Characterizations of Variable Exponent Hardy Spaces via Riesz Transforms

Let $p(\cdot ):{\mathbb{R}}^n\to (0,\mathrm{\infty })$ be a variable exponent function satisfying that there exists a constant $p_0\in (0,p_{-})$, where $p_{-}:={}_{x\in {\mathbb{R}}^n}p(x)$, such that the Hardy-Littlewood maximal operator is bounded on the variable exponent Lebesgue space $L^{p(\cdot )\mathrm{/}{p}_0}({\mathbb{R}}^n)$. 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 $p_{-}\in (\frac{n-1}{n},\mathrm{\infty })$, and via compositions of all the first order Riesz transforms when $p_{-}\in (0,\frac{n-1}{n})$.