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Let (p(cdot ): mathbb R^nrightarrow (0,infty )) be a variable exponent function satisfying that there exists a constant (p_0in (0,p_-)), where (p_-:=hbox {ess inf}_{xin 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 values 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})).

Characterizations of variable exponent Hardy spaces via Riesz transforms