  Let $p(\cdot):\ \mathbb R^n\to(0,\infty)$ be a variable exponent function
satisfying that there exists a constant $p_0\in(0,p_-)$, where
$p_-:=\mathop{\mathrm {ess\,inf}}_{x\in \mathbb R^n}p(x)$, such that the
Hardy-Littlewood maximal operator is bounded on the variable exponent Lebesgue
space $L^{p(\cdot)/p_0}(\mathbb R^n)$. In this article, via investigating
relations between boundary valued of harmonic functions on the upper half space
and elements of variable exponent Hardy spaces $H^{p(\cdot)}(\mathbb R^n)$
introduced by E. Nakai and Y. Sawano and, independently, by D. Cruz-Uribe and
L.-A. D. Wang, the authors characterize $H^{p(\cdot)}(\mathbb R^n)$ via the
first order Riesz transforms when $p_-\in (\frac{n-1}n,\infty)$, and via
compositions of all the first order Riesz transforms when
$p_-\in(0,\frac{n-1}n)$.
