with variable growth condition,
Hardy spaces $H^{p(\cdot)}(\mathbb{R}^n)$,
Besov spaces $B^{s(\cdot)}_{{p(\cdot)},{q(\cdot)}}(\mathbb{R}^n)$
and Triebel-Lizorkin spaces $F^{s(\cdot)}_{{p(\cdot)},{q(\cdot)}}(\mathbb{R}^n)$, etc.
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